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Optimal Partition Problems and Applications to Kreĭn–Feller Operators and Quantization Problems

February 2023

DOI:10.26092/elib/2011

Authors:

Aljoscha Niemann

University of Bremen

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Optimal Partition Problems and

Applications to Kre˘

ın–Feller

Operators and Quantization

Problems

Dissertation

im Fachbereich Mathematik und Informatik

zur Erlangung des Grades Dr. rer. nat.

von

Aljoscha Niemann

Vorgelegt am

20.09.2022

Institut f¨

Dynamische Systeme

Datum des Promotionskolloquiums: 26.10.2022

Erster Gutachter und zweite Gutachterin:

Prof. Dr. Marc Keßeb¨

ohmer (Universit¨

at Bremen, Germany)

Prof. Dr. Anke Pohl (Universit¨

at Bremen, Germany)

Abstract

We study the lower and upper partition entropy and the lower and upper optimized

coarse multifractal dimension with respect to certain set functions deﬁned on the

set of the dyadic cubes in

R𝑑

. For this purpose, we introduce the notion of partition

functions, generalizing the well-known

𝐿𝑞

-spectrum. We ﬁnd a formula for the upper

partition entropy in terms of the zero of the associated partition function. Further,

we establish a connection between the classical works of Solomjak and Birman

[BS66; BS74], Borzov [Bor71], and the partition entropy, improving classical

results. We give regularity conditions guaranteeing that the lower and upper partition

entropy coincide. Based on these general results, we develop a uniﬁed framework

to tackle both the computation of the upper spectral dimension of Kre

ın–Feller

operators with respect to Neumann boundary conditions and the computation of the

upper quantization dimension. Furthermore, this enables us to establish regularity

conditions, ensuring that the lower and upper spectral dimension, as well as the lower

and upper quantization dimension, coincide. The results are illustrated by several

examples; in particular, we prove that the spectral dimension and the quantization

dimension of self-conformal measures, with or without overlap, exist and can be

computed in terms of the

𝐿𝑞

-spectrum of the underlying measure. We also determine

various lower and upper bounds for the lower and upper spectral dimensions, as

well as for the lower and upper quantization dimensions in terms of the associated

𝐿𝑞

-spectrum, establishing, in particular, sharp bounds that depend only on the upper

Minkowski dimension of the support of the measure. We give ﬁrst examples in

which the lower and upper spectral dimensions do not coincide.

Zusammenfassung

Wir untersuchen die untere und obere Partitionsentropie bez

uglich bestimmter Men-

genfunktionen, die auf der Menge der

𝑑

-dimensionalen dyadischen W

urfel deﬁniert

sind. Zu diesem Zweck f

uhren wir den neuen Begriﬀder Partitionsfunktion ein,

der das bekannte

𝐿𝑞

-Spektrum verallgemeinert. Wir ﬁnden eine Formel f

ur die

obere Partitionsentropie in Form der Nullstelle der zugeh

origen Partitionsfunk-

tion. Außerdem stellen wir eine Verbindung zwischen der Partitionsentropie und

den klassischen Arbeiten von Solomjak und Birman [BS66; BS74] und Borzov

[Bor71] her und verbessern damit klassische Ergebnisse. Dar

uber hinaus stellen

wir Regularit

atsbedingungen auf, die garantieren, dass die untere und obere Par-

titionsentropie

ubereinstimmen. Aufbauend auf diesen allgemeinen Ergebnissen

sind wir in der Lage, einen einheitlichen Rahmen zur Berechnung der oberen

Spektraldimension von Kre

ın–Feller-Operatoren unter Ber

ucksichtigung Neumann-

Randbedingungen sowie der oberen Quantisierungsdimension zu entwickeln. Weiter

onnen wir so Regularit

atsbedingungen aufstellen, die sicherstellen, dass die untere

und obere Spektraldimension sowie die untere und obere Quantisierungsdimension

ubereinstimmen. Die Ergebnisse werden durch eine Reihe von Beispielen veran-

schaulicht. Insbesondere beweisen wir, dass die Spektraldimension und die Quan-

tisierungsdimension bez

uglich selbstkonformer Maße mit und ohne Separierungsbe-

dingungen existieren und mit Hilfe des

𝐿𝑞

-Spektrums berechnet werden k

onnen. Es

werden mehrere untere und obere Schranken f

ur die untere und obere Spektraldimen-

sion sowie f

ur die untere und obere Quantisierungsdimension in Abh

angigkeit des

𝐿𝑞-Spektrums des zugrunde liegenden Maßes bewiesen, insbesondere erhalten wir

scharfe Schranken in Abh

angigkeit der oberen Minkowski-Dimension des Tr

agers

des zugrunde liegenden Maßes. Des Weiteren geben wir erste Beispiele an, in denen

die obere und untere Spektraldimension nicht ¨

ubereinstimmen.

III

Contents

1. Introduction 1

1.1. Statement of the problems . . . . . . . . . . . . . . . . . . . . . 2

1.1.1. Spectral problem of Kre˘

ın–Feller operators . . . . . . . . 2

1.1.2. Quantization problem . . . . . . . . . . . . . . . . . . . . 7

1.1.3.

Optimal partition problems and optimized coarse multifrac-

taldimension........................ 9

1.2. Outline and statement of the main results . . . . . . . . . . . . . . 12

2. Preliminaries 22

2.1. DyadicPartitions .......................... 22

2.2. Form approach for Kre˘

ın–Feller operators . . . . . . . . . . . . . 23

2.2.1. Sobolev spaces and embeddings . . . . . . . . . . . . . . 23

2.2.2. Sobolev spaces and embeddings in the case 𝑑=1 . . . . . 24

2.2.3. Stein extension . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.4. Formapproach ....................... 31

2.2.5. Deﬁnition of the Kre˘

ın–Feller operator . . . . . . . . . . 32

2.2.6. Smoothing methods . . . . . . . . . . . . . . . . . . . . 37

2.3. Partition functions and 𝐿𝑞-spectra ................. 42

2.3.1. The partition function . . . . . . . . . . . . . . . . . . . 42

2.3.2. The (Dirichlet/Neumann) 𝐿𝑞-spectrum . . . . . . . . . . . 46

2.4. The spectral partition function . . . . . . . . . . . . . . . . . . . 48

2.4.1.

The spectral partition function and connections to the

𝐿𝑞

spectrum .......................... 48

2.4.2.

Relations between the Dirichlet and Neumann spectral par-

titionfunctions ....................... 51

2.4.3. Examples.......................... 55

3. Partition entropy and optimized coarse multifractal dimension 66

3.1.

Bounds for the partition entropy and optimized coarse multifractal

dimension.............................. 67

3.2. Upper bounds for the 𝔍𝜈,𝑎,𝑏-partition entropy . . . . . . . . . . . 72

3.3. Thedualproblem .......................... 75

3.4. Coarse multifractal analysis . . . . . . . . . . . . . . . . . . . . 80

Contents

Spectral dimension and spectral asymptotic for Kre

ın–Feller operators

for the one-dimensional case 89

4.1. Lower bounds for the spectral dimension . . . . . . . . . . . . . . 90

4.2. Upper bounds for the spectral dimension . . . . . . . . . . . . . . 93

4.3. Mainresults............................. 98

4.3.1.

Upper spectral dimension and lower bounds for the lower

spectral dimension . . . . . . . . . . . . . . . . . . . . . 98

4.3.2. Regularity results . . . . . . . . . . . . . . . . . . . . . . 99

4.3.3. General bounds in terms of fractal dimensions . . . . . . . 100

4.4. Examples .............................. 101

4.4.1. C1-cIFS and weak Gibbs measures . . . . . . . . . . . . 101

4.4.2. Homogeneous Cantor measures . . . . . . . . . . . . . . 121

4.4.3. Purely atomic case . . . . . . . . . . . . . . . . . . . . . 125

5. Spectral dimension for Kre˘

ın–Feller operators in higher dimensions 132

5.1. Upperbounds............................ 133

5.1.1.

Embedding constants and upper bounds for the spectral

dimension.......................... 133

5.1.2.

Upper bounds on the embedding constants and upper bounds

for the spectral dimension . . . . . . . . . . . . . . . . . 135

5.2. Lowerbounds ........................... 139

5.2.1. Lower bound on the spectral dimension . . . . . . . . . . 139

5.2.2. Lower bound on the embedding constant . . . . . . . . . 143

5.3. Mainresults............................. 145

5.3.1.

Upper spectral dimension, and lower and upper bounds for

the lower spectral dimension . . . . . . . . . . . . . . . . 146

5.3.2. Regularity results . . . . . . . . . . . . . . . . . . . . . . 147

5.3.3. General bounds in terms of fractal dimensions . . . . . . . 149

5.4. Examples .............................. 151

5.4.1. Absolutely continuous measures . . . . . . . . . . . . . . 151

5.4.2. Ahlfors–David regular measure . . . . . . . . . . . . . . 152

5.4.3. Self-conformal measures . . . . . . . . . . . . . . . . . . 153

5.4.4. Non-existence of the spectral dimension . . . . . . . . . . 153

6. Quantization Dimension 156

6.1. Upper bounds for the quantization dimension . . . . . . . . . . . 157

6.2. Lower bounds for the quantization dimension . . . . . . . . . . . 159

6.3. Mainresults............................. 162

7. Open problems and conjectures 166

7.1. Dirichlet/Neumann spectral partition function and

𝐿𝑞-spectrum............................. 166

7.2. Lower optimized coarse multifractal dimension . . . . . . . . . . 167

7.3. Spectral dimension in the critical case 𝑑=2 ............ 168

Contents

A. Appendix 170

A.1.Convexfunctions .......................... 170

A.2. Sobolev spaces, Lipschitz domains, and Stein’s extension operator 171

A.3. Sobolev spaces in the one-dimensional case . . . . . . . . . . . . 175

A.4. Self-adjoint operators and quadratic forms . . . . . . . . . . . . . 176

A.5. Fractal dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 178

Bibliography 191

List of symbols 192

Chapter 1

Introduction

In 1911, H. Weyl [Wey11] studied the following Dirichlet eigenvalue problem

Δ𝐷𝑢=−𝜆𝑢 ,

𝑢|𝜕Ω=0,

where

Ω⊂R𝑑

𝑑∈N

, is a bounded domain and

Δ𝐷

denotes the classical Laplace

operator, i.e.

Δ𝐷𝑢=𝑑

𝑖=1𝜕𝑢/𝜕𝑥𝑖

with respect to Dirichlet boundary conditions. As-

suming some regularity conditions on

, Weyl proved that the associated eigenvalue

counting function, denoted by 𝑁𝐷, obeys the following law

𝑁𝐷(𝑥)=vol𝑑(Ω)𝜔𝑑

(2𝜋)𝑑𝑥𝑑/2+𝑜𝑥𝑑/2,(1)

where

𝜔𝑑

is the

𝑑

-dimensional volume of the unit ball and

𝑜

denotes the Landau

symbol, i.e.

𝑓=𝑜(𝑔)

limsup𝑥→∞ |𝑓(𝑥)|/|𝑔(𝑥)| =

0. Nowadays, the asymptotic

expansion

(1)

is known as Weyl’s law. This result has been extended to arbitrary

bounded domains by M

etivier [M

et77]. Weyl’s pioneering works have stimulated

wide range of activities on this topic, in which many papers are concerned with

estimating the remainder term of

(1)

or investigating generalizations of the Laplace

operator in various ways. In the present thesis, we also follow this line of investi-

gation by considering a generalization of the classical Laplacian. The physicist M.

Berry [Ber79] conjectured that the remainder term of

(1)

is driven by the Hausdorﬀ

dimension of 𝜕Ω, i.e.

𝑁𝐷(𝑥)=vol𝑑(Ω)𝜔𝑑

(2𝜋)𝑑𝑥𝑑/2+𝑂𝑥dim𝐻(𝜕Ω)/2.

Here,

𝑂

denotes the Landau symbol, i.e.

𝑔=𝑂(𝑓)

limsup𝑥→∞ |𝑔(𝑥)|/|𝑓(𝑥)| <∞

Nowadays, this is known as the Weyl–Berry conjecture, which turned out to be incor-

1.1. Statement of the problems

rect, as shown by Brossard and Carmona [BC86]. Moreover, Brossard and Carmona

[BC86] suggested replacing the Hausdorﬀdimension with the upper Minkowski

dimension, which is known as the modiﬁed Weyl–Berry conjecture. A big step

forward was made by Lapidus [Lap91], who proved that under the assumption that

the upper Minkowski dimension of

𝜕Ω

lies in

(𝑑−

,𝑑]

and the Minkowski content

𝜕Ω

is ﬁnite, the remainder term

𝑜(𝑥𝑑/2)

(1)

can be replaced by

𝑂(𝑥dim𝑀(𝜕Ω)/2)

However, in the general case, the modiﬁed Weyl–Berry conjecture has been dis-

proved by Lapidus and Pomerance [LP96]. Also in this thesis, in the context of

Kre

ın–Feller operators, it turns out that the upper Minkowski dimension is also

a more appropriate concept for the description of the eigenvalue growth than the

Hausdorﬀdimension (see Theorem 4.10 and Theorem 5.15). It should be noted that

in the literature (e.g. see [Fuj87; NX20]) only cases for which the Hausdorﬀand

the upper Minkowski dimensions coincide have been considered so far, obscuring

the actual connection.

Another interesting problem is the following famous question by M. Kac [Kac66]

“Can one hear the shape of the drum?”, which asks whether it is possible to determine

the geometry of

from the eigenvalues of

Δ𝐷

. In general, the answer to this question

is “no”. In 1964, a ﬁrst counterexample for the case

𝑑=

16 was constructed by

Milnor [Mil64]. In the following decades, further counterexamples were constructed

for

𝑑≥

4 by Urakawa [Ura82] and for

𝑑=

2 by Gordon, Webb, and Wolpert

[GWW92]. Consequently, in general, we cannot expect to recover the geometry

. Nevertheless, Weyl’s law tells us that some geometric information such

as the volume of

can be inferred from the eigenvalues of

Δ𝐷

. Therefore, it is

an interesting and demanding problem to ascertain which information about

encoded by the eigenvalues of

Δ𝐷

. We also address Kac’s question in the setting

of Kre

ın–Feller operators and give partial answers, which can be found in Section

4.3.3 and Section 5.3.3.

1.1 Statement of the problems

We now introduce the problems we study in this thesis.

1.1.1 Spectral problem of Kre˘

ın–Feller operators

We start to outline the theoretical preliminaries which are necessary to deﬁne the

Kre

ın–Feller operator

Δ𝐷/𝑁

𝜈

for a given ﬁnite non-zero Borel measure

𝜈

on the ﬁxed

𝑑

-dimensional left-half open unit cube

Q≔(

]𝑑

𝑑∈N

. Let us ﬁx a bounded open

set

Ω⊂Q

with Lipschitz boundary (for the deﬁnition we refer to Appendix A.2),

for which we assume without loss of generality that

lies in the open unit cube.

The Sobolev space

𝐻1(Ω)

is the completion of

C∞

𝑏(Ω)

with respect to the metric

1.1. Statement of the problems

given by the inner product

⟨𝑓 ,𝑔⟩𝐻1(Ω)≔Ω

𝑓 𝑔 dΛ+Ω∇𝑓∇𝑔dΛ,

with

∇𝑓≔𝜕𝑓 /𝜕𝑥1, . . . , 𝜕𝑓 /𝜕𝑥𝑑𝑇

(see also Deﬁnition A.7 for an equivalent deﬁni-

tion). Further, let

𝐻1

0(Ω)

be the completion of

C∞

𝑐(Ω)

w.r.t. the same metric. Here,

denotes the

𝑑

-dimensional Lebesgue measure,

C∞

𝑐(Ω)

the vector space of smooth

functions with compact support contained in

, and

C∞

𝑏(Ω)

the vector space of

functions

𝑓

Ω→R

such that

𝑓|Ω∈ C𝑚(Ω)

for all

𝑚∈N

with

𝐷𝛼𝑓|Ω

uniformly

continuous on

for all

𝛼≔(𝛼1, . . . ,𝛼𝑑)∈N𝑑

(and therefore allowing a unique

continuous extension to Ω). We will consider the inner product

⟨𝑓 ,𝑔⟩𝐻1

0(Ω)≔Ω∇𝑓∇𝑔dΛ,

𝐻1

0(Ω)

, which gives rise to an equivalent norm as a consequence of the Poincar

inequality (see

(PI)

). Further, let

𝐿2

𝜈(Ω)

denote the standard Hilbert space (the

quotient space of the set of real-valued square-

𝜈

-integrable functions with domain

Ωwith respect to the almost-sure equivalence relation) with inner product

⟨𝑓 ,𝑔⟩𝜈≔⟨𝑓 ,𝑔⟩𝐿2

𝜈(Ω)≔Ω

𝑓 𝑔 d𝜈.

Since we mainly focus on the case of

being equal to the interior

˚=(0,1)𝑑

, we

write in this case

𝐻1≔𝐻1(Q

˚)

𝐻1

0≔𝐻1

0(Q

˚)

and

𝐿2

𝜇≔𝐿2

𝜇(Q)

with

𝜇

being a Borel

measure on

. We will assume that the canonical embedding

𝜄

of an appropriate

subspace of

𝐻1

into

𝐿2

𝜈

is continuous and has a dense image. To do so, we ﬁrst

consider the mapping

𝜄:C∞

𝑏Q,⟨·,·⟩𝐻1→𝐿2

𝜈, 𝜄(𝑢)≔𝑢 ,

which is continuous if and only if we ﬁnd a constant 𝐾>0 such that

∥𝑢∥𝐿2

𝜈≤𝐾∥𝑢∥𝐻1

in which case we can extend the operator

𝜄

𝐻1

. Now, suppose that

𝜄

is continuous

and note that its image is always dense in

𝐿2

𝜈

(cf. Proposition 2.11). If

𝜄

is not

injective, that is, if

N𝜈≔ker (𝜄)=𝑓∈𝐻1:∥𝜄(𝑓)∥𝐿2

𝜈=0

is not the null space,

then one simply restricts 𝐻1to

N⊥

𝜈≔𝑓∈𝐻1:∀𝑔∈ N𝜈:⟨𝑓 ,𝑔⟩𝐻1=0.

1.1. Statement of the problems

For the non-negative quadratic form

(𝑓 ,𝑔)↦→ 𝜄−1𝑓 ,𝜄−1𝑔𝐻1

restricted to Neumann

boundary conditions

𝜄N⊥

𝜈

we write

E𝑁

. Replacing

𝐻1

with

𝐻1

in the deﬁnition

N⊥

𝜈

gives rise to the linear subspace

N⊥

0,𝜈

𝐻1

(cf.

(2.2.1)

) and for the form

(𝑓 ,𝑔)↦→ 𝜄−1𝑓 ,𝜄−1𝑔𝐻1

, restricted to the Dirichlet boundary conditions

𝜄(N⊥

0,𝜈 )

write

E𝐷

. This allows us to deﬁne two Kre

ın–Feller operators

Δ𝐷

𝜈

(w.r.t. Dirichlet

boundary conditions) and

Δ𝑁

𝜈

(w.r.t. Neumann boundary conditions) with respect to

the two diﬀerent forms via the following characterization

𝑓∈dom Δ𝐷/𝑁

𝜈⇐⇒ ∀𝑔∈dom E𝐷/𝑁:E𝐷/𝑁(𝑓 ,𝑔)=⟨Δ𝐷/𝑁

𝜈𝑓 ,𝑔⟩𝐿2

𝜈(Ω).

For more details on this form approach, we refer to Section 2.2.5. An important

quantity for the investigation of Kre

ın–Feller operators is the

∞

-dimension of

𝜈

given by

dim∞(𝜈)≔liminf

𝑛→∞

max𝑄∈D𝑁

𝑛log (𝜈(𝑄))

−𝑛log(2),

where D𝑁

𝑛denotes a partition of Qby cubes of the form 𝑄≔𝑑

𝑖=1𝐼𝑖with

𝐼𝑖≔(𝑘𝑖2−𝑛,(𝑘𝑖+1)2−𝑛]

for some 𝑘𝑖∈ {0, . . . , 2𝑛−1}. Now, if the Hu–Lau–Ngai condition [HLN06]

dim∞(𝜈)>𝑑−2 (♠)

is fulﬁlled, then a result of Maz’ya [Maz85] adapted to the dyadic grid (see Lemma

5.3) ensures that the embedding

𝜄

is compact and

Δ𝐷/𝑁

𝜈

admits a countable set of

eigenfunctions spanning

𝐿2

𝜈

with a non-negative and non-decreasing sequence of

eigenvalues

𝜆𝐷/𝑁

𝑛,𝜈 𝑛∈N

tending to inﬁnity, which correspond to the orthonormal

system of eigenfunctions

𝜑𝐷/𝑁

𝑛,𝜈 𝑛∈N

. As mentioned above, the Hu–Lau–Ngai

condition already appeared implicitly in [Tri97, Theorem 30.2 (Isotropic fractal

drum)] in the context of Ahlfors–David regular measures, for which we provide

more details in Section 5.4.2. The lower and upper exponent of divergence of the

eigenvalue counting function

𝑁𝐷/𝑁

𝜈(𝑥)≔sup 𝑛∈N:𝜆𝐷/𝑁

𝑛,𝜈 ≤𝑥

𝑥≥

0, are given

𝑠𝐷/𝑁

𝜈≔liminf

𝑥→∞

log 𝑁𝐷/𝑁

𝜈(𝑥)

log(𝑥)and 𝑠𝐷/𝑁

𝜈≔limsup

𝑥→∞

log 𝑁𝐷/𝑁

𝜈(𝑥)

log(𝑥)(1.1.1)

and we refer to these numbers as the lower and upper spectral dimension of

E𝐷/𝑁

(or of

Δ𝐷/𝑁

or just of

𝜈

), respectively. If the two values coincide, then we denote

the common value by

𝑠𝐷/𝑁

𝜈

and call it the Dirichlet (respect. Neumann) spectral

dimension. There exists a constant

𝐶

such that for all

𝑘∈N

we have

𝜆𝑁

𝑘,𝜈 ≤𝐶𝜆𝐷

𝑘,𝜈

1.1. Statement of the problems

(see Lemma 2.18). This shows that we always have

𝑠𝐷

𝜈≤𝑠𝑁

𝜈and 𝑠𝐷

𝜈≤𝑠𝑁

𝜈.

The spectral dimension also provides some essential information on the domains of

the associated quadratic form and the Kre

ın–Feller operator, namely via the spectral

representation (see for instance [Tri92, Section 4.5.]) given by

•dom E𝐷/𝑁=𝑛∈N𝑎𝑛𝜑𝐷/𝑁

𝑛,𝜈 :𝑛∈N𝑎2

𝑛𝜆𝐷/𝑁

𝑛,𝜈 <∞,

•dom Δ𝐷/𝑁

𝜈=𝑛∈N𝑎𝑛𝜑𝐷/𝑁

𝑛,𝜈 :𝑛∈N𝑎2

𝑛𝜆𝐷/𝑁

𝑛,𝜈 2

<∞.

The knowledge of the growth rate of the eigenvalues leads to many further applica-

tions. For instance, it can be used to study heat kernel estimates [GHN20], stochastic

heat/wave equations deﬁned by Kre

ın–Feller operators [Ehn19; EH21], the approxi-

mation order of Kolmogorov diameters [Eva+09; KN22a], and logarithmic

𝐿2

-small

ball asymptotics [Naz06].

The Kre˘

ın–Feller operator for the one-dimensional case was introduced in [Kre51;

Fel57] and, since the late 1950’s, has been studied by various authors [KK58;

Kac59; UH59; MR62; BS70; KW82; Fuj87; SV95; Vol05; Nga11; Fag12; Arz15;

DN15; FW17; NTX18; Arz14; FM20; Min20; NX20; PS21; JT22]. For dimensions

𝑑>

1 however the situation is quite diﬀerent; in general, it is not even possible to

deﬁne the Kre

ın–Feller operator for a given Borel measure

𝜈

. This is due to the fact

that, in general, there is no continuous embedding of the Sobolev space of weakly

diﬀerentiable functions into the Hilbert space of square-

𝜈

-integrable functions

𝐿2

𝜈

(for example when

𝜈

has atoms). For Dirichlet boundary conditions, in [HLN06] a

suﬃcient condition in terms of the maximal asymptotic direction of the

𝐿𝑞

-spectrum

𝜈

(the

∞

-dimension of

𝜈

) has been established, as provided in (

♠

), which ensures

a compact embedding of the relevant Sobolev space into

𝐿2

𝜈

. It is worth pointing out

that Triebel already stated this condition implicitly in 1997 in [Tri97]. In 2003 (see

[Tri03; Tri04]) he also indicated that there should be a subtle connection between

the multifractal concept of the

𝐿𝑞

-spectrum and analytic properties of the associated

“fractal” operators, a conjecture that we conﬁrm in this thesis.

In contrast to the one-dimensional case, the spectral dimension of Kre

ın–Feller

operators is so far known only for very limited number of singular measures.

The spectral dimension of Kre

ın–Feller operators for higher dimensions was ﬁrst

computed by Birman and Solomjak [BS70, Theorem 5.1] for absolutely continuous

measures, by Naimark and Solomjak [NS95; Sol94] for self-similar measures under

the open set condition (OSC), by Triebel [Tri97, Theorem 30.2] in the setting of

Ahlfors–David regular measures, and, recently, by Ngai and Xie [NX21] for a class

of graph-directed self-similar measures satisfying the graph open set condition. In

1.1. Statement of the problems

[NX21, Sec. 5] Ngai and Xie pointed out that it would also be interesting to study

self-similar measures deﬁned by iterated function systems with overlaps on

R𝑑

𝑑≥

1. Indeed, as an application of our general results from Section 5.3, we are able

to extend these achievements to self-conformal measures without any restriction on

the separation conditions.

In the remainder of this section, we discuss some important results regarding the

spectral dimension for the case

𝑑=

1. Note that we always have

𝑠𝐷/𝑁

𝜈≤

2 (see

[BS66; BS67]). The case for measures with a non-zero absolutely continuous part

was completely solved in an elegant way in [BS70] (see also [MR62]) using a

variational approach. In this case, we have for a ﬁnite Borel measure

𝜈

(

)

with absolutely continuous part 𝜎Λand singular part 𝜂,

lim

𝑥→∞

𝑁𝐷/𝑁

𝜂+𝜎Λ(𝑥)

𝑥1/2=1

𝜋[0,1]

√𝜎dΛ,

and particularly if

𝜎

is non-vanishing, then the spectral dimension exists and equals

2. Besides these estimates, many partial results have been obtained showing that

there is a subtle connection between spectral properties and geometric data of

𝜈

which is a major line of investigation since the famous result by H. Weyl [Wey11].

Another important example is the case of self-similar measures

𝜈

under the open

set condition (OSC) with contractions

𝑟1, . . . ,𝑟𝑛∈(0,1)

and probability weights

𝑝1, . . . ,𝑝𝑛∈(0,1)

. It has been shown in [Fuj87; Sol94; UH59] that in this case the

spectral dimension 𝑠𝐷/𝑁

𝜈is given by the unique number 𝑞>0 fulﬁlling

𝑛



𝑖=1(𝑝𝑖𝑟𝑖)𝑞=1.(1.1.2)

Arzt [Arz14] generalized this result to a class of homogeneous Cantor measures.

In Section 4.4.2, we will use a similar construction to ﬁnd an example for which

the spectral dimension does not exist (see Example 4.48 and Example 5.29). Re-

cently, building on the ideas of Arzt [Arz14] and Freiberg, Hambly, and Hutchinson

[FHH17], Minorics [Min20; Min17] computed the spectral dimension of random

𝑉

-variable Cantor measures and random recursive Cantor measures. Another way

to generalize the classical self-similar setting under OSC is to drop the assumption

of the OSC. Special classes of self-similar measures with overlap have been investi-

gated by Ngai [Nga11], Ngai, Tang, and Xie [NTX18], and Ngai and Xie [NX20].

We will make use of the following notation. For any two functions

𝑓 ,𝑔

R≥0→R

we write

𝑓≪𝑔

if there exist positive constants

𝑐, 𝑥0

such that

𝑐 𝑓 (𝑥) ≤𝑔(𝑥)

for all

𝑥≥𝑥0

; we write

𝑓≍𝑔

if both

𝑓≪𝑔

and

𝑔≪𝑓

hold. The asymptotic behavior of

𝑁𝐷/𝑁

𝜈

strongly depends on the measure

𝜈

and it should be noted that the signiﬁcant

diﬀerence between the Kre

ın–Feller operator and the classical Laplace operator

is that the leading term of the eigenvalue counting function of the Kre

ın–Feller

1.1. Statement of the problems

operator may oscillate, as has been pointed out by Triebel [Tri97]. In general, one

cannot even expect that

𝑁𝐷/𝑁

𝜈

obeys a power law with a positive exponent

𝑠>

i.e.

𝑁𝐷/𝑁

𝜈(𝑥) ≍ 𝑥𝑠

(for counter examples see, e.g. [Arz14], Example 5.29, and

Example 4.51). Therefore, determining the leading term of

𝑁𝐷/𝑁

𝜈

is a challenging

problem. However, if we restrict our attention to the spectral dimension, then this

problem becomes easier attackable. Surprisingly, we are able to treat arbitrary Borel

measures on

(

)

and determine the upper spectral dimension solely from the data

provided by the measure-geometric information carried by the

𝐿𝑞

-spectrum of

𝜈

Under mild regularity conditions on the measure we can guarantee the existence of

the spectral dimension (see Corollary 4.12). Also, with the help of the

𝐿𝑞

-spectrum

𝜈

we are able to construct ﬁrst examples for which the spectral dimension does

not exist (see Section 4.4.2). In this way we give a partial answer to Kac’s question

in terms of the spectral dimension revealing how the measure theoretic properties

𝜈

and the topological properties of its fractal support determine the spectral

dimension (see Theorem 4.10 and Corollary 4.17). This striking connection is the

subject of this thesis.

1.1.2 Quantization problem

Quantization refers to the operation of converting input from a continuous or

large set of values (e.g. a continuous signal) into a representation space of lower

cardinality than the input (e.g. a discrete signal).

0.511.522.533.544.5

0.5

1.5

2.5



𝑓(𝑞)

𝑞

𝑓(𝑞)

Figure 1.1.1 Simple quantization of the “signal”

𝑓

by averaging

𝑓

over the intervals

[𝑖/2,(𝑖+1)/2],𝑖=0, . . ., 9, denoted by 

𝑓(gray).

The quantization problem for probability measures originates from information

theory, in particular, image compression and data compression. Recently, this theory

has attracted increasing attention in applications such as optimal transport problems

[JP22], numerical integration [ELP22; Pag15], and mathematical ﬁnance [PW12;

Hof+14; BFP16; FPS19; BPW10]. From a mathematical point of view, one is

concerned with the asymptotics of the errors in approximating a given random

1.1. Statement of the problems

variable with a quantized version of that random variable (i.e. taking only ﬁnitely

many values), in the sense of 𝑟-means, 𝑟>0.

We start with a stochastic formulation of the quantization problem. Let

𝑋

a bounded

R𝑑

-valued random variable on a probability space

(Ω,A,P)

and set

𝜈≔P◦𝑋−1

. For a given

𝑛∈N

, let

F𝑛

denote the set of all Borel measurable

functions

𝑓

R𝑑→R𝑑

with

card 𝑓R𝑑≤𝑛

. Our goal is to approximate

𝑋

with a

quantized version of

𝑋

, i.e.

𝑋

will be approximated by elements of the form

𝑓(𝑋)

with 𝑓∈ F𝑛, with respect to the 𝐿𝑟-quasinorm for some 𝑟>0, that is,

𝔢𝑛,𝑟 (𝜈)≔inf

𝑓∈F𝑛Ω|𝑋−𝑓(𝑋)|𝑟dP1/𝑟

=inf

𝑓∈F𝑛|𝑥−𝑓(𝑥)|𝑟d𝜈(𝑥)1/𝑟

We call 𝔢𝑛,𝑟 (𝜈)the 𝑛-th quantization error for 𝜈of order 𝑟>0.

In the following considerations we assume that

𝜈

is a compactly supported Borel

probability measure. For every

𝑛∈N

, we write

A𝑛

={𝛼⊂R𝑑

: 1

≤card(𝛼) ≤ 𝑛}

In [GL00b, Lemma 3.1] an equivalent formulation of the

𝑛

-th quantization error for

𝜈of order 𝑟is given by

𝔢𝑛,𝑟 (𝜈)=inf

𝛼∈A𝑛𝑑(𝑥, 𝛼)𝑟d𝜈(𝑥)1/𝑟

, 𝑟 >0,(1.1.3)

where

𝑑(𝑥, 𝛼)≔min𝑦∈𝛼∥𝑥−𝑦∥

and

∥ · ∥

denotes the Euclidean norm on

R𝑑

. By

[GL00b, Lemma 6.1], we have

𝔢𝑛,𝑟 (𝜈) →

0 for

𝑛→ ∞

. In fact, it is well known that

𝔢𝑛,𝑟 (𝜈)=𝑂(𝑛−1/𝑑)

and

𝔢𝑛,𝑟 (𝜈)=𝑜(𝑛−1/𝑑)

for

𝑛→ ∞

𝜈

is singular with respect to

the Lebesgue measure (see also Proposition 6.1). Hence, it is natural to ask for the

“optimal exponent” of convergence. The calculation of this exponent will be one of

the main achievement of this thesis. For this purpose we deﬁne the lower and upper

quantization dimension for 𝜈of order 𝑟by

𝐷𝑟(𝜈)≔liminf

𝑛→∞

log(𝑛)

−log𝔢𝑛,𝑟 (𝜈)and 𝐷𝑟(𝜈)≔limsup

𝑛→∞

log(𝑛)

−log𝔢𝑛,𝑟 (𝜈).

𝐷𝑟(𝜈)=𝐷𝑟(𝜈)

, then we call the common value the quantization dimension of

𝜈

of order

𝑟

and denote it by

𝐷𝑟(𝜈)

. The quantization dimension reﬂects the

exponential rate of this convergence and has been studied by various authors, for

example [Del+04; Gra02; LM02; Zhu15a; Zhu15b; ZZS16; KZ16; ZZS17; KZ15;

KZ17; Zhu18; Zhu20; ZZ21; Roy13]. A detailed introduction to the mathematical

foundations of the quantization problem can be found in [GL00b]. As pointed out

for instance in [LM02], “the problem of determining the quantization dimension

function for a general probability is open”. In this thesis we close this gap for the

upper quantization dimension and, under additional regularity conditions, also for

the lower quantization dimension. Building on a result of [PS00; Fen07], we conﬁrm

1.1. Statement of the problems

the existence of the quantization dimension of self-conformal measures with respect

to conformal iterated function systems without any separation conditions.

The following theorem by Zador is a classical result from quantization theory. It

was proposed in [Zad82] and then generalized by Bucklew and Wise [BW82]; we

refer to [GL00b, Theorem 6.2] for a rigorous proof.

Let

𝜈

be a Borel probability measure with bounded support and let

ℎ

denote the density of the absolutely continuous part of 𝜈. Then

lim

𝑛→∞𝑛−𝑟/𝑑𝔢𝑛,𝑟 (𝜈)𝑟=𝐶(𝑟, 𝑑 )ℎ𝑑

𝑑+𝑟(𝑥)d𝑥𝑑+𝑟

𝑑

, 𝑟 >0,

where 𝐶(𝑟,𝑑 )is a constant independent of 𝜈.

Interestingly, there is a similar result for polyharmonic operators (see for instance

[BS70]). While engineers are mainly dealing with absolutely continuous distribu-

tions, from a mathematical point of view, the quantization problem is signiﬁcant for

all Borel probability measures with bounded support.

Another important example is the case of self-similar measures

𝜌

under OSC with

contractions

𝑟1, . . . ,𝑟𝑛∈(0,1)

and probability weights

𝑝1, . . . ,𝑝𝑛∈(0,1)

. By Graf

and Luschgy [GL00a], the quantization dimension exists and is uniquely determined

by 𝑛



𝑖=1𝑝𝑖𝑟𝑟

𝑖𝐷𝑟(𝜈)/(𝑟+𝐷𝑟(𝜈)) =1.(1.1.4)

Graf and Luschgy’s work on the quantization dimension was the starting point of

many further generalizations investigating more general classes of fractal measures

such as self-aﬃne measures on Bedford-McMullen carpets [KZ16], self-conformal

measures [LM02], and inhomogeneous self-similar measures [Zhu08a; Zhu08b]. It

is worth mentioning that in the case

𝑟=𝑑=

1 the formula of the spectral dimension

in the self-similar case under OSC is quite similar to the formula of

𝐷1(𝜈)

. In

fact, this is no coincidence; we will prove that for general measures

𝜈

the spectral

dimension and upper quantization dimension are closely related (see Corollary 6.8).

1.1.3

Optimal partition problems and optimized coarse multifractal

dimension

Motivated by the study of upper bounds of the spectral dimension of Kre

ın–Feller

operators, polyharmonic operators [KN22b; KN22a; KN22c], and quantization

dimension [KNZ22], we are interested in the following general combinatorial

problem which plays a major role for the investigations in this thesis. Let

𝔍

D →

R≥0with D≔𝑛∈ND𝑁

𝑛satisfying the following natural assumptions:

1.1. Statement of the problems

•𝔍is monotone, that is, 𝔍(𝑄′) ≤ 𝔍(𝑄)for all 𝑄′,𝑄 ∈ D with 𝑄′⊂𝑄.

•𝔍is uniformly vanishing, i.e. lim𝑛→∞ sup𝑄∈D𝑚,𝑛 ≤𝑚𝔍(𝑄)=0.

•𝔍

is locally non-vanishing, i.e. if

𝔍(𝑄)>

0 for

𝑄∈ D

, then there exists

𝑄′⊊𝑄,𝑄′∈ D with 𝔍(𝑄′)>0.

We are particularly interested in the following class of set functions

𝔍𝜈,𝑎,𝑏 (𝑄)≔









sup

𝑄∈D(𝑄)𝜈

𝑄𝑏log Λ

𝑄, 𝑎 =0,

sup

𝑄∈D(𝑄)𝜈

𝑄𝑏

Λ

𝑄𝑎

, 𝑎 ≠0,

where

𝑏≥

𝑎∈R

D(𝑄)≔

𝑄∈ D :

𝑄⊂𝑄

, and

𝜈

is a ﬁnite Borel measure on

, which we call spectral partition function with parameters

𝑎

𝑏

. The spectral

partition function arises naturally in the investigation of Kre

ın–Feller operators (for

𝑎=(

−𝑑)/𝑑

𝑏=

1) and the quantization problem (for

𝑎>

𝑏=

1). Our goal is to

control the asymptotic behavior of

M𝔍(𝑥)≔inf card (𝑃):𝑃∈Π𝔍: max

𝑄∈𝑃𝔍(𝑄)<1/𝑥, 𝑥 >1/𝔍(Q),

where

Π𝔍

denotes the set of ﬁnite collections of dyadic cubes such that for all

𝑃∈Π𝔍

there exists a partition



𝑃

by dyadic cubes from

with

𝑃={𝑄∈

𝑃

𝔍(𝑄)>

}

An important quantity for measuring the growth rate of

M𝔍

is the lower, resp. upper

𝔍-partition entropy deﬁned by

ℎ𝔍≔liminf

𝑥→∞

log M𝔍(𝑥)

log(𝑥), ℎ𝔍≔lim sup

𝑥→∞

log M𝔍(𝑥)

log(𝑥).(1.1.5)

Under mild conditions on

𝔍

, it is closely related to its dual problem (see Proposition

3.11), which is concerned with the control of the asymptotic behavior of

𝛾𝔍,𝑛 ≔inf

𝑃∈Π𝔍,

card(𝑃) ≤𝑛

max

𝑄∈𝑃𝔍(𝑄).

For the special choice

𝔍≔𝔍𝜈,𝑎,1

with

𝑎>

0 and

𝜈

being a ﬁnite Borel measure on

(or, more generally, a superadditive function, see Section 3.3), the dual problem has

attracted much attention in numerous papers by Birman and Solomjak [BS67; BS70],

Borzov [Bor71], and more recently by Davydov, Kozynenko, and Skorokhodov

[DKS20] and by Hu, Kopotun, and Yu [HKY00]. The study of

𝛾𝔍,𝑛

in [BS67] was

motivated by the study of integral operators (see for instance [Bor71]). The classical

1.1. Statement of the problems

result by Birman and Solomjak [BS67, Theorem 2.1] states that

𝛾𝔍,𝑛 =𝑂𝑛−(1+𝑎),

which under the additional assumption that

𝜈

is singular with respect to the Lebesgue

measure was improved by Borzov [Bor71] to

𝛾𝔍,𝑛 =𝑜𝑛−(1+𝑎)

. It should be noted

that in the early 1970s a ﬁrst attempt was made to ﬁnd the “right exponent” in the

case

𝑑=

1; however, the estimate obtained in [Bor71, p. 41] depends only on the

support of

𝜈

. Consequently, this approach ignores important information about the

involved measure

𝜈

, resulting in an inaccurate estimate of the exponent. In this

thesis, we close this gap by giving the exact exponent (see Corollary 3.21).

Motivated by the study of lower estimates of the spectral dimension and quantization

dimension (see Section 4.1, Section 5.2 and Section 6.2) we borrow ideas from the

coarse multifractal analysis (see [Nga97; Fal14; Rie95] and [Fal97, Chapter 11]),

which, roughly speaking, is concerned with the study of global (coarse) properties

of compactly supported Borel measures on small dyadic cubes. In contrast to the

coarse multifractal analysis in which only bounded Borel measures are considered,

we generalize this idea to set functions

𝔍

under the assumptions formulated above.

To be more precise, for all 𝑛∈Nand 𝛼>0 we deﬁne

N𝐷/𝑁

𝔍,𝛼 (𝑛)≔card 𝑀𝐷/𝑁

𝑛(𝛼), 𝑀𝐷/𝑁

𝔍,𝑛 (𝛼)≔𝑄∈ D𝐷/𝑁

𝑛:𝔍(𝑄)≥2−𝛼𝑛 ,

with

D𝐷

𝑛≔𝑄∈ D𝑁

𝑛:𝜕Q∩𝑄=∅

, an object motivated by the study of Kre

ın–

Feller operators with respect to Dirichlet boundary conditions (see also Section 2.1).

We set

𝐹𝐷/𝑁

𝔍(𝛼)≔limsup

𝑛→∞

log+N𝐷/𝑁

𝔍,𝛼 (𝑛)

log(2𝑛)and 𝐹𝐷/𝑁

𝔍(𝛼)≔liminf

𝑛→∞

log+N𝐷/𝑁

𝔍,𝛼 (𝑛)

log(2𝑛),

with

𝑥≥

0 and

log+(𝑥)≔max{

,log(𝑥)}

(where we use the convention that

log(0)≔−∞), and refer to the quantities

𝐹𝐷/𝑁

𝔍≔sup

𝛼>0

𝐹𝐷/𝑁

𝔍(𝛼)

𝛼and 𝐹𝐷/𝑁

𝔍≔sup

𝛼>0

𝐹𝐷/𝑁

𝔍(𝛼)

𝛼

as the upper, resp. lower optimized (Dirichlet/Neumann) coarse multifractal dimen-

sion with respect to

𝔍

. In Chapter 3, we see that the

𝔍

-partition entropy and the

optimized coarse multifractal dimension with respect to

𝔍

are strongly linked by

ideas from the theory of large deviations.

1.2. Outline and statement of the main results

1.2 Outline and statement of the main results

This thesis is dedicated to the study of general optimal partition problems and

their applications for the determination of the spectral dimension of Kre

ın–Feller

operators and the quantization dimension. The main achievement of this thesis is

the development of a uniﬁed framework to tackle both the computation of the upper

spectral dimension and the upper quantization dimension.

The thesis is divided into ﬁve main parts. In Chapter 2, we provide some preliminary

considerations. In Section 2.2, under the Hu–Lau–Ngai condition (

♠

), we deﬁne

Kre

ın–Feller operators

Δ𝐷/𝑁

𝜈

via a form approach with respect to Dirichlet/Neumann

boundary conditions. We prove a slight modiﬁcation of the well-known min-max

principle (Proposition 2.17) for the representation of the eigenvalues. The min-max

principle is a powerful tool which enables us to reduce the eigenvalue counting

problem to the optimal partition problem described in Section 1.1.3. We conclude

Section 2.2 by constructing smooth functions via molliﬁers. We will use this

construction to prove that the condition

dim∞(𝜈)<𝑑−

2 implies that there is no

continuous embedding of

𝐻1(Q)

into

𝐿2

𝜈(Q)

. As a consequence, if

dim∞(𝜈)<𝑑−

then it is impossible to deﬁne the Kre

ın–Feller operator. Section 2.3 is dedicated to

the introduction of the new concept of partition functions, which, to a certain extent,

is borrowed from the coarse multifractal analysis; for a non-negative, monotone set

function 𝔍:D → R≥0, the Dirichlet/Neumann partition function of 𝔍is given by

𝜏𝐷/𝑁

𝔍(𝑞)≔limsup

𝑛→∞

log(2𝑛)log 

𝑄∈D𝐷/𝑁

𝑛,

𝔍(𝑄)>0

𝔍(𝑄)𝑞, 𝑞 ≥0.

The function

𝜏𝑁

𝔍

encodes important information about

𝔍

; it provides a quantitative

description of the global ﬂuctuation of

𝔍

. Furthermore, under mild conditions on

𝔍

(see Lemma 2.25), we have the following important representation of the zero of

𝜏𝑁

𝔍as a critical value:

𝑞𝑁

𝔍≔inf{𝑞>0 : 𝜏𝑁

𝔍(𝑞)<0}=inf 





𝑞>0 : 

𝑄∈D

𝔍(𝑄)𝑞<∞





.

This representation will be crucial for establishing upper bounds of the

𝔍

-partition

entropy. An important special case is the Neumann partition function of

𝔍 = 𝜈

which

is known as the

𝐿𝑞

-spectrum of

𝜈

. Throughout this thesis we write

𝛽𝐷/𝑁

𝜈=𝜏𝐷/𝑁

𝜈

The

𝐿𝑞

-spectrum of

𝜈

provides important information about

𝜈

, for example

𝛽𝑁

𝜈(

)

equal to the upper Minkowski dimension of the support of

𝜈

and

lim𝑞→∞ 𝛽𝑁

𝜈(𝑞)/𝑞=

−dim∞(𝜈)

. In Section 2.4.2, we discuss conditions which guarantee that the Dirich-

let partition function and Neumann partition function coincide. We conclude that

1.2. Outline and statement of the main results

chapter by computing the (Dirichlet/Neumann) partition function for the spectral par-

tition function for leading examples; we consider absolutely continuous measures,

product measures, Ahlfors-David regular measures, and self-conformal measures.

Chapter 3 can be seen as the ﬂagship of this thesis. We develop a machinery

which enables us to tackle the problem of the computation of the upper

𝔍

-partition

entropy under mild assumptions on

𝔍

(see Section 3.4). Section 3.1 is devoted to

establishing lower and upper bounds of the lower and upper

𝔍

-partition entropy

in terms of the zero of

𝜏𝑁

𝔍

and the lower and upper optimized coarse multifractal

dimension with respect to

𝔍

, respectively. This will enable us to use an adaptive

approximation algorithm to construct certain partitions of dyadic cubes to estimate

the

𝔍

-partition entropy from above. A detailed motivation for this approach is given

in the beginning of Section 3.1. Based on that, in Section 3.2, we show that the

𝔍𝜈,𝑎,𝑏-partition entropy can be bounded from above by

𝑞𝑁

𝔍𝑎,𝑏 =inf{𝑞>0 : 𝜏𝑁

𝔍𝑎,𝑏 (𝑞)<0}

whenever

𝑏dim∞(𝜈) +𝑎𝑑 >

0. Section 3.3 is devoted to the study of the corre-

sponding dual problem. We begin with a presentation of an adaptive approximation

algorithm by Birman and Solomjak [BS67] to reproduce known results. We then

demonstrate how one can use the results of Section 3.1 to improve known upper

bounds in terms of

𝑞𝑁

𝔍

(Proposition 3.11). Section 3.4 contains the main results of

that chapter. The basic idea of that section is to apply large derivation theory (see

Lemma 3.17) to estimate the upper optimized coarse multifractal dimension with

respect to

𝔍

from below. The ﬁrst main result is given by Corollary 3.21. It states

that

𝐹𝐷

𝔍=𝑞𝐷

𝔍

and

𝐹𝑁

𝔍=ℎ𝔍=𝑞𝑁

𝔍.

The second main result is concerned with the question under which conditions

we can ensure that

𝐹𝑁

𝔍

and

ℎ𝔍

exist as limits (i.e.

ℎ𝔍=ℎ𝔍

and

𝐹𝑁

𝔍=𝐹𝑁

𝔍

). In

Corollary 3.23, we impose a suﬃcient condition (see Deﬁnition 3.22); if

𝔍

Dirichlet/Neumann partition function regular, that is, if

𝜏𝐷/𝑁

𝔍

is diﬀerentiable and

exists as a limit in

𝑞𝐷/𝑁

𝔍

, or

𝜏𝐷/𝑁

𝔍

exists as a limit on a left-sided neighborhood of

𝑞𝐷/𝑁

𝔍, then

𝐹𝐷

𝔍=𝐹𝐷

𝔍=𝑞𝐷

𝔍

and

𝐹𝑁

𝔍=𝑞𝑁

𝔍=ℎ𝔍=ℎ𝔍.

1.2. Outline and statement of the main results

Chapter 4 is dedicated to studying the spectral dimension of Kre

ın–Feller operators

for the case

𝑑=

1 with respect to non-zero Borel measures on

(

)

. It turns out that

the spectral partition function with parameters (2−𝑑)/𝑑, 1, given by

𝔍𝜈(𝑄)= 𝔍𝜈,1,1(𝑄)=𝜈(𝑄)Λ(𝑄), 𝑄 ∈ D,

is a central object for the calculation of the upper spectral dimension. Its importance

stems from the fact that

𝔍𝜈

appears as an embedding constant (which, in fact, is

equivalent to the best constant) for the embedding of

𝐻1

0(𝑄)

into

𝐿2

𝜈(𝑄)

. Combined

with the min-max principle, we can reduce the original problem of the computation

of the lower and upper spectral dimension to the combinatorial problems with

respect to

𝔍𝜈

considered in Chapter 3. In Section 4.1, we establish lower bounds for

the lower and upper spectral dimension in terms of the lower and upper optimized

coarse multifractal dimension with respect to

𝔍𝜈

, respectively. Section 4.2 is devoted

to the study of upper bounds for the lower and upper spectral dimension. We show

that the lower and upper spectral dimension is bounded from above by the lower and

upper

𝔍𝜈

-partition entropy, respectively. Further, we obtain upper bounds for the

lower spectral dimension in terms of the lower Minkowski dimension of

supp(𝜈)

denoted by dim𝑀(𝜈), and the ∞-dimension of 𝜈as follows:

𝑠𝐷/𝑁

𝜈≤ℎ𝔍𝜈≤dim𝑀(𝜈)

1+dim∞(𝜈).

In Section 4.3, we present the main results of that chapter. By combining the results

of Section 4.1 and Section 4.2 we are able to compute the upper spectral dimension.

The ﬁrst main result (Theorem 4.10) reads as follows:

𝐹𝑁

𝔍𝜈≤𝑠𝐷/𝑁

𝜈≤𝑠𝐷/𝑁

𝜈=𝐹𝑁

𝔍𝜈=𝑞𝑁

𝔍𝜈=ℎ𝔍𝜈.

This reveals an interesting connection between the upper spectral dimension, the

optimized coarse multifractal dimension with respect to

𝔍𝜈

, the

𝔍𝜈

-partition entropy,

and the spectral partition function. Since the partition function of

𝔍𝜈

is equal to

𝑞↦→ 𝛽𝑁

𝜈(𝑞) −𝑞

, we obtain an interesting geometric interpretation of the upper

spectral dimension. If

𝑞𝑁

𝔍𝜈

0, then the upper spectral dimension is given by the

ﬁxed point of the 𝐿𝑞-spectrum of the corresponding measure 𝜈(see Figure 1.2.1).

Section 4.3.2 is concerned with the determination of the conditions that ensure the

spectral dimension exists. For this purpose we use the regularity results of Section

3.4 applied to

𝔍𝜈

. This leads to the following regularity condition, which ensures

the existence of the spectral dimension (see Corollary 4.12):

𝔍𝜈

is Neumann partition function regular, then the spectral dimen-

sion exists and is given by 𝑠𝐷/𝑁

𝜈=𝑞𝑁

𝔍𝜈.

1.2. Outline and statement of the main results

dim𝑀(𝜈)

𝛽𝑁

𝜈(𝑞)

𝑠𝐷/𝑁

𝜈

dim𝐻(𝜈)

𝑞

Figure 1.2.1 The intersection point

𝑞𝑁

𝔍𝜈

of the

𝐿𝑞

-Spectrum

𝛽𝑁

𝜈

with respect to

𝜈

and

the identity map. Here

𝜈

is chosen to be the

(0.05,0.95)

-Salem measure with full sup-

port

supp (𝜈)=[

]

. The intersection of

𝛽𝑁

𝜈

with the

𝑦

-axis yields the Minkowski

dimension of

supp (𝜈)

, namely 1, and the intersection with the (dotted) tangent to

𝛽𝑁

𝜈

(0,1)

yields the Hausdorﬀdimension

dim𝐻(𝜈)

of the measure

𝜈

, which equals

(0.05 log (0.05)+0.95log (0.95))/log (2).

Fortunately, this regularity condition is usually easy to verify. For instance, we will

apply this result for weak Gibbs measures without any separation conditions. In

Section 4.3.3, we derive lower and upper bounds for the lower and upper spectral

dimension, respectively. For the lower and upper spectral dimension we have

the following general bounds depending on the topological support of

𝜈

, namely

dim𝑀(𝜈), and the right and left derivative 𝜕−𝛽𝑁

𝜈,𝜕+𝛽𝑁

𝜈of 𝛽𝑁

𝜈at 1:

−𝜕+𝛽𝑁

𝜈(1)

1−𝜕−𝛽𝑁

𝜈(1)≤𝑠𝐷/𝑁

𝜈≤𝑠𝐷/𝑁

𝜈≤dim𝑀(𝜈)

1+dim𝑀(𝜈)≤1

and

𝑠𝐷/𝑁

𝜈=dim𝑀(𝜈)

1+dim𝑀(𝜈)⇐⇒ −𝜕−𝛽𝑁

𝜈(1)=dim𝑀(𝜈).

We conclude this chapter with three leading examples in Section 4.4. In the ﬁrst

example we investigate weak Gibbs measures with respect to a

-IFS (with or

without overlap). Thereby, we generalize the classical result for the self-similar

setting under OSC (1.1.2) in three ways:

•

In Section 4.4.1.1, we provide a ﬁrst contribution to the nonlinear setting in a

broad sense. More precisely, we consider weak Gibbs measures on fractals

which are generated by non-trivial C1-IFS’s under OSC. It turns out that the

1.2. Outline and statement of the main results

spectral dimension is given by the zero of the associated pressure function

(see (4.4.3)), which constitutes a natural generalization of (1.1.2).

•

As a second novelty in Section 4.4.1.3, we drop the assumption of the OSC

and allow overlaps. In this situation the computation of the spectral dimension

is much more complex compared to

(1.1.2)

. However, by ideas of [Fen07;

PS00], we are able to prove the existence of the

𝐿𝑞

-spectrum on

(

]

(see

Proposition 4.45). This implies that

𝔍𝜈

is Neumann partition function regular.

Consequently, Corollary 4.12 yields the existence of the spectral dimension

given as the ﬁxed point of the associated 𝐿𝑞-spectrum.

•

Our ﬁnal contribution to the nonlinear setting concerns Gibbs measures on

fractals generated by

C1+𝛾

-IFS’s under OSC. For this class, using renewal

theory in a dynamical context (see for instance [Kom18; KK17]), we are able

to prove the spectral asymptotics (see Theorem 4.42)

𝑁𝐷

𝜈(𝑡) ≍𝑡𝑧𝜈,

where 𝑧𝜈is the unique zero of the pressure function as deﬁned in (4.4.3).

In Section 4.4.2, we construct a ﬁrst example with non-converging

𝐿𝑞

-spectrum for

which the spectral dimension does not exist, with the help of homogeneous Cantor

measures (see Example 4.48). We end Chapter 4 with the investigation of purely

atomic measures whose spectral dimension exists and attains values in [0,1/2].

In Chapter 5, we discuss the spectral dimension of Kre

ın–Feller operators

Δ𝐷/𝑁

𝜈

with respect to Dirichlet and Neumann boundary conditions for the case

𝑑>

where 𝜈is a non-zero Borel measure on Qwith dim∞(𝜈)>𝑑−2. This chapter can

be seen as generalization of some results for the one-dimensional case. However,

in contrast to the one-dimensional case, there are some diﬃculties. There is no

continuous embedding of the Sobolev space

𝐻1(Q)

into

C𝑏(Q)

and, in general, we

cannot guarantee that the spectral dimensions with respect to Dirichlet and Neu-

mann boundary conditions coincide. Thus, many proofs from the one-dimensional

case cannot be directly adopted. Again, the main strategy is to use the min-max

principle to reduce the problem of the computation of the spectral dimension to the

combinatorial problems investigated in Chapter 3, where

𝔍

is chosen to be equal to

𝔍𝜈(𝑄)≔𝔍𝜈,(2−𝑑)/𝑑,1(𝑄)=









sup

𝑄∈D(𝑄)𝜈

𝑄log Λ

𝑄, 𝑑 =2,

sup

𝑄∈D(𝑄)𝜈

𝑄Λ

𝑄(2−𝑑)/𝑑

, 𝑑 >2,

with

𝑄∈ D

. In Section 5.1, we discuss upper bounds for the lower and upper

spectral dimension. Motivated by the ideas of [NS95] (see also Remark 5.2) and the

proof of Proposition 4.4, we start with an important observation on the embedding

1.2. Outline and statement of the main results

constants on sub-cubes of

and the upper spectral dimension in Section 5.1.1. The

main result of that section reads as follows:

Suppose there exists a non-negative, uniformly vanishing, monotone

set function

𝔍

such that for all

𝑄∈ D

and all

𝑢∈ C∞

𝑏(𝑄)

with

𝑄𝑢dΛ=0, we have

∥𝑢∥2

𝐿2

𝜈(𝑄)≤𝔍(𝑄)∥∇𝑢∥2

𝐿2

Λ(𝑄).(1.2.1)

Then we have

𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤ℎ𝔍

and

𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤ℎ𝔍.

In contrast to the case

𝑑=

1 (see Lemma 2.2), the best embedding constant on dyadic

sub-cubes of

from the embedding of

𝐻1(𝑄)

into

𝐿2

𝜈(𝑄)

is hard to compute for

general measures. Fortunately, the best embedding constant for the embedding of

𝐿𝑡

𝜈|𝑄R𝑑

with

𝑡>

2 into

𝐻1R𝑑

has been computed by Maz’ya and Preobrazenskii

[Maz11, p. 83] for

𝑑=

2 and by Adam [Ada71; Ada73] (see also [Maz11, p. 67]) for

𝑑>

2. Using this and the Stein extension established in Lemma 2.8, we show that

(1.2.1)

is valid for

𝔍 = 𝔍𝜈, (2−𝑑)/𝑑,2/𝑡

with 2

<𝑡<

dim∞(𝜈)/(𝑑−

)

(see Lemma

5.5). By combining the results above, the main results of Section 5.1.2 are the

following chains of inequalities:

𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤lim

𝑡↓2ℎ𝔍𝜈,𝑡 (2/𝑑−1)/2,1≤𝑞𝑁

𝔍𝜈

and

𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤lim

𝑡↓2ℎ𝔍𝜈,𝑡 (2/𝑑−1)/2,1.

Section 5.2 is devoted to establishing lower bounds for the lower and upper spectral

dimension. Motivated by the proof of Proposition 4.1 for the one-dimensional

setting, the lower estimate of the spectral dimension is based on the following

general principle which connects the optimized coarse multifractal dimension and

the spectral dimension (see Proposition 5.9 for the deﬁnition):

Assume there exists a non-negative monotone set function

𝔍

with

dim∞(𝔍)>

0(see Section 2.3.1) such that for every

𝑄∈ D

with

𝔍(𝑄)>

0there exists a non-negative and non-zero function

𝜓𝑄∈ C∞

𝑐R𝑑

with

support contained in 𝑄

˚3such that

𝜓𝑄2

𝐿2

𝜈≥𝔍(𝑄)∇𝜓𝑄2

𝐿2

Λ(R𝑑),(1.2.2)

1.2. Outline and statement of the main results

where

𝑄

˚3

denotes the cube centered and parallel with respect to

𝑄

such that

𝑄

˚3=𝑇(𝑄

˚)+(

−

)𝑥0

with

𝑇(𝑥)=

𝑥, 𝑥 ∈R𝑑

and

𝑥0∈R𝑑

is the center of 𝑄. Then

𝐹𝑁

𝔍≤𝑠𝑁

𝜈, 𝐹 𝑁

𝔍≤𝑠𝑁

𝜈, 𝐹𝐷

𝔍≤𝑠𝐷

𝜈and 𝐹𝐷

𝔍≤𝑠𝐷

𝜈.(1.2.3)

In Section 5.1.2, as an application of the general principle above, we use the results

of Section 2.2.6 to construct appropriate functions from

C∞

𝑐(R𝑑)

and the min-max

principle to demonstrate that (1.2.3) is valid for 𝔍 = 𝔍𝜈.

By combining the lower and upper bounds of the spectral dimension presented in

Section 5.2 and Section 5.1, we are able to calculate the upper spectral dimension

with respect to Neumann boundary conditions (see Section 5.3). More precisely, we

show equality of the optimized coarse multifractal dimension with respect to

𝔍𝜈

the 𝔍𝜈-partition entropy, and the unique zero of 𝜏𝑁

𝔍𝜈:

ℎ𝔍𝜈=𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈=𝐹𝑁

𝔍𝜈.

Surprisingly, it turns out that in the case

𝑑=

2, the formula above simpliﬁes to

𝑠𝑁

𝜈=

1, and under the assumption

𝜈(Q

˚)>

0, we also have

𝑠𝐷

𝜈=

1. Thus, in the case

𝑑=

2, the upper spectral dimension contains no information about the underlying

measure

𝜈

. This can be explained by the simple fact that the

𝐿𝑞

-spectrum always

has a zero at 1.

Another important question is under which conditions we can ensure that the upper

spectral dimension with respect to Dirichlet and Neumann boundary conditions

coincide. We show that if

𝜏𝑁

𝔍𝜈(𝑞𝐷

𝔍𝜈)=

0, or equivalently

𝐹𝑁

𝔍𝜈=𝐹𝐷

𝔍𝜈

, then the upper

Dirichlet and Neumann spectral dimensions fulﬁll

𝑠𝐷

𝜈=𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈

. Motivated by this

observation, we impose conditions on the Minkowski dimension of

supp(𝜈) ∩𝜕Q

and on the growth rate of the boundary cubes (see

(5.3.2)

) to guarantee

𝜏𝑁

𝔍𝜈(𝑞𝐷

𝔍𝜈)=

Further, we address the question of the existence of the spectral dimension. As

in the one-dimensional case, we make use of the regularity conditions imposed in

Proposition 3.24 to establish the following regularity results.

•

𝔍𝜈

is Neumann partition function regular, then the spectral dimension

𝑠𝑁

𝜈

exists.

•

𝔍𝜈

is Dirichlet partition function regular and

𝜏𝑁

𝔍𝜈(𝑞𝐷

𝔍𝜈)=

0, then both the

Dirichlet and Neumann spectral dimension exist and coincide, i.e. 𝑠𝐷

𝜈=𝑠𝑁

𝜈.

Additionally, using the formula for

𝑠𝑁

𝜈

, we estimate the upper spectral dimension

with respect to Neumann boundary conditions in terms of the upper Minkowski

1.2. Outline and statement of the main results

dimension of supp(𝜈)and ∞-dimension of 𝜈as follows:

𝑑

2≤dim𝑀(𝜈)

dim𝑀(𝜈)−𝑑+2≤𝑠𝑁

𝜈≤dim∞(𝜈)

dim∞(𝜈)−𝑑+2.(1.2.4)

We conclude Section 5.3 by discussing Kac’s question in view of

(1.2.4)

. Finally,

we end Chapter 5 with four examples in Section 5.4. Here, we study absolutely

continuous measures, Ahlfors-David regular measures, self-conformal measures

without any separation conditions, and we present an example for which the spectral

dimension does not exist.

Chapter 6 is dedicated to the study of the lower and upper quantization dimension

with respect to a ﬁnite Borel measure

𝜈

. Here, we use the same strategy as in

Chapter 4 and Chapter 5; we reduce the computation of the quantization dimension

to the auxiliary combinatorial problems investigated in Chapter 3 for the special

choice

𝔍𝜈,𝑟/𝑑(𝑄)≔𝔍𝜈,𝑟/𝑑,1(𝑄)=𝜈(𝑄)Λ(𝑄)𝑟/𝑑, 𝑄 ∈ D.

More precisely, we will link the lower and upper quantization dimension to the

dual problem studied in Section 3.3 with respect to

𝔍𝜈,𝑟/𝑑

. Section 6.1 and Section

6.2 are devoted to the study of lower and upper bounds for the lower and upper

quantization dimension. Section 6.3 contains the main results of that chapter. By

combining the estimates obtained in Section 6.1 and Section 6.2, we are able to

compute the upper quantization dimension for the ﬁrst time. The ﬁrst main result

(see Theorem 6.5) reads as

𝑟 𝐹 𝑁

𝔍𝜈,𝑟 /𝑑

1−𝐹𝑁

𝔍𝜈,𝑟 /𝑑≤𝐷𝑟(𝜈)≤

𝑟ℎ𝔍𝜈,𝑟 /𝑑

1−ℎ𝔍𝜈,𝑟 /𝑑≤𝐷𝑟(𝜈)=𝑟ℎ𝔍𝜈,𝑟 /𝑑

1−ℎ𝔍𝜈,𝑟 /𝑑

𝑟𝑞𝑁

𝔍𝜈,𝑟 /𝑑

1−𝑞𝑁

𝔍𝜈,𝑟 /𝑑

=𝑟 𝐹 𝑁

𝔍𝜈,𝑟 /𝑑

1−𝐹𝑁

𝔍𝜈,𝑟 /𝑑

(1.2.5)

Interestingly, if

sup𝑥∈(0,1)𝛽𝑁

𝜈(𝑥)>

0, then the upper quantization dimension coin-

cides with the upper R´

enyi dimension at 𝑞𝑁

𝔍𝜈,𝑟 /𝑑, given by

ℜ𝜈(𝑞)≔𝛽𝑁

𝜈(𝑞)

1−𝑞, 𝑞 ≠1.

This perspective sheds new light on the connection between the quantization prob-

lem and other concepts from fractal geometry in that we obtain a one-to-one corre-

spondence of the upper quantization dimension and the

𝐿𝑞

-spectrum restricted to

(0,1)

. Further, as a consequence of the formula for

𝐷𝑟(𝜈)

, we derive an alternative

1.2. Outline and statement of the main results

representation for 𝐷𝑟(𝜈)as a critical value:

𝐷𝑟(𝜈)

𝐷𝑟(𝜈)+𝑟=inf 





𝑞>0 : 

𝑄∈D Λ(𝑄)𝑟/𝑑𝜈(𝑄)𝑞

<∞





.

This further results in a surprising connection with the upper spectral dimension for

the case 𝑑=𝑟=1 via

𝐷1(𝜈)

𝐷1(𝜈)+1=𝑠𝐷/𝑁

𝜈.

It is worth mentioning that this formula demonstrates a striking connection between

two, at ﬁrst sight, distinct ﬁelds of mathematics (the spectral problem of Kre

ın–

Feller Operators and the quantization of compactly supported Borel measures). As

a further application we conﬁrm a conjecture of Lindsay [Lin01] which states that

𝑟↦→ 𝐷𝑟(𝜈),is continuous for 𝑟>0.

The second main Theorem 6.5 addresses the existence of the quantization dimension.

Based on the regularity conditions imposed in Proposition 3.24 and

(1.2.5)

, we

prove the following regularity result:

𝜏𝑁

𝔍𝜈,𝑟 /𝑑is Neumann partition regular =⇒𝐷𝑟(𝜈)=𝐷𝑟(𝜈)=

𝑟𝑞𝑁

𝔍𝜈,𝑟 /𝑑

1−𝑞𝑁

𝔍𝜈,𝑟 /𝑑

We conclude Chapter 6 with a corollary aﬃrming the existence of the quantization

dimension for self-conformal measure without any separation conditions.

Certain background information (for instance the relation between self-adjoint

operators and quadratic forms) is provided in the Appendix A and referenced

throughout this thesis. A list of symbols we use in this thesis is appended, including

standard notation (e.g. the set of natural numbers).

This thesis is based on the following publications:

•

M. Kesseb

ohmer and A. Niemann. Spectral dimensions of Kre

ın–Feller

operators and

𝐿𝑞

-spectra. Adv. Math. 399 (2022), Paper No. 108253. doi:

10.1016/j.aim.2022.108253

•

M. Kesseb

ohmer and A. Niemann. Approximation order of Kolmogorov

diameters via

𝐿𝑞

-spectra and applications to polyharmonic operators. J. Funct.

Anal. 283.7 (2022), Paper No. 109598. doi:

10.1016/j.jfa.2022.109598

•

M. Kesseb

ohmer and A. Niemann. Spectral asymptotics of Kre

ın-Feller

operators for weak Gibbs measures on self-conformal fractals with overlaps.

Adv. Math. 403 (2022), Paper No. 108384. doi:

10. 1016/j. aim.2022.

108384

1.2. Outline and statement of the main results

•

M. Kesseb

ohmer and A. Niemann. Spectral dimensions of Kre

ın-Feller

operators in higher dimensions. arXiv: 2202.05247 (2022)

•

M. Kesseb

ohmer, A. Niemann, and S. Zhu. Quantization dimensions of

of compactly supported probability measures via R

enyi dimensions. arXiv:

2205.15776 (2022). To appear in: Trans. AMS

Chapter 2

Preliminaries

2.1 Dyadic Partitions

Throughout this thesis we will frequently use dyadic cubes contained in

Q=(

]𝑑

Therefore, we list important basic notations from the introduction:

•D𝑁

𝑛=𝑑

𝑖=1(𝑘𝑖2−𝑛,(𝑘𝑖+1)2−𝑛]:𝑘𝑖=0, . . . , 2𝑛−1,

•D𝐷

𝑛=𝑄∈ D𝑁

𝑛:𝜕Q∩𝑄=∅,

•D=𝑛∈ND𝑁

𝑛,

•D(𝑄)=

𝑄∈ D :

𝑄⊂𝑄with 𝑄∈ D.

We remark that

D𝑁

𝑛+1

is a reﬁnement of

D𝑁

𝑛

for each

𝑛∈N

, this means that each

element of

D𝑁

𝑛

can be decomposed into 2

𝑑

disjoint elements of

D𝑁

𝑛+1

. Hence,

it is easy to see that

is a semiring. The reason why the deﬁnition of

D𝐷

𝑛

is appropriate becomes clear in the constructing certain functions with compact

support contained in

for the proof for the lower bounds in the Dirichlet case (see

proof of Proposition 5.9). For this purpose, a certain distance to the boundary of

the

is needed. The notation

D𝑁

𝑛

is motivated by the proof of the upper estimate

of the spectral dimension of the Kre

ın–Feller with respect to Neumann boundary

conditions. We end this section with some simple facts about Dand D𝐷/𝑁

𝑛.

Lemma 2.1. Let

𝑄1,𝑄2∈ D

with

𝑄1∩𝑄2≠∅

. Then

𝑄1⊂𝑄2

𝑄2⊂𝑄1

. Further,

for all 𝑛∈N, we have

card D𝑁

𝑛\D𝐷

𝑛=2𝑑𝑛 −(2𝑛−2)𝑑.

2.2. Form approach for Kre˘

ın–Feller operators

2.2 Form approach for Kre˘

ın–Feller operators

In this section, for ﬁxed

𝑑∈N

, we deﬁne the Kre

ın–Feller operator with respect to a

non-zero ﬁnite Borel measure

𝜈

Q=(

]𝑑

. Note that we allow that

Q\Q

may

have positive

𝜈

-measure. It should be noted that this assumption is of a technical

character; in fact, the following considerations are also valid if we allow positive

measure on

[

]𝑑\(

)𝑑

. However, this results in some technical diﬃculties for

the deﬁnition of the dyadic cubes of

[

]𝑑

(for details see [KN22d]). Further, let

Ω⊂Qbe a bounded Lipschitz domain in R𝑑. Throughout this chapter, we assume

card(supp(𝜈)∩ Ω)=∞

, or equivalently that

𝐿2

𝜈(Ω)

is an inﬁnite dimensional vector

space.

2.2.1 Sobolev spaces and embeddings

We start by recalling the form approach following ideas in [HLN06]. The space

𝐻1(Ω)

with the bilinear form

⟨·,·⟩𝐻1(Ω)

deﬁnes a Hilbert space and

𝐻1

0(Ω)

is a

closed subspace. For all

𝑢∈𝐻1

0(Ω)

, or

𝑢∈𝑓∈𝐻1(Ω):Ω𝑓dΛ=0

, the Poincar´e

inequality, respect. Poincar´e–Wirtinger inequality (see [Rui12, Lemma 3, p. 500]

and Lemma 2.7), reads, for some constant 𝑐>0, as follows

∥𝑢∥𝐿2

Λ(Ω)≤𝑐∥∇𝑢∥𝐿2

Λ(Ω).(PI)

Since

is a bounded Lipschitz domain, the norm induced by the form

⟨·,·⟩𝐻1

0(Ω)

therefore equivalent to the norm induced by

⟨·,·⟩𝐻1(Ω)

𝐻1

0(Ω)

. We will consider

only those ﬁnite Borel measures

𝜈

on the closure of

for which the following

𝜈-Poincar´e inequality holds for some 𝑐1>0:

∥𝑢∥𝐿2

𝜈(Ω)≤𝑐1∥𝑢∥𝐻1(Ω)for all 𝑢∈𝐶∞

𝑏Ω.(𝜈PI)

This then guarantees a continuous embedding of the Sobolev spaces

𝐻1(Ω)

into

𝐿2

𝜈(Ω)

and

𝐻1

0(Ω)

into

𝐿2

𝜈(Ω)

. In fact, since

𝐶∞

𝑏(Ω)

lies dense in

𝐻1(Ω)

(this

follows e.g. from Proposition A.1), for every

𝑢∈𝐻1(Ω)

there exists a sequence

(𝑢𝑛)𝑛

of elements of

𝐶∞

𝑏(Ω)

such that

𝑢𝑛→𝑢

with respect to the norm of

𝐻1(Ω)

Now,

(𝜈𝑃 𝐼)

implies that

(𝑢𝑛)𝑛

is also a Cauchy sequence in

𝐿2

𝜈(Ω)

, hence there

exists

𝑢∈𝐿2

𝜈(Ω)

such that

𝑢𝑛→𝑢

𝐿2

𝜈(Ω)

. It is easy to see that this limit is

independent of the particular choice of

(𝑢𝑛)𝑛

and we therefore obtain in this way a

bounded linear operator

𝜄≔𝜄𝜈≔𝜄Ω,𝜈 :𝐻1(Ω)→𝐿2

𝜈Ω, 𝑓 ↦→ 𝑓 ,

2.2. Form approach for Kre˘

ın–Feller operators

with

𝜄(𝑢)=𝑢

for all

𝑢∈𝐶∞

𝑏(Ω)

. If

𝜄

is also injective, then we may regard

𝐻1(Ω)

a subspace of

𝐿2

𝜈(Ω)

. In the case the map is not injective we consider the following

closed subspace of 𝐻1(Ω)

N𝜈≔N𝜈,Ω≔ker (𝜄)=𝑓∈𝐻1(Ω):∥𝜄(𝑓)∥𝐿2

𝜈(Ω)=0

and have the natural embedding

N⊥

𝜈≔𝑓∈𝐻1(Ω):∀𝑔∈ N𝜈:⟨𝑓 ,𝑔⟩𝐻1(Ω)=0↩→𝐿2

𝜈Ω,

which is again given by

𝜄

. In particular, there exists a sequence

(𝑢𝑛)𝑛∈𝐶∞

𝑏(Ω)

such

that

𝑢𝑛→𝑢

𝐻1(Ω)

and

𝜄(𝑢𝑛)=𝑢𝑛→𝜄(𝑢)

𝐿2

𝜈(Ω)

. The extended version of

the Poincar´

e inequality (𝜈𝑃 𝐼 )reads as

∥𝜄(𝑢)∥𝐿2

𝜈(Ω)≤𝑐1∥𝑢∥𝐻1(Ω)for all 𝑢∈𝐻1(Ω).

Recall that

𝐶∞

𝑐(Ω) ⊂ 𝐶∞

𝑏(Ω)

lies dense in

𝐻1

0(Ω)

. Hence, for all

𝑢∈𝐻1

0(Ω)

, it

follows

∥𝜄(𝑢)∥𝐿2

𝜈(Ω)=∥𝜄(𝑢)∥𝐿2

𝜈(Ω).

Consequently,

N𝜈∩𝐻1

0(Ω)=𝑓∈𝐻1

0(Ω):∥𝜄(𝑓)∥𝐿2

𝜈(Ω)=0.

Therefore, this embedding carries over to

N⊥

0,𝜈 ≔𝑓∈𝐻1

0(Ω):∀𝑔∈ N𝜈∩𝐻1

0(Ω):⟨𝑓 ,𝑔⟩𝐻1

0(Ω)=0↩→𝐿2

𝜈(Ω)(2.2.1)

and by (PI), we have respectively

∥𝜄(𝑢)∥𝐿2

𝜈(Ω)≤𝑐2∥𝑢∥𝐻1

0(Ω)for all 𝑢∈𝐻1

0(Ω),

for some 𝑐2>0.

2.2.2 Sobolev spaces and embeddings in the case 𝑑=1

Here, we consider the case

𝑑=

1 and

Ω=(𝑎,𝑏)

with

𝑎,𝑏 ∈R

and 0

≤𝑎<𝑏≤

Due to Lemma A.15 and Proposition A.17 the Sobolev space

𝐻1(𝑎,𝑏)≔𝐻1((𝑎,𝑏))

is compact embedded into

𝐶([𝑎,𝑏])

, where

𝐶([𝑎,𝑏])

denotes the vector space

of continuous functions on

[𝑎,𝑏]

. Therefore, for elements

𝑓∈𝐻1(𝑎,𝑏)

, we will

always choose the continuous representative of

𝑓

. Moreover, by Lemma A.15,

𝐻1

0(𝑎,𝑏)≔𝐻1

0((𝑎,𝑏))

can be identiﬁed by

{𝑓∈𝐻1(𝑎,𝑏)

𝑓(𝑎)=𝑓(𝑏)=

}

. Since

𝐶([𝑎,𝑏]) ⊂ 𝐿2

𝜈([𝑎,𝑏])

for any ﬁnite Borel measure

𝜈

[𝑎,𝑏]

, the situation in the

2.2. Form approach for Kre˘

ın–Feller operators

one-dimensional case becomes much simpler.

Lemma 2.2. For every 𝑓∈𝐻1(𝑎, 𝑏), we have

∥𝑓∥∞≔sup

𝑥∈[𝑎,𝑏 ]|𝑓(𝑥)| ≤ (𝑏−𝑎)1/2+ (𝑏−𝑎)−1/2∥∇𝑓∥2

𝐿2

Λ((𝑎,𝑏)) +∥𝑓∥2

𝐿2

Λ((𝑎,𝑏)) 1/2

and for all 𝑓∈𝐻1

0(𝑎,𝑏)and for every interval 𝐼with 𝐼

˚=(𝑎,𝑏),

𝐼

𝑓2d𝜈≤𝜈(𝐼)(𝑏−𝑎)𝐼

˚(∇𝑓)2dΛ.

Proof.

The ﬁrst inequality was proved in [Kan+09, Lemma 1.4]. Further, for all

𝑓∈𝐻1

0(𝑎,𝑏), an application of the Cauchy-Schwarz inequality yields

𝐼

𝑓(𝑥)2d𝜈(𝑥)=𝐼(𝑓(𝑥) −𝑓(𝑎))2d𝜈(𝑥)

=𝐼(𝑎,𝑥)∇𝑓dΛ2d𝜈(𝑥)

≤𝐼(𝑥−𝑎)(𝑎,𝑥)(∇𝑓)2dΛd𝜈(𝑥)

≤ (𝑏−𝑎)𝜈(𝐼)(𝑎,𝑏)(∇𝑓)2dΛ.□

Corollary 2.3. Let 𝑓∈𝐻1(𝑎, 𝑏 )with (𝑎,𝑏)𝑓dΛ=0, then we have

||𝑓||𝐿2

𝜈(𝐼)≤5·𝜈(𝐼)Λ(𝐼)||∇𝑓||𝐻1

0(𝑎,𝑏)

with 𝐼

˚=(𝑎,𝑏).

Proof. By [Arz15, Lemma 2.3.1], for all 𝑔∈𝐻1(0,1)with (0,1)𝑔dΛ=0, we have

||𝑔||2

𝐿2

Λ((0,1)) ≤||∇𝑔||2

𝐿2

Λ((0,1))

and by Lemma 2.2,

sup

𝑦∈[0,1]|𝑔(𝑥)|2≤4||𝑔||2

𝐻1(0,1).

Let

ℎ∈ (

)

and

𝑏∈R

be such that

𝑇((0,1)) =𝐼

with

𝑇(𝑥)≔ℎ𝑥 +𝑏

𝑥∈ [

]

2.2. Form approach for Kre˘

ın–Feller operators

Hence, for 𝑓∈𝐻1(𝑎, 𝑏)with (𝑎,𝑏)𝑓dΛ=0, we deduce

𝐼

𝑓2d𝜈≤𝜈(𝐼)sup

𝑥∈𝐼|𝑓(𝑥)|2

=𝜈(𝐼)sup

𝑥∈[0,1]|𝑓◦𝑇(𝑥)|2

≤4𝜈(𝐼)||𝑓◦𝑇||2

𝐻1(0,1)

≤4𝜈(𝐼)||∇(𝑓◦𝑇)||2

𝐿2

Λ(0,1)1+1

4

=5𝜈(𝐼)Λ(𝐼)||∇(𝑓)||2

𝐿2

Λ(0,1).

□

The following lemma shows that

(𝜈𝑃 𝐼)

holds for any ﬁnite Borel measure

𝜈

[𝑎,𝑏]

and

𝜄𝜈, (𝑎,𝑏)(𝑢)

coincides (in the

𝐿2

𝜈([𝑎,𝑏])

sense) with the natural choice of

the continuous representative of 𝐻1(𝑎,𝑏).

Lemma 2.4. Let

𝜈

be a ﬁnite Borel measure on

[𝑎,𝑏]

. Then for all

𝑢∈𝐻1(𝑎,𝑏)

, we

have [𝑎,𝑏]

𝑢2

𝑐d𝜈≤(𝑏−𝑎)1/2+(𝑏−𝑎)−1/22

𝜈([𝑎,𝑏])||𝑢||2

𝐻1(𝑎,𝑏),

and

𝜄𝜈, (𝑎,𝑏)(𝑢)=𝑢𝑐𝜈-almost surely,

where 𝑢𝑐denotes the unique continuous representative of 𝑢in 𝐻1(𝑎,𝑏).

Proof. By Lemma 2.2, for all 𝑢∈𝐻1(𝑎,𝑏 ), we obtain

||𝑢𝑐||2

∞≤𝐶2||𝑢||2

𝐻1(𝑎,𝑏),

with 𝐶≔(𝑏−𝑎)1/2+(𝑏−𝑎)−1/2. This leads to

[𝑎,𝑏]

𝑢2

𝑐d𝜈≤𝜈([𝑎, 𝑏])||𝑢𝑐||2

∞≤𝐶2𝜈([𝑎,𝑏])||𝑢||2

𝐻1(𝑎,𝑏),

which proves the ﬁrst claim. Now let

𝑢∈𝐻1(𝑎,𝑏)

and

(𝑢𝑛)𝑛∈N∈𝐶∞

𝑏([𝑎,𝑏])

such

that

𝑢𝑛→𝑢

in the norm of

𝐻1(𝑎,𝑏)

. Clearly, we have

𝑢𝑐−𝑢𝑛∈𝐻1(𝑎,𝑏) ∩𝐶([𝑎, 𝑏])

which implies [𝑎,𝑏](𝑢𝑐−𝑢𝑛)2d𝜈≤𝐶||𝑢𝑐−𝑢𝑛||2

𝐻1(𝑎,𝑏).

2.2. Form approach for Kre˘

ın–Feller operators

This gives 𝑢𝑛→𝑢𝑐for 𝑛→ ∞ in 𝐿2

𝜈([𝑎,𝑏]), allowing us to conclude that

𝜄𝜈, (𝑎,𝑏)(𝑢)=𝑢𝑐.□

In order to determine

N0,𝜈

, we introduce the following subset of continuous func-

tions on [𝑎, 𝑏 ]:

𝐶𝜈([𝑎,𝑏]) ≔{𝑓∈𝐶([𝑎,𝑏]) :𝑓is aﬃne linear on the components of [𝑎,𝑏] \supp(𝜈)}.

Further, we deﬁne the orthogonal complement of

𝐶𝜈([𝑎,𝑏])

with respect to

𝐻1

0(𝑎,𝑏)

𝐶𝜈([𝑎, 𝑏 ]) ∩𝐻1

0(𝑎,𝑏)⊥≔𝑓∈𝐻1

0(𝑎,𝑏):∀𝑔∈𝐶𝜈([𝑎,𝑏])∩𝐻1

0(𝑎,𝑏):⟨𝑓 , 𝑔⟩𝐻1

0(𝑎,𝑏)=0.

Proposition 2.5. Let 𝜈be a Borel measure on (𝑎,𝑏). Then we have

N0,𝜈 =𝐶𝜈([𝑎, 𝑏]) ∩𝐻1

0(𝑎,𝑏)⊥,

or equivalently N⊥

0,𝜈 =𝐶𝜈([𝑎,𝑏])∩𝐻1

0(𝑎,𝑏).

Proof. Pick 𝑓∈𝐶𝜈([𝑎, 𝑏]) ∩𝐻1

0(𝑎,𝑏)⊥. Then we deﬁne for 𝑥∈supp(𝜈)∩ (𝑎,𝑏)

𝑔𝑥(𝑦)≔𝑓(𝑥)(𝑦−𝑎)

𝑥−𝑎

[𝑎,𝑥 ]+𝑓(𝑥)𝑏−𝑦

𝑏−𝑥

(𝑥,𝑏 ].

Hence, using 𝑓(𝑎)=𝑓(𝑏)=0, we obtain

0=⟨𝑓 ,𝑔𝑥⟩𝐻1

0(𝑎,𝑏)=[𝑎,𝑥 ]∇𝑓∇𝑔𝑥dΛ+(𝑥,𝑏]∇𝑓∇𝑔𝑥dΛ

=𝑓(𝑥)

𝑥−𝑎(𝑓(𝑥)− 𝑓(𝑎)) + −𝑓(𝑥)

𝑏−𝑥(𝑓(𝑏)−𝑓(𝑥)) =𝑓(𝑥)2

𝑥−𝑎+𝑓(𝑥)2

𝑏−𝑥,

and consequently, for all

𝑥∈supp(𝜈) ∩(𝑎,𝑏),

we have

𝑓(𝑥)=

0. In particular, we

have

∥𝑓∥𝐿2

𝜈((𝑎,𝑏)) =

0. On the other hand, for

𝑓∈𝑔∈𝐻1

0(𝑎,𝑏):∥𝑔∥𝐿2

𝜈((𝑎,𝑏)) =0

𝑓

vanishes

𝜈

-a.e. and, using the continuity of

𝑓

, we obtain

𝑓=

0 on

supp (𝜈)

. To

simplify notation, assume

𝑎,𝑏 ∈supp(𝜈)

. Now, with

𝑖∈𝐼(𝑎𝑖,𝑏𝑖)=[𝑎, 𝑏]\supp(𝜈)

one easily veriﬁes that

∇𝑓=𝑖∈𝐼

(𝑎𝑖,𝑏𝑖)∇𝑓∈𝐿2

Λ((𝑎,𝑏))

and we obtain that for all

2.2. Form approach for Kre˘

ın–Feller operators

𝑔∈𝐶𝜈([𝑎,𝑏])∩𝐻1

0(𝑎,𝑏),

⟨𝑓 ,𝑔⟩𝐻1

0(𝑎,𝑏)=[𝑎,𝑏 ]∇𝑓∇𝑔dΛ

=

𝑖∈𝐼(𝑎𝑖,𝑏𝑖)∇𝑓∇𝑔dΛ

=

𝑖∈𝐼

𝑔(𝑏𝑖)−𝑔(𝑎𝑖)

𝑏𝑖−𝑎𝑖(𝑓(𝑏𝑖)−𝑓(𝑎𝑖)) =0.

This gives 𝑓∈𝐶𝜈([𝑎,𝑏]) ∩𝐻1

0(𝑎,𝑏)⊥.□

2.2.3 Stein extension

We begin with the deﬁnition of a Stein extension.

Deﬁnition 2.6. We say a bounded domain

𝐴⊂R𝑑

permits a Stein extension if there

exists a continuous linear operator

𝔈Ω

𝐻1(𝐴)→𝐻1R𝑑

such that

𝔈Ω(𝑓)|𝐴=𝑓

and

𝔈𝐴:𝐶∞

𝑏𝐴→𝐶∞

𝑐R𝑑with 𝔈(𝑓)|𝐴=𝑓 .

Necessarily, we then have that

𝐶∞

𝑏𝐴

lies dense in

𝐻1(𝐴)

. The second property

above is not standard in the literature but follows from [Ste70, Sec. 3.2 and 3.3]

(see Appendix A.2 for a more detailed presentation). Note that every bounded

Lipschitz domain permits a Stein extension (see Theorem A.14 in Appendix A.2),

in particular

as a bounded convex open set, see e.g. [Gri85, Corollary 1.2.2.3] or

[Ste70, Example 2, p. 189] is a bounded Lipschitz domain, thus the Stein extension

𝔈Q

with the above properties is well deﬁned. Note that for any cube

𝑄∈ D

, by the

deﬁnition of the weak derivatives, we have 𝐻1(𝑄)=𝐻1(𝑄

˚).

Lemma 2.7. There exists a constant

𝐷Q>

0such that for all half-open cubes

𝑄⊂Q

with edges parallel to the coordinate axes and 𝑢∈𝐻1(𝑄),

𝐷Q∥𝑢∥2

𝐻1(𝑄)≤∥∇𝑢∥2

𝐿2

Λ(𝑄)+1

Λ(𝑄)𝑄

𝑢dΛ2

≤∥𝑢∥2

𝐻1(𝑄).

Proof. Clearly, by the Cauchy-Schwarz inequality, we have for all 𝑢∈𝐻1(𝑄)

Λ(𝑄)𝑄

𝑢dΛ2

≤∥𝑢∥2

𝐿2

Λ(𝑄),

proving the second inequality. From [NS01, Lemma 3, p. 500] we obtain that there

2.2. Form approach for Kre˘

ın–Feller operators

exists 𝐶Q>0 such that for all 𝑣∈𝐻1(Q)

𝐶QQ

𝑣2dΛ≤∥∇𝑣∥2

𝐿2

Λ(Q)+Q

𝑣dΛ2

Let

𝑇

R𝑑→R𝑑, 𝑥 ↦→ 𝑥0+ℎ𝑥,

with

ℎ∈(0,1)

𝑥0∈Q

, such that

𝑄=𝑇(Q)

. Note that

𝑢◦𝑇∈𝐻1(Q)

and

∥∇(𝑢◦𝑇)∥2

𝐿2

Λ(Q)=ℎ2−𝑑∥∇𝑢∥2

𝐿2

Λ(𝑄)

, for all

𝑢∈𝐻1(𝑄)

, leading to

𝐶Q

ℎ𝑑𝑄

𝑢2dΛ=𝐶QQ

𝑢2◦𝑇dΛ

≤∥∇(𝑢◦𝑇)∥2

𝐿2

Λ(Q)+Q

𝑢◦𝑇dΛ2

=ℎ2−𝑑∥∇𝑢∥2

𝐿2

Λ(𝑄)+ℎ−2𝑑𝑄

𝑢dΛ2

Hence, using ℎ<1, we obtain

𝐶Q𝑄

𝑢2dΛ+∥∇𝑢∥2

𝐿2

Λ(𝑄)≤ℎ2+𝐶Q∥∇𝑢∥2

𝐿2

Λ(𝑄)+ℎ−𝑑𝑄

𝑢dΛ2

≤ (1+𝐶Q)∥∇𝑢∥2

𝐿2

Λ(𝑄)+1

Λ(𝑄)𝑄

𝑢dΛ2.□

Lemma 2.8. Let

𝑑≥

2and

𝑇

R𝑑→R𝑑, 𝑥 ↦→ 𝑥0+ℎ𝑥,

with

ℎ∈(0,1)

𝑥0∈Q

, such

that the cube 𝑄≔𝑇(Q)belongs to D. Then we have:

1. 𝔈𝑄:𝐻1(𝑄)→𝐻1R𝑑, 𝑢 ↦→ 𝔈Q(𝑢◦𝑇) ◦𝑇−1deﬁnes a Stein extension with

𝔈𝑄≤𝔈Q/ℎ2,

2. 𝔈𝑄|𝑁Λ(𝑄)≤𝔈Q/𝐷Qwith 𝑁Λ(𝑄)≔𝑢∈𝐻1(𝑄):𝑄𝑢dΛ=0.

Proof.

We only prove the second claim. Fix

𝑇

𝑥↦→𝑥0+ℎ𝑥

such that

𝑇(Q)=𝑄

and assume

𝑄𝑢

Λ=

0. For a vector space

𝑉

we write

𝑉★≔𝑉\{0}

. Hence, we

2.2. Form approach for Kre˘

ın–Feller operators

obtain

𝔈𝑄|𝑁Λ(𝑄)=sup

𝑢∈𝑁Λ(𝑄)★𝔈𝑄(𝑢)𝐻1(R𝑑)

∥𝑢∥𝐻1(𝑄)

=sup

𝑢∈𝑁Λ(𝑄)★𝔈Q(𝑢◦𝑇)◦𝑇−1𝐻1(R𝑑)

∥𝑢∥𝐻1(𝑄)

≤sup

𝑢∈𝑁Λ(𝑄)★∇𝔈Q(𝑢◦𝑇)◦𝑇−12dΛ+𝔈Q(𝑢◦𝑇)◦𝑇−12dΛ1

𝑄∇(𝑢◦𝑇) ◦𝑇−12dΛ+1

Λ(𝑄)𝑄𝑢dΛ21

=sup

𝑢∈𝑁Λ(𝑄)★ℎ𝑑−2∇𝔈Q(𝑢◦𝑇)2dΛ+ℎ𝑑𝔈(𝑢◦𝑇)2dΛ1

ℎ𝑑−2Q(∇(𝑢◦𝑇))2dΛ1

≤sup

𝑢∈𝑁Λ(𝑄)★ℎ𝑑−2∇𝔈Q(𝑢◦𝑇)2dΛ+𝔈(𝑢◦𝑇)2dΛ1

ℎ𝑑−2Q(∇(𝑢◦𝑇))2dΛ1

≤sup

𝑢∈𝑁Λ(𝑄)★𝔈Q(𝑢◦𝑇)𝐻1(R𝑑)

𝐷Q∥𝑢◦𝑇∥𝐻1(Q)≤𝔈Q

𝐷Q

where in the last inequality we used the fact that

Q𝑢◦𝑇

Λ=

ℎ𝑑<ℎ𝑑−2

, and

Lemma 2.7. □

Lemma 2.9. Assuming that the following Poincar´e inequality

∥𝑢∥𝐿2

𝜈(R𝑑)≤𝑐1∥𝑢∥𝐻1(R𝑑)for all 𝑢∈𝐶∞

𝑐R𝑑

holds for some 𝑐1>0. Let 𝜄R𝑑:𝐻1R𝑑→𝐿2

𝜈R𝑑denote the embedding and

ℜΩ:𝐿2

𝜈R𝑑→𝐿2

𝜈Ω, 𝑓 ↦→ 𝑓|Ω

the restriction operator. Then we have 𝜄Ω= ℜΩ◦𝜄R𝑑◦𝔈Ω.

Proof.

First note that

𝜄R𝑑

restricted to

𝐶∞

𝑐R𝑑

is the identity. Now, using the fact

that

𝔈Ω

𝐶∞

𝑏Ω→𝐶∞

𝑐R𝑑

combined with the above observation we ﬁnd for all

𝑢∈𝐶∞

𝑏Ω,

ℜΩ𝜄R𝑑◦𝔈Ω(𝑢)= ℜΩ(𝔈Ω(𝑢)) =𝑢|Ω=𝜄Ω(𝑢).

Since 𝜄Ωis continuous and 𝐶∞

𝑏(Ω)lies dense in 𝐻1(Ω), the claim follows. □

2.2. Form approach for Kre˘

ın–Feller operators

2.2.4 Form approach

Since

𝜄≔𝜄Ω

maps

N⊥

𝜈

bijectively to

dom E𝑁≔dom E𝑁

Ω≔𝜄N⊥

𝜈

and

N⊥

0,𝜈

dom E𝐷≔dom E𝐷

Ω≔𝜄N⊥

0,𝜈 

, we may deﬁne the relevant corresponding

forms by the push forward

E𝑁(𝑢,𝑣 )≔E𝑁

Ω(𝑢,𝑣 )≔𝜄−1𝑢, 𝜄−1𝑣𝐻1(Ω),for 𝑢, 𝑣 ∈dom(E𝑁)

and

E𝐷(𝑢,𝑣 )≔E𝐷

Ω(𝑢,𝑣 )≔𝜄−1𝑢, 𝜄−1𝑣𝐻1

0(Ω),for 𝑢, 𝑣 ∈dom(E𝐷).

In the latter case (Dirichlet case), we always assume that

card (Ω∩supp(𝜈))=∞.

Lemma 2.10. C∞

𝑐(Ω)lies dense in 𝐿2

𝜈(Ω)and C∞

𝑏(Ω)lies dense in 𝐿2

𝜈(Ω).

Proof.

We start to show that indicator functions

𝐴

with

𝐴∈𝔅(Ω)

, where

𝔅(Ω)

denotes the Borel

𝜎

-algebra of

, can be approximated by functions of

C∞

𝑐(Ω)

Since

𝜈

is a ﬁnite Borel measure on

, by [Els11, 1.16 Satz von Ulam], for ﬁxed

𝜀>

0, there exist a compact set

𝐾

and an open set

𝑈⊂Ω

with

𝐾⊂𝐴⊂𝑈

such that

𝜈(𝑈\𝐾)<𝜀.

Now, using molliﬁers (see e.g. Section 2.2.6), we see that there exists

𝑓∈ C∞

𝑐(𝑈)

with 𝑓|𝐾=1 and 0 ≤𝑓≤1. Hence,

|

𝐴−𝑓|d𝜈≤𝜈(𝑈\𝐾)<𝜀.

Notice, that the simple functions lie dense in

𝐿2(Ω)

, proving the ﬁrst claim. To prove

the second claim, consider a bounded open set

𝑂

such that

Ω⊂𝑂

. Then repeating

the previous argument, we obtain that

C∞

𝑐(𝑂)

lies dense in

𝐿2

𝜈(𝑂)

. Furthermore,

for each

𝑔∈ C∞

𝑐(𝑂)

, we have

𝑔|Ω∈ C∞

𝑏(Ω)

. Using

supp(𝜈) ⊂ Ω

, we deduce that

C∞

𝑏(Ω)lies dense in 𝐿2

𝜈(Ω).□

Proposition 2.11. The set

dom E𝐷

lies dense in

𝐿2

𝜈(Ω)

and

dom E𝑁

lies dense

in 𝐿2

𝜈(Ω).

Proof.

Here we follow the arguments of [HLN06]. By Lemma 2.10, it follows that

C∞

𝑐(Ω)

lies dense in

𝐿2

𝜈(Ω)

. This carries over to the orthogonal projection onto

𝜄N⊥

0,𝜈 

since

𝜄N0,𝜈 

is the zero space in

𝐿2

𝜈(Ω)

. Similarly, for the Neumann case,

by Lemma 2.10, we obtain that

C∞

𝑏(Ω)

lies dense in

𝐿2

𝜈(Ω)

which carries over to

𝜄N⊥

𝜈.□

2.2. Form approach for Kre˘

ın–Feller operators

Proposition 2.12. Assuming

(𝜈𝑃 𝐼)

, we have that

dom E𝐷/𝑁

equipped with the

inner product

⟨𝑓 ,𝑔⟩𝜈+E𝐷/𝑁(𝑓 ,𝑔)

deﬁnes a Hilbert spaces, i.e.

E𝐷/𝑁

is a closed

form with respect to 𝐿2

𝜈(Ω)and 𝐿2

𝜈(Ω), respectively.

Proof.

Since both cases can be treated completely analogously, we only consider

the ﬁrst case: We ﬁrst observe that

N⊥

𝜈

is a closed linear subspace with respect to

⟨·,·⟩𝐻1(Ω)

, which by

(𝜈𝑃 𝐼)

induces a norm that is equivalent to the norm induced

⟨·,·⟩𝐻1(Ω)+⟨𝜄·,𝜄·⟩𝜈

. Therefore,

N⊥

𝜈,⟨·,·⟩𝐻1(Ω)+⟨𝜄·,𝜄·⟩𝜈

is a Hilbert space and

since E𝑁+⟨·,·⟩𝜈is the push-forward of ⟨·,·⟩𝐻1(Ω)+⟨𝜄·,𝜄·⟩𝜈, the claim follows. □

2.2.5 Deﬁnition of the Kre˘

ın–Feller operator

Recall that the Hu–Lau–Ngai condition (♠) is given by

dim∞(𝜈)=liminf

𝑛→∞

max𝑄∈D𝑁

𝑛log(𝜈(𝑄))

−𝑛log(2)>𝑑−2.

We note that for

𝑄∈ D

with

𝜈(𝑄)>

0 we have

dim∞(𝜈) ≤ dim∞𝜈|𝑄

and hence

the condition (

♠

) carries over to the restricted measure

𝜈|𝑄

. We also remark that our

deﬁnition of

dim∞(𝜈)

is consistent with the usual deﬁnition in terms of balls rather

than cubes from a uniform lattice (see e.g. [Str93]). Obviously, we always have

dim∞(𝜈)≤𝑑

, and the assumption

dim∞(𝜈)>

0 excludes the possibility of

𝜈

having

atoms in higher dimensional case.

Under the Hu–Lau–Ngai condition

(♠)

, from Proposition 2.11 and Proposition 2.12,

we deduce that

E𝐷/𝑁,dom E𝐷/𝑁

is a densely deﬁned closed form on

𝐿2

𝜈(Ω)

and

𝐿2

𝜈(Ω)

, respectively. Now we are in the position to deﬁne the Kre

ın–Feller operator

with respect to Dirichlet/Neumann boundary conditions:

For

E𝐷,dom E𝐷

, by Lemma A.22 and Theorem A.23, there exists a non-negative

self-adjoint operator Δ𝐷

𝜈on 𝐿2

𝜈(Ω)such that

𝑓∈dom Δ𝐷

𝜈⊂dom Δ𝐷

𝜈1/2=dom E𝐷

if and only if 𝑓∈dom E𝐷and there exists 𝑢∈𝐿2

𝜈(Ω)such that

E𝐷(𝑓 ,𝑔)=⟨𝑢,𝑔⟩𝐿2

𝜈(Ω), 𝑔 ∈dom E𝐷.

Further, for

E𝑁,dom E𝑁

, by Theorem A.23 and Lemma A.22, there exists a

non-negative self-adjoint operator Δ𝑁

𝜈on 𝐿2

𝜈(Ω)such that

𝑓∈dom Δ𝑁

𝜈⊂dom Δ𝑁

𝜈1/2=dom E𝑁

2.2. Form approach for Kre˘

ın–Feller operators

if and only if 𝑓∈dom E𝑁and there exists 𝑢∈𝐿2

𝜈(Ω)such that

E𝑁(𝑓 ,𝑔)=⟨𝑢,𝑔⟩𝐿2

𝜈(Ω), 𝑔 ∈dom E𝑁.

In this case, we have

𝑢=Δ𝐷/𝑁

𝜈𝑓

. Furthermore, we call

Δ𝐷/𝑁

𝜈

Kre˘ın–Feller operator

with respect to Dirichlet/Neumann boundary conditions and

𝑓∈dom Δ𝐷

𝜈★

(Dirichlet) eigenfunction with eigenvalue 𝜆∈Rif

E𝐷(𝑓 ,𝑔)=𝜆Ω

𝑓 𝑔 d𝜈

for all

𝑔∈dom E𝐷

. Further, we call

𝑓∈dom Δ𝑁

𝜈★

an (Neumann) eigenfunction

with eigenvalue 𝜆∈Rif

E𝑁(𝑓 ,𝑔)=𝜆Ω

𝑓 𝑔 d𝜈

for all

𝑔∈dom E𝑁

. Notice in the case

Ω=Q

, since

𝜈

is Borel measure on

, we

have

E𝑁(𝑓 ,𝑔)=𝜆Q

𝑓 𝑔 d𝜈.

In order to prove that the embeddings

dom E𝐷,E𝐷↩→𝐿2

𝜈(Ω)and dom E𝑁,E𝑁↩→𝐿2

𝜈(Ω)

are compact under the assumption

dim∞(𝜈)>𝑑−

2, we need the following result

due to Maz’ya [Maz85, Theorem 3, p. 386] and [Maz85, Theorem 4, p. 387].

Theorem 2.13. For

𝑑>

2and

𝑡>

2, the set

𝑢∈𝐶∞

𝑐(R𝑑):∥𝑢∥𝐻1(R𝑑)≤1

is pre-

compact in 𝐿𝑡

𝜈(R𝑑)if and only if

lim

𝑟↓0sup

𝑥∈R𝑑,𝜌 ∈(0,𝑟)

𝜌(1−𝑑/2)𝜈𝐵𝜌(𝑥)1/𝑡=0,

and for

𝑑=

𝑢∈𝐶∞

𝑐(R𝑑):∥𝑢∥𝐻1(R𝑑)≤1

is precompact in

𝐿𝑡

𝜈(R2)

if and only if

lim

𝑟↓0sup

𝑥∈R2,𝜌 ∈(0,𝑟 )|log(𝜌)|1/2𝜈𝐵𝜌(𝑥)1/𝑡=0,

where 𝐵𝜌(𝑥)denotes the open unit ball with radius 𝜌>0and center 𝑥in R𝑑.

Proposition 2.14. The assumption

dim∞(𝜈)>𝑑−

2implies

(𝜈𝑃 𝐼)

and the embed-

dings dom E𝐷,E𝐷↩→𝐿2

𝜈(Ω)and dom E𝑁,E𝑁↩→𝐿2

𝜈Ω

are compact.

2.2. Form approach for Kre˘

ın–Feller operators

Proof.

For the case

𝑑=

1, the claim follows from Proposition A.17. For

𝑑>

2 and

𝑡∈ (2,2 dim∞(𝜈)/(𝑑−2)), the assumption (𝜈 𝑃 𝐼 )implies

lim

𝑟↓0sup

𝑥∈R𝑑,𝜌 ∈(0,𝑟)

𝜌(1−𝑑/2)𝜈𝐵𝜌(𝑥)1/𝑡=0,

and for 𝑑=2,

lim

𝑟↓0sup

𝑥∈R2,𝜌 ∈(0,𝑟 )|log(𝜌)|1/2𝜈𝐵𝜌(𝑥)1/𝑡=0

(see [HLN06]). Hence, by [Maz85, Theorem 3 and Theorem 4, p. 583] we know

that

𝑢∈𝐶∞

𝑐R𝑑:∥𝑢∥𝐻1(R𝑑)≤1

is precompact in

𝐿𝑡

𝜈R𝑑

. Thus, there exists

𝑐>0 such that for all 𝑢∈𝐶∞

𝑐R𝑑, we have

∥𝑢∥𝐿2

𝜈(R𝑑)≤∥𝑢∥𝐿𝑡

𝜈(R𝑑)≤𝑐·∥𝑢∥𝐻1(R𝑑),

which implies that the identity map

𝐶∞

𝑐R𝑑→𝐿2

𝜈R𝑑

permits a unique continuous

continuation

𝜄R𝑑

𝐻1R𝑑→𝐿2

𝜈R𝑑

. To see that

𝜄R𝑑

is compact, ﬁx a bounded

sequence

(𝑢𝑛)𝑛

𝐻1R𝑑

. We ﬁnd another sequence

(𝑣𝑛)𝑛

𝐶∞

𝑐R𝑑

such that

∥𝑣𝑛−𝑢𝑛∥𝐻1(R𝑑)→

0 for

𝑛→ ∞

(see for instance [Ada75, Corollary 3.19 ]). The

aforementioned precompactness leads to a subsequence

(𝑣𝑛𝑘)𝑘

converging in

𝐿2

𝜈(R𝑑)

to an element 𝑣. Then 𝜄R𝑑(𝑢𝑛𝑘)𝑘also converges to 𝑣in 𝐿2

𝜈(R𝑑), since

𝑣−𝜄R𝑑(𝑢𝑛𝑘)𝐿2

𝜈(R𝑑)=𝑣−𝑣𝑛𝑘+𝑣𝑛𝑘−𝜄R𝑑(𝑢𝑛𝑘)𝐿2

𝜈(R𝑑)

≤𝑣−𝑣𝑛𝑘𝐿2

𝜈(R𝑑)+𝜄R𝑑(𝑣𝑛𝑘−𝑢𝑛𝑘)𝐿2

𝜈(R𝑑)

≤𝑣−𝑣𝑛𝑘𝐿2

𝜈(R𝑑)+𝑐·𝑣𝑛𝑘−𝑢𝑛𝑘𝐻1(R𝑑)→0,

for

𝑘

tending to inﬁnity. This shows that

𝜄R𝑑{𝑢∈𝐻1(R𝑑):∥𝑢∥𝐻1(R𝑑)≤𝑘}

precompact for every

𝑘>

0. Since for the Stein extension

𝔈Ω

for all

𝑢∈𝐻1(Ω)

have

∥𝔈Ω(𝑢)∥𝐻1(R𝑑)≤∥𝔈Ω∥ ∥𝑢∥𝐻1(Ω),

it follows that 𝜄R𝑑(𝔈Ω({𝑢∈𝐻1(Ω):𝑢∥𝐻1(Ω)≤1)) is precompact as a subset of

𝜄R𝑑𝑢∈𝐻1R𝑑:∥𝑢∥𝐻1(R𝑑)≤∥𝔈Ω∥.

Applying the restriction operator ℜΩfrom Lemma 2.9, we ﬁnd that

𝑢∈dom E𝑁:𝜄−1𝑢𝐻1(Ω)≤1⊂𝜄𝑢∈𝐻1(Ω):∥𝑢∥𝐻1(Ω)≤1

= ℜΩ𝜄R𝑑◦𝔈Ω𝑢∈𝐻1:∥𝑢∥𝐻1(Ω)≤1.

Therefore,

{𝑢∈dom E𝑁

𝜄−1𝑢𝐻1(Ω)≤

}

is relative compact in

𝐿2

𝜈(Ω)

as a

2.2. Form approach for Kre˘

ın–Feller operators

subset of a continuous image of a relatively compact set. In particular, there exists

𝑐1>0 such that for all 𝑢∈dom E𝑁

∥𝑢∥𝐿2

𝜈(Ω)≤𝑐1𝜄−1𝑢𝐻1(Ω).

The Dirichlet case follows by almost the same means without the use of the extension

operator (see also [HLN06]). □

As a direct consequence of Theorem A.26 and Proposition 2.14, we obtain the

following important corollary.

Corollary 2.15. Assume

dim∞(𝜈)>𝑑−

2. Then the operator

Δ𝐷/𝑁

has compact

resolvent and there exists a complete orthonormal basis of eigenvectors

𝑓𝐷/𝑁

𝑘𝑘∈N

with eigenvalues 𝜆𝐷/𝑁

𝑛,𝜈 𝑛∈Nwith 𝜆𝐷/𝑁

𝑛,𝜈 ≤𝜆𝐷/𝑁

𝑛+1,𝜈 tending to inﬁnity.

By Corollary 2.15 we can refer to the lower and upper spectral dimension with

respect to Dirichlet and Neumann boundary conditions deﬁned in (1.1.1).

Remark 2.16.Let

be a closed form with domain

dom (E)

densely deﬁned

H ∈ {𝐿2

𝜈(Ω),𝐿2

𝜈(Ω)}

, in particular

dom (E)

deﬁnes a Hilbert space with re-

spect to

(𝑓 ,𝑔)E≔⟨𝑓 ,𝑔⟩H+ E(𝑓 ,𝑔)

. Moreover, assume that the inclusion from

dom (E),⟨·,·⟩E

into

is compact. Then the Poincar´e–Courant–Fischer–Weyl

min-max principle is applicable, that is for the

𝑖

-th eigenvalue

𝜆𝑖(E)

𝑖∈N

, we

have (see also Theorem A.27 or [Dav95; KL93])

𝜆𝑖(E)=inf sup 𝑅(𝜓):𝜓∈𝐺★:𝐺<𝑖dom (E),⟨·,·⟩E,

where we write

𝐺<𝑖(𝐻, ⟨·,·⟩)

𝐺

is a linear subspace of the Hilbert space

𝐻

with

inner product

⟨·,·⟩

and the vector space dimension of

𝐺

is equal to

𝑖∈N

; for the

Rayleigh–Ritz quotient given by 𝑅(𝜓)≔E(𝜓,𝜓 )/⟨𝜓,𝜓 ⟩H.

The following proposition will be crucial for the proof of the upper bound of the

spectral dimension as stated in Corollary 5.6.

Proposition 2.17. We have for all 𝑖∈N,

𝜆𝐷

𝑖,𝜈 =inf sup 𝑅𝐻1

0(Ω)(𝜓):𝜓∈𝐺★:𝐺<𝑖N⊥

0,𝜈,⟨·,·⟩𝐻1

0(Ω)

=inf sup 𝑅𝐻1

0(Ω)(𝜓):𝜓∈𝐺★:𝐺<𝑖𝐻1

0(Ω),⟨·,·⟩𝐻1

0(Ω),

where the relevant Rayleigh–Ritz quotient is given by

𝑅𝐻1

0(Ω)(𝜓)≔⟨𝜓 ,𝜓 ⟩𝐻1

0(Ω)/⟨𝜄𝜓, 𝜄𝜓⟩𝜈.

The same result holds true for

E𝑁

with

N⊥

0,𝜈

replaced by

N⊥

𝜈

and

𝐻1

0(Ω)

𝐻1(Ω)

2.2. Form approach for Kre˘

ın–Feller operators

Proof. The ﬁrst equality follows by the min-max principle and the fact that

dom E𝐷≃ N⊥

0,𝜈 .

The part ‘

≥

’ for the second equality follows from the inclusion

N⊥

0,𝜈 ⊂𝐻1

0(Ω)

. For

the reverse inequality we consider an 𝑖-dimensional subspace

𝐺=span(𝑓1, . . . , 𝑓𝑖) ⊂ 𝐻1

0(Ω).

There exists a unique decomposition

𝑓𝑗=𝑓1,𝑗 +𝑓2,𝑗

with

𝑓1,𝑗 ∈ N⊥

0,𝜈

and

𝑓2,𝑗 ∈ N0,𝜈

𝑗=

, . . . ,𝑖

. Suppose that

𝑓𝑗,1𝑗=1,...,𝑖

are not linearly independent, then there exists

a non-zero element

𝑔∈𝐺∩N0,𝜈

. To see this, we ﬁx

(𝜆1, . . . ,𝜆𝑛)≠(

, . . . ,

)

with

𝜆1𝑓1,1+···+𝜆𝑖𝑓1,𝑖 =0. Then

𝜆1𝑓1,1+𝑓2,1+··· +𝜆𝑖𝑓1,𝑖 +𝑓2,𝑖 

ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ

∈𝐺★

=𝜆1𝑓2,1+···+𝜆𝑖𝑓2,𝑖

ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ

∈N0,𝜈

≕𝑔.

Using E𝐷(𝑔,𝑔)>0, we get in this case

sup 𝑅𝐻1

0(Ω)(𝜓):𝜓∈𝐺★=∞.

Otherwise, using the assumption

𝑓1,𝑗 ∈ N⊥

0,𝜈

and

𝑓2,𝑗 ∈ N0,𝜈

and particularly

𝜄(𝑓2,𝑗 )=

0, we have for every vector 𝑎𝑗𝑗∈R𝑖\{0}

𝑅𝐻1

0(Ω)

𝑗

𝑎𝑗𝑓1,𝑗 +

𝑗

𝑎𝑗𝑓2,𝑗 =𝑗𝑎𝑗𝑓1,𝑗 ,𝑗𝑎𝑗𝑓1, 𝑗 𝐻1

0(Ω)+𝑗𝑎𝑗𝑓2,𝑗 ,𝑗𝑎𝑗𝑓2,𝑗 𝐻1

0(Ω)

⟨𝜄𝑗𝑎𝑗𝑓1, 𝑗 ,𝜄 𝑗𝑎𝑗𝑓1, 𝑗 ⟩𝜈

≥𝑅𝐻1

0(Ω)

𝑗

𝑎𝑗𝑓1,𝑗 .

Note that

span(𝑓1,1, . . . , 𝑓1,𝑖) ⊂ N⊥

0,𝜈

is also an

𝑖

-dimensional subspace in

𝐻1

0(Ω)

Hence, in any case the reverse inequality follows. □

Lemma 2.18. There exists 𝐶>0such that we have for all 𝑖∈N

𝜆𝑁

𝑖,𝜈 ≤𝐶𝜆𝐷

𝑖,𝜈 .

Proof. Using (PI) and 𝑐>0 as deﬁned therein, we obtain for all 𝑢∈𝐻1

0(Ω)that

⟨𝑢,𝑢⟩𝐻1(Ω)≤ (𝑐+1)⟨𝑢,𝑢⟩𝐻1

0(Ω).

Since

𝐻1

0(Ω) ⊂ 𝐻1(Ω)

, the claim follows from Proposition 2.17 with

𝐶≔𝑐+

□

2.2. Form approach for Kre˘

ın–Feller operators

The leading idea to obtain lower bounds on

𝑁𝐷/𝑁

𝜈

is to construct appropriate ﬁnite

dimensional subspaces of

𝐻1

0(Ω)

and

𝐻1(Ω)

, respectively. This will be subject of

the following lemma.

Lemma 2.19. Let

{𝑓1, . . . , 𝑓𝑛}⊂𝐻1

0(Ω)★

such that

{𝑓1, . . . , 𝑓𝑛}

is orthogonal in

𝐻1

0(Ω)

and

{𝜄(𝑓1), . . . ,𝜄 (𝑓𝑛)}

is orthogonal in

𝐿2

𝜈(Ω)

. Further, assume there exists

𝐶>0such that for all 𝑖=1, . . . ,𝑛 , we have

⟨𝑓𝑖, 𝑓𝑖⟩𝐻1

0(Ω)

⟨𝜄(𝑓𝑖),𝜄 (𝑓𝑖)⟩𝜈≤𝐶 .

Then,

𝑁𝐷

𝜈(𝐶) ≥𝑛

. The same result holds if we replace

𝐻1

0(Ω)

and

𝐿2

𝜈(Ω)

𝐻1(Ω)

and 𝐿2

𝜈(Ω).

Proof. For every (𝑐1, . . . , 𝑐𝑛) ∈ R𝑛with 𝑛

𝑖=1𝑐𝑖𝑓𝑖∈𝐻1

0(Ω)★, we have

𝑛

𝑖=1𝑐𝑖𝑓𝑖,𝑛

𝑖=1𝑐𝑖𝑓𝑖𝐻1

0(Ω)

𝑛

𝑖=1𝑐𝑖𝜄(𝑓𝑖),𝑛

𝑖=1𝑐𝑖𝜄(𝑓𝑖)𝜈

=𝑛

𝑖=1𝑐2

𝑖⟨𝑓𝑖, 𝑓𝑖⟩𝐻1

0(Ω)

𝑛

𝑖=1𝑐2

𝑖⟨𝜄(𝑓𝑖),𝜄 (𝑓𝑖)⟩𝜈≤𝐶𝑛

𝑖=1𝑐2

𝑖⟨𝜄(𝑓𝑖),𝜄 (𝑓𝑖)⟩𝜈

𝑛

𝑖=1𝑐2

𝑖⟨𝜄(𝑓𝑖),𝜄 (𝑓𝑖)⟩𝜈

=𝐶

Thus, by Proposition 2.17, we have 𝜆𝐷/𝑁

𝑛,𝜈 ≤𝐶.□

2.2.6 Smoothing methods

To obtain lower estimates of the lower and upper spectral dimension, it is crucial to

construct appropriate ﬁnitely dimensional subspaces of

C∞

𝑐(R𝑑)

(see Proposition

5.9). In this section, we address this demand via molliﬁers.

In the following, we assume that each cube has edges parallel to the coordinate

axes. For

𝑠>

0 let

⟨𝑄⟩𝑠

denote the cube centered and parallel with respect to the

cube

𝑄⊂R𝑑

such that

Λ⟨𝑄⟩𝑠=Λ(𝑄)𝑠𝑑

𝑠>

0 (i.e.

⟨𝑄⟩𝑠=𝑇(𝑄)+(

−𝑠)𝑥0

where

𝑇(𝑥)=𝑠𝑥, 𝑥 ∈R𝑑and 𝑥0∈R𝑑is the center of 𝑄). Note that we have

⟨𝑄⟩1/𝑠𝑠=𝑠−1𝑄+ (1−𝑠−1)𝑥0𝑠=𝑄

and if 𝑄=𝑑

𝑖=1(𝑎𝑖,𝑏𝑖], then

⟨𝑄⟩𝑠=

𝑑



𝑖=1−𝑠𝑏𝑖−𝑎𝑖

2+𝑎𝑖+𝑏𝑖

2,𝑎𝑖+𝑏𝑖

2+𝑠𝑏𝑖−𝑎𝑖

2.

Lemma 2.20. For

𝑚>

1and

𝑟>

0, let

𝑄⊂R𝑑

be a cube with side length

𝑚𝑟

and

𝑄′≔⟨𝑄⟩1/𝑚

the centered and parallel sub-cube with side length

𝑟

. Then there

exists 𝜑𝑄,𝑚 ∈𝐶∞

𝑐(R𝑑)which satisﬁes the following properties:

2.2. Form approach for Kre˘

ın–Feller operators

1. 0≤𝜑𝑄,𝑚 (𝑥) ≤ 1for all 𝑥∈R𝑑,

2. supp(𝜑𝑄 ,𝑚) ⊂ 𝑄

˚,

3. 𝜑𝑄,𝑚 (𝑥)=1for all 𝑥∈𝑄′,

4. there exists a constant 𝐶>0(depending on 𝑑) such that

(𝜕/𝜕𝑥𝑖)𝜑𝑄 ,𝑚 (𝑥)≤𝐶(𝑚−1)−1𝑟−1

for all 𝑖=1, . . . ,𝑑 and 𝑥∈𝑄.

𝑄′

𝑄′′

𝑄′′ +𝐵1(0)

𝑄

Figure 2.2.1 Illustration of the construction of 𝜑𝑄,𝑚 for the case 𝑑=2, 𝑚=5/2 and 𝑟=4.

Proof.

Let

𝜓

𝑥↦→

𝐵1(0)(𝑥)𝑐2exp(

/(∥𝑥∥2−

))

be the normalized Friedrichs’

molliﬁer with

𝑐2≔

/𝐵1(0)exp(

/(∥𝑥∥2−

))

𝑥

and

𝜓𝜖(𝑥)≔𝜓(𝑥/𝜖)/𝜖𝑑

be the

molliﬁer with radius of molliﬁcation

𝜖≔(𝑚−1)𝑟/

6. For the centered and parallel

sub-cube

𝑄′′ ≔⟨𝑄⟩(𝑚+2)/(3𝑚)⊂𝑄

with side length

(𝑚+2)𝑟/

3 we deﬁne

𝜑𝑄,𝑚

the convolution

𝜑𝑄,𝑚 (𝑥)≔

𝑄′′ ★𝜓𝜖(𝑥)≔R𝑑

𝑄′′ (𝑦)𝜓𝜖(𝑥−𝑦)dΛ(𝑦), 𝑥 ∈R𝑑.

Since we assume that each cube has edges parallel to the coordinate axes, we can

write

𝑄=𝑑

𝑖=1𝐼𝑖

and

𝑄′′ =𝑑

𝑖=1𝐼′

𝑖

with

𝐼

˚𝑖=(𝑎𝑖,𝑏𝑖)

and

𝐼

˚′

𝑖=(𝑐𝑖,𝑑𝑖)

. Thus, we have

𝑄′′ +𝐵𝜀(0) ⊂

𝑑



𝑖=1[𝑐𝑖−𝜀,𝑑𝑖+𝜀].

Moreover, note that by the deﬁnition of

𝑄′′

(recall

𝑄′′

is centered with respect to

2.2. Form approach for Kre˘

ın–Feller operators

𝑄), we have (𝑎𝑖+𝑏𝑖)/2=(𝑐𝑖+𝑑𝑖)/2. Therefore, we ﬁnd that

[𝑐𝑖−𝜀,𝑑𝑖+𝜀]=𝑐𝑖+𝑑𝑖

2−𝑑𝑖−𝑐𝑖

2−(𝑚−1)𝑟

6,𝑐𝑖+𝑑𝑖

2+𝑑𝑖−𝑐𝑖

2+(𝑚−1)𝑟

6

=𝑎𝑖+𝑏𝑖

2−(𝑚+2)𝑟

6−(𝑚−1)𝑟

6,𝑎𝑖+𝑏𝑖

2+(𝑚+2)𝑟

6+(𝑚−1)𝑟

6

=𝑎𝑖+𝑏𝑖

2−(2𝑚+1)𝑟

6,𝑎𝑖+𝑏𝑖

2+(2𝑚+1)𝑟

6

⊂𝑎𝑖+𝑏𝑖

2−𝑟𝑚

2,𝑎𝑖+𝑏𝑖

2+𝑟𝑚

2

=𝑎𝑖+𝑏𝑖

2−𝑏𝑖−𝑎𝑖

2,𝑎𝑖+𝑏𝑖

2+𝑏𝑖−𝑎𝑖

2

=(𝑎𝑖,𝑏𝑖).

It follows that supp 𝜑𝑄,𝑚 ⊂𝑄′′ +𝐵𝜀(0) ⊂ 𝑄

˚. Since

𝑟

2+𝜀=𝑟𝑚+2

it follows analogously as above that for each 𝑥∈𝑄′, we have

𝑥−𝐵𝜀(0) ⊂ 𝑄′+𝐵𝜀(0) ⊂ 𝑄′′.

Hence, for each 𝑥∈𝑄′,

𝜑𝑄,𝑚 (𝑥)=R𝑑

𝑄′′ (𝑦)𝜓𝜖(𝑥−𝑦)dΛ(𝑦)

=𝑥−𝐵𝜀(0)

𝑄′′ (𝑦)𝜓𝜖(𝑥−𝑦)dΛ(𝑦)

=𝑥−𝐵𝜀(0)

𝜓𝜖(𝑥−𝑦)dΛ(𝑦)

=𝐵𝜀(0)

𝜓𝜖(𝑦)dΛ(𝑦)=1.

2.2. Form approach for Kre˘

ın–Feller operators

Further, for 𝑥∈𝑄′′ +𝐵𝜀(0), we have

(𝜕/𝜕𝑥𝑖)𝜑𝑄 ,𝑚 (𝑥)=𝑄′′

𝑄′′ (𝑦)(𝜕/𝜕𝑥𝑖)𝜓𝜖(𝑥−𝑦)dΛ(𝑦)

≤𝑄′′

𝑄′′ (𝑦)2|𝑥𝑖−𝑦𝑖|

𝜀2𝑥−𝑦

𝜀2−12𝜓𝜖(𝑥−𝑦)dΛ(𝑦)

≤1

𝜀𝑄′′∩𝐵𝜀(𝑥)

𝑄′′ (𝑦)2

𝑥−𝑦

𝜀2−12𝜓𝜖(𝑥−𝑦)dΛ(𝑦)

≤1

𝜀𝐵𝜀(0)+𝑥

𝑥−𝑦

𝜀2−12𝜓𝜖(𝑥−𝑦)dΛ(𝑦)

𝜀𝐵𝜀(0)

𝑦

𝜀2−12𝜓𝜖(𝑦)dΛ(𝑦)

𝜀𝐵1(0)

∥𝑦∥2−12𝜓(𝑦)dΛ(𝑦).

Observing 𝐵1(0)∥𝑦∥2−1−2

𝜓(𝑦)dΛ(𝑦)<∞, the claim follows. □

Lemma 2.21. Let

𝑄

be a cube with side length

𝑚𝑟 >

𝑚>

1. Then there exists a

constant

𝐶>

0depending on

𝑚>

1and

𝑑

such that for

𝜑𝑄,𝑚

as deﬁned in Lemma

2.20 we have

∇𝜑𝑄,𝑚2dΛ

𝜑𝑄,𝑚2d𝜈≤𝐶(𝑚−1)−2𝑚1−2/𝑑

Λ⟨𝑄⟩1/𝑚1−2/𝑑

𝜈⟨𝑄⟩1/𝑚.

Proof. Using Lemma 2.20, we ﬁnd

∇𝜑𝑄,𝑚2dΛ

𝜑2

𝑄,𝑚 d𝜈≤𝑑𝐶2Λ(𝑄)

(𝑚−1)2𝑟2𝜈⟨𝑄⟩1/𝑚≤𝑑𝐶2

(𝑚−1)2𝑚2

Λ(𝑄)1−2/𝑑

𝜈⟨𝑄⟩1/𝑚

=𝑑𝐶2(𝑚−1)−2𝑚4−2/𝑑

Λ⟨𝑄⟩1/𝑚1−2/𝑑

𝜈⟨𝑄⟩1/𝑚.□

The following proposition applies only in the case 𝑑>2.

Proposition 2.22. If dim∞(𝜈)<𝑑−2and 𝑑>2, then the identity operator

C∞

𝑏Q,⟨·,·⟩𝐻1→𝐿2

𝜈

2.2. Form approach for Kre˘

ın–Feller operators

is not continuous.

Proof. First note that

dim∞(𝜈)=liminf

𝑛→∞

max𝑄∈D𝑁

𝑛log(𝜈(𝑄))

−𝑛log(2)<𝑑−2

implies that there exists a sequence of cubes

(𝑄𝑛)∈ DN

with strictly decreasing

diameters such that

𝜈(𝑄𝑛)≥Λ(𝑄𝑛)𝑎/𝑑

𝑛∈N

, for some

𝑎∈(dim∞(𝜈),𝑑 −2)

. Now

we have for 𝑢𝑛≔Λ⟨𝑄𝑛⟩21/𝑑−1/2𝜑⟨𝑄𝑛⟩2,2with 𝐶>0 given in Lemma 2.20

∥𝑢𝑛∥2

𝐻1=Λ⟨𝑄𝑛⟩22/𝑑−1Q∇𝜑⟨𝑄𝑛⟩2,22dΛ+Q𝜑⟨𝑄𝑛⟩2,22dΛ

≤Λ⟨𝑄𝑛⟩22/𝑑−12𝐶Λ⟨𝑄𝑛⟩2

Λ(𝑄𝑛)2/𝑑+Q𝜑⟨𝑄𝑛⟩2,22dΛ

≤Λ⟨𝑄𝑛⟩22/𝑑−14𝐶Λ⟨𝑄𝑛⟩2−2/𝑑+1+Λ⟨𝑄𝑛⟩2≤4(𝐶+1).

On the other hand we have for 𝑛tending to inﬁnity

∥𝑢𝑛∥2

𝐿2

𝜈≥Λ⟨𝑄𝑛⟩22/𝑑−1𝜈(𝑄𝑛)=22−𝑑Λ(𝑄𝑛)2/𝑑−1𝜈(𝑄𝑛)

≥22−𝑑Λ(𝑄𝑛)(𝑎+2−𝑑)/𝑑→ ∞.

This proves the claim. □

2.3. Partition functions and 𝐿𝑞-spectra

2.3 Partition functions and 𝐿𝑞-spectra

In this chapter, we investigate the new notion of partition functions with respect to a

non-negative monotone function

𝔍

deﬁned on the set of dyadic cubes

. Further,

we assume

𝔍

is locally non-vanishing, that is, if

𝔍(𝑄)>

0 for

𝑄∈ D

, then there

exists

𝑄′⊊𝑄

𝑄′∈ D

with

𝔍(𝑄′)>

0. Note that this assumption is satisﬁed for each

speciﬁc choice for

𝔍

that we consider. Of particular interests is the case when

𝔍

chosen to be

𝔍(𝑄)≔𝜈(𝑄)

𝑄∈ D

, where

𝜈

is a ﬁnite Borel measure on

; in this

case we obtain the well-known

𝐿𝑞

-spectrum of

𝜈

. The

𝐿𝑞

-spectrum has been studied

by various authors, for example Ngai [Nga97], Ngai and Lau [LN98], Heurteaux

[Heu07], Hochman [Hoc14], Shmerkin [Shm19], and Ngai and Xie [NX19].

2.3.1 The partition function

We start with recalling the deﬁnition of the partition function from the introduction.

The Dirichlet/Neumann partition function of 𝔍is given by

𝜏𝐷/𝑁

𝔍(𝑞)=limsup

𝑛→∞

𝜏𝐷/𝑁

𝔍,𝑛 (𝑞), 𝜏𝐷/𝑁

𝔍,𝑛 (𝑞)=

log 𝐶∈D𝐷/𝑁

𝑛𝔍(𝐶)𝑞

log(2𝑛).

Here, we use the convention that 0

0, that is for

𝑞=

0 we neglect the summands

with 𝔍(𝑄)=0. Further, recall from the introduction

𝑞𝐷/𝑁

𝔍=inf 𝑞≥0 : 𝜏𝐷/𝑁

𝔍(𝑞)<0

and

𝜅𝔍=inf 





𝑞≥0 : 

𝑄∈D

𝔍(𝑄)𝑞<∞





.

Let us begin with some general observations for which we need the following

objects:

supp (𝔍)≔

𝑘∈N

𝑛≥𝑘𝑄:𝑄∈ D𝑁

𝑛,𝔍(𝑄)>0

and

dim∞(𝔍)≔liminf

𝑛→∞

max𝑄∈D𝑁

𝑛log (𝔍(𝑄))

−log(2𝑛).

We call

dim∞(𝔍)

the

∞

-dimension of

𝔍

which generalizes the

∞

-dimension for

𝜈

Obviously, the following holds.

Lemma 2.23. If dim∞(𝔍)>0, then 𝔍is uniformly vanishing, i.e. we have

lim

𝑛→∞ max

𝐶∈D𝑁

𝑛

𝔍(𝐶)=0.

2.3. Partition functions and 𝐿𝑞-spectra

Remark 2.24.Note that due to the monotonicity of 𝔍the assumption

lim

𝑛→∞ max

𝐶∈D𝑁

𝑛

𝔍(𝐶)=0

is equivalent to lim𝑛→∞ sup𝐶∈𝑘≥𝑛D𝑁

𝑘𝔍(𝐶)=0.

In the following lemma we use the convention −∞·0=0.

Lemma 2.25. For 𝑞≥0, we have

−dim∞(𝔍)𝑞≤𝜏𝑁

𝔍(𝑞) ≤ dim𝑀(supp (𝔍)) −dim∞(𝔍)𝑞. (2.3.1)

In particular,

𝑞𝑁

𝔍≤dim𝑀(supp (𝔍))/dim∞(𝔍).

Further, we have

dim∞(𝔍)>0⇐⇒ 𝑞𝑁

𝔍<∞.

and

𝑞𝑁

𝔍<∞=⇒𝜅𝔍=𝑞𝑁

𝔍.

Proof. The ﬁrst claim follows from the following simple inequalities

𝑞·max

𝑄∈D𝑁

𝑛

log 𝔍(𝑄)≤log 

𝐶∈D𝑁

𝑛

𝔍(𝐶)𝑞

≤log 

𝐶∈D𝑁

𝑛,𝔍(𝐶)>0

1+𝑞max

𝑄∈D𝑁

𝑛

log (𝔍(𝑄)) .

Now, assume

𝑞𝑁

𝔍<∞

. It follows there exists

𝑞>

0 such that

𝜏𝑁

𝔍(𝑞)<

0. Conse-

quently, from

(2.3.1)

we obtain

−dim∞(𝔍)𝑞≤𝜏𝑁

𝔍(𝑞)<

0, which yields

dim∞(𝔍)>

0. Reversely, suppose

dim∞(𝔍)>

0. In the case

dim∞(𝔍)=∞

, using

(2.3.1)

we have

𝑞𝑁

𝔍=

0 due to

𝜏𝑁

𝔍(𝑞)=−∞

for

𝑞>

0. It is left to consider the case

0<dim∞(𝔍)<∞. Then it follows from (2.3.1) that

𝜏𝑁

𝔍(𝑞)<0 for all 𝑞>dim𝑀(supp (𝔍))

dim∞(𝔍),

proving the implication. Now, assume

𝑞𝑁

𝔍<∞

. Thus, we have

𝜏𝑁

𝔍(𝑞)<

0 for all

𝑞>𝑞𝑁

𝔍

, and therefore, for every

𝜀>

0 with

𝜏𝑁

𝔍(𝑞)<−𝜀<

0 and

𝑛

large enough, we

obtain 

𝑄∈D𝑁

𝑛

𝔍(𝑄)𝑞≤2−𝑛𝜀,

implying 𝑄∈D 𝔍(𝑄)𝑞<∞. This shows inf 𝑞≥0 : 𝑄∈D 𝔍(𝑄)𝑞<∞≤𝑞𝑁

𝔍.

For the reversed inequality we note that if

𝑞𝑁

𝔍=

0, then the claimed equality is clear.

2.3. Partition functions and 𝐿𝑞-spectra

If, on the other hand,

𝑞𝑁

𝔍>

0, then we necessarily have

dim∞(𝔍)<∞

. Therefore,

𝜏𝑁

𝔍

is decreasing, convex (and therefore continuous), and proper (i.e.

𝜏𝑁

𝔍(𝑞)>−∞

for all 𝑞≥0). Hence, it follows that 𝑞𝑁

𝔍is a zero of 𝜏𝑁

𝔍and for all 0 <𝑞<𝑞𝑁

𝔍

0<𝜏𝑁

𝔍(𝑞).

This implies that for every 0 <𝛿<𝜏𝑁

𝔍(𝑞), there is a subsequence (𝑛𝑘)𝑘such that

2𝑛𝑘𝛿≤

𝑄∈D𝑁

𝑛𝑘

𝔍(𝑄)𝑞

and therefore,

∞=

𝑘∈N

𝑄∈D𝑁

𝑛𝑘

𝔍(𝑄)𝑞≤

𝑄∈D

𝔍(𝑄)𝑞

and consequently 𝑞𝑁

𝔍≤inf 𝑞≥0 : 𝑄∈D 𝔍(𝑄)𝑞<∞.□

Remark 2.26.Note that in the case

dim∞(𝔍) ≤

0, from Lemma 2.25 we deduce that

𝜏𝑁

𝔍(𝑞)

is non-negative for

𝑞≥

0, hence

𝑞𝑁

𝔍=∞

. However, it is possible that

𝜅𝔍<∞

Indeed, consider

𝔍(𝑄)≔









𝑛,if 𝑄=0,1

2𝑛𝑑

0,otherwise.

Then it follows 𝜅𝔍=1<𝑞𝑁

𝔍=∞and dim∞(𝔍)=0.

Lemma 2.27. If

dim∞(𝔍) ∈ (

,∞)

, then

𝜏𝑁

𝔍

is convex and strictly decreasing on

R≥0. In particular, if 𝑞𝑁

𝔍>0, then 𝑞𝑁

𝔍is the only zero of 𝜏𝑁

𝔍.

Proof. First, note that Lemma 2.25 implies 𝜏𝑁

𝔍(𝑞) ∈ Rfor all 𝑞≥0 and

lim

𝑞→∞𝜏𝑁

𝔍(𝑞)=−∞.

Since

dim∞(𝔍)>

0 it follows from Lemma 2.23 that for

𝑛

large we have

𝔍(𝑄)<

, 𝑄 ∈ D𝑁

𝑛

. Hence, it follows that

𝜏𝑁

𝔍

is decreasing and as pointwise limit superior

of convex functions again convex.

Now, we show that

𝜏𝑁

𝔍

is strictly decreasing. Assume there exist

𝑞1,𝑞2

with 0

<𝑞1<

𝑞2

such that

𝜏𝑁

𝔍(𝑞1)=𝜏𝑁

𝔍(𝑞2)

. Since

𝜏𝑁

𝔍

is decreasing, we obtain

𝜏𝑁

𝔍(𝑞1)=𝜏𝑁

𝔍(𝑞)

for all

𝑞∈ [𝑞1,𝑞2]

. Fix

𝑞′′ ∈ (𝑞1,𝑞2)

. Since

𝜏𝑁

𝔍

is convex, for all

𝑞′>𝑞′′

Theorem

A.5 implies

𝜏𝑁

𝔍(𝑞′′) −𝜏𝑁

𝔍(𝑞1)

𝑞′′ −𝑞1≤

𝜏𝑁

𝔍(𝑞′)−𝜏𝑁

𝔍(𝑞1)

𝑞′−𝑞1≤0,

2.3. Partition functions and 𝐿𝑞-spectra

which implies

𝜏𝑁

𝔍(𝑞)=𝜏𝑁

𝔍(𝑞1)

for all

𝑞>𝑞1

which contradicts

lim𝑞→∞𝜏𝑁

𝔍(𝑞)=−∞

For the second claim note that, since

𝜏𝑁

𝔍

is convex and ﬁnite on

R>0

, it follows that

𝜏𝑁

𝔍

is continuous on

R>0

. Hence, we obtain

𝜏𝑁

𝔍(𝑞𝑁

𝔍)=

0. Finally, the uniqueness

follows from the fact that 𝜏𝑁

𝔍is a ﬁnite strictly decreasing function. □

We now summarise the above and mention a few more basic characteristics.

Fact 2.28. We make the following elementary observations under the assumption

dim∞(𝔍)∈(0,∞):

1. lim𝑞→∞𝜏𝑁

𝔍(𝑞)/𝑞=−dim∞(𝔍).

2. 𝜏𝑁

𝔍(𝑞)>−∞ for all 𝑞≥0.

3. 𝜏𝑁

𝔍(0)=dim𝑀(supp (𝔍)) ≤𝑑.

4. If 𝜏𝑁

𝔍(1)≥0 and 𝑞𝑁

𝔍>1 hold, then

𝑞𝑁

𝔍≤

dim∞(𝔍)+𝜏𝑁

𝔍(1)

dim∞(𝔍).

5. If 𝑞𝑁

𝔍>1,then

dim𝑀(supp (𝔍))

dim𝑀(supp (𝔍))−𝜏𝑁

𝔍(1)≤𝑞𝑁

𝔍.

6. If 𝑞𝑁

𝔍<1, then

𝑞𝑁

𝔍≤dim𝑀(supp (𝔍))

dim𝑀(supp (𝔍))−𝜏𝑁

𝔍(1).

7. If supp (𝔍)⊂Q

˚, then we have 𝜏𝐷

𝔍(𝑞)=𝜏𝑁

𝔍(𝑞).

The partition function is scale invariant, i.e. for

𝑐>

0, we have

𝜏𝐷/𝑁

𝑐𝔍=𝜏𝐷/𝑁

𝔍

Proof. We only give a proof of the assertion in 3, namely

𝜏𝑁

𝔍(0)=dim𝑀(supp (𝔍)).

First, we observe that if

𝑄∈ D𝑁

𝑛, 𝑄 ∩supp(𝔍)≠∅

, then there exists

𝑄′∈ D𝑁

𝑛

with

𝑄′∩𝑄≠∅

and

𝔍(𝑄′)>

0. This can be seen as follows: For

𝑥∈𝑄∩supp(𝔍)

there

exists a subsequence

(𝑛𝑘)𝑘

such that

𝑥∈𝑄𝑛𝑘

𝑄𝑛𝑘∈ D𝑛𝑘

and

𝔍(𝑄𝑛𝑘)>

0. For

𝑘∈N

such that

𝑛𝑘≥𝑛

there exists exactly one

𝑄′∈ D𝑁

𝑛

with

𝑄𝑛𝑘⊂𝑄′

. Further, we

have

𝑥∈𝑄𝑛𝑘⊂𝑄′

implies

𝑄′∩𝑄≠∅

and since

𝔍

is monotone, we have

𝔍(𝑄′)>

2.3. Partition functions and 𝐿𝑞-spectra

Furthermore, for each

𝑄∈ D𝑁

𝑛

, we have

card 𝑄′′ ∈ D𝑁

𝑛:𝑄′′ ∩𝑄≠∅≤

𝑑

Combining these two observations, we obtain

card 𝑄∈ D𝑁

𝑛:𝑄∩supp(𝔍)≠∅

≤card 𝑄∈ D𝑁

𝑛:∃𝑄′∈ D𝑁

𝑛, 𝑄′∩𝑄≠∅,𝔍(𝑄′)>0

≤3𝑑card 𝑄∈ D𝑁

𝑛:𝔍(𝑄)>0,

implying 𝜏𝑁

𝔍(0)≥dim𝑀(supp (𝔍)).

For the reversed inequality, we ﬁrst show that for

𝑄∈ D𝑁

𝑛

with

𝔍(𝑄)>

0, we

have

𝑄∩supp(𝔍)≠∅

. Indeed, since

𝔍

is locally non-vanishing there exists a

subsequence

(𝑛𝑘)𝑘

with

𝑄𝑛𝑘∈ D𝑁

𝑛𝑘

𝔍𝑄𝑛𝑘>

0 and

𝑄𝑛𝑘⊂𝑄𝑛𝑘−1⊂𝑄

. Since

𝑄𝑛𝑘𝑘is a nested sequence of non-empty compact subsets of 𝑄, we have

∅≠

𝑘∈N

𝑄𝑛𝑘⊂supp(𝔍) ∩𝑄 .

Therefore, we complete the proof by observing

card 𝑄∈ D𝑁

𝑛:𝔍(𝑄)>0≤card 𝑄∈ D𝑁

𝑛:𝑄∩supp(𝔍)≠∅

≤3𝑑card 𝑄∈ D𝑁

𝑛:𝑄∩supp(𝔍)≠∅.□

2.3.2 The (Dirichlet/Neumann) 𝐿𝑞-spectrum

In this section, we collect some important facts about the Dirichlet/Neumann

𝐿𝑞

spectrum. The Dirichlet/Neumann 𝐿𝑞-spectrum of 𝜈is given by

𝛽𝐷/𝑁

𝜈(𝑞)≔𝜏𝐷/𝑁

𝜈(𝑞)=limsup

𝑛→∞

𝛽𝐷/𝑁

𝑛(𝑞)

with

𝛽𝐷/𝑁

𝑛(𝑞)≔𝛽𝐷/𝑁

𝜈,𝑛 (𝑞)≔1

log(2𝑛)log 

𝑄∈D𝐷/𝑁

𝑛

𝜈(𝑄)𝑞

, 𝑞 ∈R.

The Neumann

𝐿𝑞

-spectrum we also simply call the

𝐿𝑞

-spectrum of

𝜈

. In the Dirichlet

case, we will assume

𝜈(Q

˚)>

0, implying that there exists a sub-cube

𝑄∈ D

with

𝑄⊂Q

𝜈(𝑄)>

0 and hence

−∞<𝛽𝑁

𝜈|𝑄≤𝛽𝐷

. In the following, we list some standard

facts about the 𝐿𝑞-spectrum.

Fact 2.29. We make the following elementary observations:

2.3. Partition functions and 𝐿𝑞-spectra

1. 𝛽𝑁

𝜈(0)=dim𝑀(supp(𝜈)),

where

dim𝑀(𝐴)

denotes the upper Minkowski

dimension of 𝐴⊂R𝑑.

2. dim∞(𝜈) ≤𝑑.

3. 𝛽𝑁

𝜈(1)=0 and if 𝜈(Q

˚)>0 , then also 𝛽𝐷

𝜈(1)=0.

4. For the Dirichlet 𝐿𝑞-spectrum we have 𝛽𝐷

𝜈=𝛽𝐷

𝜈|Q

˚.

5. For all 𝑞≥0, we have −𝑞𝑑 ≤𝛽𝑁

𝜈(𝑞).

6. If supp(𝜈) ⊂ Q

˚, then we have 𝛽𝐷

𝜈=𝛽𝑁

𝜈.

𝜈

is absolutely continuous with density

ℎ∈𝐿𝑡

Λ(Q)

for some

𝑡>𝑑/

2, then

𝛽𝐷

𝜈(𝑞)=𝛽𝑁

𝜈(𝑞), for all 𝑞∈[0,𝑡 ].

The condition

dim∞(𝜈)>𝑑−

2 implies that the upper Minkowski dimen-

sion

dim𝑀(supp(𝜈))

and the Hausdorﬀdimension

dim𝐻(𝜈)

must also lie in

(𝑑−2,𝑑]

. This in particular rules out the possibility of atomic parts of

𝜈

𝑑≥2.

Fact 2.30. The function

𝛽𝑁

𝜈

will not alter when we take

𝛿

-adic cubes instead of

dyadic ones (see, e.g. [Rie95, Proposition 2 and Remarks, p. 466] or [Rie93,

Proposition 1.6] and note that the deﬁnition in [Rie93, Proposition 1.6] coincides

with our deﬁnition for 𝑞≥0). More precisely, for ﬁxed 𝛿>0, set

𝐺𝜈,𝛿 ≔𝐺𝛿≔𝑑



𝑖=1((𝑘𝑖−1)𝛿,𝑘𝑖𝛿]:𝑘𝑖∈Z, 𝜈 𝑑



𝑖=1((𝑘𝑖−1)𝛿,𝑘𝑖𝛿]>0

and let

(𝛿𝑛)𝑛

be an admissible sequence, i.e.

𝛿𝑛∈ (

𝛿𝑛→

0 and there exists a

constant 𝐶>0 such that 𝐶𝛿𝑛≤𝛿𝑛+1≤𝛿𝑛for all 𝑛∈N. Then for 𝑞≥0 we have

limsup

𝛿↓0

−log(𝛿)log 

𝐶∈𝐺𝛿

𝜈(𝐶)𝑞=limsup

𝑚→∞

−log(𝛿𝑚)log 

𝐶∈𝐺𝛿𝑚

𝜈(𝐶)𝑞.

In particular, for

𝛿𝑚≔

−𝑚

, the above expression coincides with the deﬁnition of

𝛽𝑁

𝜈(𝑞).

Fact 2.31. The function

𝛽𝐷/𝑁

𝜈

as a pointwise limit superior of convex functions is

again convex and we have (see also [Rie95, Corollary 11])

𝛽𝑁

𝜈(0)=dim𝑀(supp(𝜈))≤𝑑and 𝛽𝐷/𝑁

𝜈(1)=0.

The function 𝛽𝑁

𝜈is non-increasing and non-negative on R<1and

liminf

𝑛→∞ 𝛽𝑁

𝑛(𝑞) ≥ −𝑑𝑞

2.4. The spectral partition function

for all 𝑞≥0.

Fact 2.32. If

𝜈

has an atomic part, then

𝛽𝑁

𝜈(𝑞)=

0, for all

𝑞≥

1. If

𝜈|Q

has an

atomic part, then also 𝛽𝐷

𝜈(𝑞)=0 for all 𝑞≥1.

Proof.

We consider only the ﬁrst case. Assume that

𝜈

has an atom in

𝑥0∈Q

and

let

𝑞>

1. Then, for every

𝑛∈N

, we have 0

<𝜈({𝑥0})𝑞≤𝐶∈D𝑁

𝑛𝜈(𝐶)𝑞

, implying

0≤𝛽𝑁

𝜈(𝑞) ≤ 𝛽𝑁

𝜈(1)=0. □

2.4 The spectral partition function

In this section, we study the spectral partition function which will play an important

role in the study of the spectral dimension as well as for the quantization dimension.

2.4.1

The spectral partition function and connections to the

𝐿𝑞

-spectrum

This section is devoted to the special case

𝔍 = 𝔍𝜈,𝑎,𝑏

, where for

𝑏≥

𝑎∈R

and

𝑄∈ D,

𝔍𝜈,𝑎,𝑏 (𝑄)=









sup

𝑄∈D(𝑄)𝜈

𝑄𝑏log Λ

𝑄, 𝑎 =0,

sup

𝑄∈D(𝑄)𝜈

𝑄𝑏Λ

𝑄𝑎

, 𝑎 ≠0.

Recall that

𝜏𝐷/𝑁

𝔍𝜈=𝜏𝐷/𝑁

𝔍𝜈, (2/𝑑−1),1

. We call

𝜏𝐷/𝑁

𝔍𝜈,𝑎,𝑏

the (Dirichlet/Neumann) spectral

partition function of

𝜈

with parameters

𝑎

𝑏

. For the Dirichlet case we always

assume 𝜈(Q

˚)>0.

We now elaborate some connections between the

𝐿𝑞

-spectrum and the spectral

partition function.

Proposition 2.33. Fix 𝑎∈R,𝑎>0with 𝑏dim∞(𝜈)+𝑎𝑑 >0.

1. If 𝑎>0, then 𝛽𝐷/𝑁

𝜈(𝑏𝑞) −𝑎𝑑𝑞 =𝜏𝐷/𝑁

𝔍𝜈,𝑎,𝑏 (𝑞)for 𝑞≥0.

𝑎<

0, then

𝛽𝐷/𝑁

𝜈(𝑏𝑞) − 𝑎𝑑𝑞 ≤𝜏𝐷/𝑁

𝔍𝜈,𝑎,𝑏 (𝑞) ≤ 𝛽𝐷/𝑁

𝜈(𝑞(𝑏+𝑎𝑑/dim∞(𝜈)))

for

𝑞≥0, and in particular, 𝜏𝐷/𝑁

𝔍𝜈,𝑎,𝑏 (0)=𝛽𝐷/𝑁

𝜈(0).

Proof.

We only consider the case

𝑎<

0. We have for every

−𝑎𝑑/𝑏<𝑠<dim∞(𝜈)

and 𝑛large enough

𝜈(𝐶) ≤ 2−𝑠𝑛,

with 𝐶∈ D𝑁

𝑛. This leads to 𝑛≤ −log2(𝜈(𝐶))/𝑠. Hence, for 𝑞≥0, we obtain

𝜈(𝐶)𝑏𝑞Λ(𝐶)𝑞𝑎 =𝜈(𝐶)2−𝑎𝑑𝑞𝑛 ≤𝜈(𝐶)𝑏𝑞 2𝑎𝑑𝑞 log2(𝜈(𝐶))/𝑠=𝜈(𝐶)𝑞(𝑏+𝑎𝑑 /𝑠).

2.4. The spectral partition function

We get 𝜈(𝐶)𝑏𝑞 Λ(𝐶)𝑞𝑎 ≤𝔍𝜈,𝑎,𝑏 (𝐶)𝑞≤𝜈(𝐶)𝑞(𝑏+𝑎𝑑/𝑠)and

𝜏𝐷/𝑁

𝔍𝜈,𝑎,𝑏 (𝑞) ≤ 𝛽𝐷/𝑁

𝜈(𝑞(𝑏+𝑎𝑑/𝑠)).

Finally, the continuity of 𝛽𝐷/𝑁

𝜈gives

𝜏𝐷/𝑁

𝔍𝜈,𝑎,𝑏 (𝑞) ≤ 𝛽𝐷/𝑁

𝜈(𝑞(𝑏+𝑎𝑑/dim∞(𝜈))).□

Corollary 2.34. Let

𝑎≠

0. Assume

𝑏dim∞(𝜈)+𝑎𝑑 >

0and

𝛽𝑁

𝜈

is linear on

[

,∞)

Then, for all 𝑞≥0, we have

𝜏𝑁

𝔍𝜈,𝑎,𝑏 (𝑞)=𝛽𝑁

𝜈(𝑏𝑞) −𝑎𝑑𝑞 =dim𝑀(𝜈)−𝑞(𝑏dim𝑀(𝜈)+𝑎𝑑 ).

Proposition 2.35. Assume dim∞(𝜈)>0. Then for all 𝑏>0and 𝑞≥0, we have

𝛽𝐷/𝑁

𝜈(𝑏𝑞)=𝜏𝐷/𝑁

𝔍𝜈,0,𝑏 (𝑞).

Furthermore, if 𝛽𝐷/𝑁

𝜈(𝑏𝑞)exists as a limit, then

𝛽𝐷/𝑁

𝜈(𝑏𝑞)=lim inf

𝑛→∞ 𝜏𝐷/𝑁

𝔍𝜈,0,𝑏,𝑛 (𝑞).

Proof. Let 𝑞>0. For dim∞(𝜈)>𝜀>0,we have

𝜈(𝐶) ≤ 2−𝜀𝑛

for

𝑛

large enough and all

𝐶∈ D𝐷/𝑁

𝑛

. Hence, for every 0

<𝛿<𝑏

and

𝑛

large, we

obtain

|log (Λ(𝐶))|𝑞=(𝑑log(2)𝑛)𝑞≤2𝑛𝑞𝛿𝜀 ≤𝜈(𝐶)−𝑞𝛿 .

Recall that we neglect the summands with 𝜈(𝐶)=0. This leads to

log(2)𝑑

𝐶∈D𝐷/𝑁

𝑛

𝜈(𝐶)𝑏𝑞 ≤

𝐶∈D𝐷/𝑁

𝑛

𝔍𝜈,0,𝑏 (𝐶)𝑞

≤

𝐶∈D𝐷/𝑁

𝑛

sup

𝑄∈D(𝐶)

𝜈(𝑄)−𝑞𝛿𝜈(𝑄)𝑏𝑞 .

=

𝐶∈D𝐷/𝑁

𝑛

𝜈(𝐶)𝑞(𝑏−𝛿).

Hence,

𝛽𝐷/𝑁

𝜈(𝑞𝑏) ≤ 𝜏𝐷/𝑁

𝔍𝜈,0,𝑏 (𝑞) ≤ 𝛽𝐷/𝑁

𝜈(𝑞(𝑏−𝛿))

and for

𝛿↘

0, the continuity of

𝛽𝐷/𝑁

𝜈

gives

𝛽𝐷/𝑁

𝜈(𝑞𝑏)=𝜏𝐷/𝑁

𝔍𝜈,0,𝑏 (𝑞)

. Under the

2.4. The spectral partition function

assumption that 𝛽𝐷/𝑁

𝜈exists as a limit, we infer

𝛽𝐷/𝑁

𝜈(𝑏𝑞)≤lim inf

𝑛→∞ 𝜏𝐷/𝑁

𝔍𝜈,0,𝑏,𝑛 (𝑞)

≤liminf

𝑛→∞ 𝛽𝐷/𝑁

𝑛(𝑞(𝑏−𝛿)) ≤ 𝛽𝐷/𝑁

𝜈(𝑞(𝑏−𝛿)).

Now, for 𝛿↘0, the continuity of 𝛽𝐷/𝑁

𝜈proves the claim. □

Corollary 2.36. If

𝑑=

2and

dim∞(𝜈)>

0, then

𝜏𝑁

𝔍𝜈(1)=𝛽𝑁

𝜈(1)=

0, or equivalently,

𝑞𝑁

𝔍𝜈=

1. If additionally

𝜈(Q

˚)>

0, then

𝜏𝐷

𝔍𝜈(1)=𝛽𝐷

𝜈(1)=

0, or equivalently,

𝑞𝐷

𝔍𝜈=

By virtue of Proposition 2.33 and Proposition 2.35, we arrive at the following list of

facts.

Fact 2.37. Assuming

𝑏dim∞(𝜈)+𝑎𝑑 >0

, then the following list of properties of

the spectral partition function applies:

1. supp 𝔍𝜈,𝑎,𝑏 =supp (𝜈).

2. dim∞(𝔍𝜈,𝑎,𝑏)=𝑏dim∞(𝜈)+𝑎𝑑 >0.

We have that

𝑞𝑁

𝔍𝜈,𝑎,𝑏

is the unique zero of

𝜏𝑁

𝔍𝜈,𝑎,𝑏

and if

𝑎<

0, then by Proposi-

tion 2.33, we have

𝑞𝑁

𝔍𝜈,𝑎,𝑏 ≤dim∞(𝜈)

𝑏dim∞(𝜈)+𝑎𝑑 .

If 𝑎>0, then

𝑞𝑁

𝔍𝜈,𝑎,𝑏 ≤dim𝑀(𝜈)

𝑏dim𝑀(𝜈)+𝑎𝑑 .

4. We have dim∞(𝜈)≤dim𝑀(supp (𝜈)).

5. If 𝑑>1, then

𝑑

2≤dim𝑀(supp (𝜈))

dim𝑀(supp (𝜈))−𝑑+2≤𝑞𝑁

𝜈Λ(2/𝑑−1)≤𝑞𝑁

𝔍𝜈.(5)

If additionally dim∞(𝜈)=dim𝑀(supp (𝜈)), then

𝑞𝑁

𝜈Λ(2/𝑑−1)=𝑞𝑁

𝔍𝜈=dim𝑀(supp (𝜈))

dim𝑀(supp (𝜈))−𝑑+2.

𝜈

is absolutely continuous with density

ℎ∈𝐿𝑟

Λ(Q)

for some

𝑟>𝑑/

2, then

𝜏𝐷

𝔍𝜈(𝑞)=𝜏𝑁

𝔍𝜈(𝑞)=𝛽𝑁

𝜈(𝑞)+(𝑑−2)𝑞, for all 𝑞∈[0,𝑟 ].

7. For the Dirichlet spectral partition function we have 𝜏𝐷

𝔍𝜈,𝑎,𝑏 =𝜏𝐷

𝔍𝜈|Q

˚,𝑎,𝑏 .

2.4. The spectral partition function

For

𝑐>

0, we have

𝜏𝐷/𝑁

𝔍𝑐𝜈,𝑎,𝑏 =𝜏𝐷/𝑁

𝔍𝜈,𝑎,𝑏

and without loss of generality we can

assume that 𝜈is a probability measure.

Proof.

We only need to prove assertation (5). The convexity of

𝛽𝑁

𝜈

implies for all

𝑞>1

𝑑(1−𝑞)+(𝑑−2)𝑞≤dim𝑀(𝜈)(1−𝑞)+(𝑑−2)𝑞≤𝛽𝑁

𝜈(𝑞)+(2−𝑑)𝑞≤𝜏𝑁

𝔍𝜈(𝑞),

implying the claim. □

2.4.2

Relations between the Dirichlet and Neumann spectral partition

functions

In this section, we investigate under which conditions we can guarantee that for

given 𝑞≥0, we have 𝜏𝐷

𝔍𝜈(𝑞)=𝜏𝑁

𝔍𝜈(𝑞). As auxiliary quantities we need

dim𝑁\𝐷

∞(𝜈)≔liminf

𝑛→∞

log max𝑄∈D𝑁

𝑛\D𝐷

𝑛𝜈(𝑄)

−log (2𝑛)

and

dim𝐷

∞(𝜈)≔liminf

𝑛→∞

log max𝑄∈D𝐷

𝑛𝜈(𝑄)

−log (2𝑛).

In the following, we assume dim∞(𝜈)>𝑑−2.

Lemma 2.38. For any 𝑞≥0such that

dim𝑀(supp (𝜈)∩𝜕Q)−𝑞dim𝑁\𝐷

∞(𝜈)−𝑑+2<𝜏𝑁

𝔍𝜈(𝑞),

we have

𝜏𝐷

𝔍𝜈(𝑞)=𝜏𝑁

𝔍𝜈(𝑞).

In particular, since dim∞(𝜈)≤dim𝑁\𝐷

∞(𝜈), if

dim𝑀(supp (𝜈)∩𝜕Q)−𝑞(dim∞(𝜈)−𝑑+2)<𝜏𝑁

𝔍𝜈(𝑞),

then 𝜏𝐷

𝔍𝜈(𝑞)=𝜏𝑁

𝔍𝜈(𝑞).

Remark 2.39.Using dim𝑀(𝜈)/(dim𝑀(𝜈)−𝑑+2) ≤𝑞𝑁

𝔍𝜈, we ﬁnd that

dim𝑀(supp (𝜈)∩𝜕Q)<dim𝑀(𝜈)dim∞(𝜈)−𝑑+2

dim𝑀(𝜈)−𝑑+2

implying 𝜏𝑁

𝔍𝜈𝑞𝑁

𝔍𝜈=𝜏𝐷

𝔍𝜈𝑞𝑁

𝔍𝜈=0.

2.4. The spectral partition function

Proof. First, we consider the case 𝑑>2. Notice that



𝑄∈D𝐷

𝑛

𝔍𝜈(𝑄)𝑞≤

𝑄∈D𝑁

𝑛

𝔍𝜈(𝑄)𝑞=

𝑄∈D𝐷

𝑛

𝔍𝜈(𝑄)𝑞+

𝑄∈D𝑁

𝑛\D𝐷

𝑛

𝔍𝜈(𝑄)𝑞.

Set 𝜏𝑁\𝐷(𝑞)≔limsup𝑛→∞ 1/log(2𝑛)log(𝑄∈D𝑁

𝑛\D𝐷

𝑛𝔍𝜈(𝑄)𝑞). Then for 𝑞≥0

𝜏𝐷

𝔍𝜈(𝑞)≤𝜏𝑁

𝔍𝜈(𝑞)=max 𝜏𝑁\𝐷(𝑞),𝜏𝐷

𝔍𝜈(𝑞).

Further, we always have

0<dim∞(𝜈)−𝑑+2≤𝐴≔lim inf

𝑛→∞

log max𝑄∈D𝑁

𝑛\D𝐷

𝑛𝔍𝜈(𝑄)

−𝑛log(2)

=lim

𝑞→∞

𝜏𝑁\𝐷(𝑞)

−𝑞.

By the deﬁnition of 𝔍𝜈we have

dim𝑁\𝐷

∞(𝜈)−𝑑+2≥𝐴

and

dim𝑁\𝐷

∞(𝜈)−𝑑−

≥dim∞(𝜈)−𝑑+

0. Fix

𝑑−

<𝑠<dim𝑁\𝐷

∞(𝜈)

, then we

obtain for all 𝑛large and 𝑄∈ D𝑁

𝑛\D𝐷

𝑛,

𝜈(𝑄)Λ(𝑄)2/𝑑−1≤2𝑛(𝑑−2−𝑠).

Therefore,

𝐴≥𝑠−𝑑+

2, which yields

𝐴=dim𝑁\𝐷

∞(𝜈)−𝑑+

2. By the deﬁnition of

𝜏𝑁\𝐷, we have

𝜏𝑁\𝐷(𝑞)≤dim𝑀(supp(𝜈)∩𝜕Q)−𝑞𝐴.

Hence, by our assumption dim𝑀(supp(𝜈)∩𝜕Q) −𝑞𝐴 <𝜏𝑁

𝔍𝜈(𝑞), we obtain

𝜏𝑁\𝐷(𝑞)<𝜏𝑁

𝔍𝜈(𝑞).

This gives

𝜏𝑁\𝐷(𝑞)<𝜏𝑁

𝔍𝜈(𝑞)=max 𝜏𝑁\𝐷(𝑞),𝜏𝐷

𝔍𝜈(𝑞)=𝜏𝐷

𝔍𝜈(𝑞).

For the case 𝑑≤2, notice that by Proposition 2.35, we have

𝜏𝐷/𝑁

𝔍𝜈(𝑞)=𝛽𝐷/𝑁

𝜈(𝑞)+(𝑑−2)𝑞

for 𝑞≥0. Hence, this case follows in a similar way. □

In the next section we will see that many examples, which have been investigated in

2.4. The spectral partition function

the literature (see [NS95; Tri97; NX21]), fulﬁll

𝜏𝑁

𝔍𝜈=𝜏𝐷

𝔍𝜈

. It is worth pointing out

that in the one-dimensional case, the situation becomes considerably simpler, which

follows from the fact that the boundary only contains

{

}

. The rest of this section

is devoted to the proof of 𝛽𝐷

𝜈=𝛽𝑁

𝜈for the case 𝑑=1 whenever 𝜈({0,1}) =0.

Proposition 2.40. For 𝑑=1and 𝜈({0,1}) =0,

dim𝑁\𝐷

∞(𝜈)≥dim∞(𝜈)=dim𝐷

∞(𝜈).

Proof.

First we consider the case

dim𝐷

∞(𝜈)>

0. Then, for

dim𝐷

∞(𝜈)>𝑠>

0 and

𝑛

large, we obtain

𝜈(𝑄) ≤ 2−𝑠𝑛, 𝑄 ∈ D𝐷

𝑛.

Hence, using 𝜈({0,1}) =0, it follows

𝜈((0,2−𝑛]) =∞



𝑘=0𝜈0,2−(𝑛+𝑘)−𝜈0,2−(𝑛+𝑘+1)

=∞



𝑘=0

𝜈2−(𝑛+𝑘+1),2−(𝑛+𝑘)

≤∞



𝑘=0

2−𝑠(𝑛+𝑘+1)≤2−𝑠𝑛 ∞



𝑘=0

2−𝑠𝑘

and

𝜈2𝑛−1

2𝑛,1=∞



𝑘=0

𝜈2𝑛+𝑘−1

2𝑛+𝑘,1−𝜈2𝑛+𝑘+1−1

2𝑛+𝑘+1,1

=∞



𝑘=0

𝜈2𝑛+𝑘−1

2𝑛+𝑘,2𝑛+𝑘+1−1

2𝑛+𝑘+1

≤2−𝑠𝑛 ∞



𝑘=0

2−𝑠𝑘 .

Hence, we obtain

dim𝑁\𝐷

∞(𝜈)≥dim𝐷

∞(𝜈).

Now, we observe

2.4. The spectral partition function

log max𝑄∈D𝑁

𝑛𝜈(𝑄)

−log(2𝑛)=

max𝑘∈{𝐷,𝑁 \𝐷}log max𝑄∈D𝑘

𝑛𝜈(𝑄)

−log(2𝑛)

=min

𝑘∈{𝐷,𝑁 \𝐷}

log max𝑄∈D𝑘

𝑛𝜈(𝑄)

−log(2𝑛)

=min 









log max𝑄∈D𝑁\𝐷

𝑛𝜈(𝑄)

−log(2𝑛),

log max𝑄∈D𝐷

𝑛𝜈(𝑄)

−log(2𝑛)







.

Thus, we obtain

dim∞(𝜈) ≥ min dim𝑁\𝐷

∞(𝜈),dim𝐷

∞(𝜈)=dim𝐷

∞(𝜈)≥dim∞(𝜈).

dim𝐷

∞(𝜈)=

0, then clearly

dim∞(𝜈)=

0. Thus, in any cases, we obtain

dim∞(𝜈)=

dim𝐷

∞(𝜈).□

Proposition 2.41. Let 𝑑=1and 𝜈({0,1}) =0. Then, for all 𝑞≥0, we have

𝛽𝐷

𝜈(𝑞)=𝛽𝑁

𝜈(𝑞).

Proof. If for some 𝑞>0 we have

−𝑞dim𝑁\𝐷

∞(𝜈)≤ −𝑞dim∞(𝜈)<𝛽𝑁

𝜈(𝑞),

then 𝛽𝐷

𝜈(𝑞)=𝛽𝑁

𝜈(𝑞)by Lemma 2.38. Hence, for all

𝑞<𝛼≔inf 𝑠>0 : 𝛽𝑁

𝜈(𝑠)>−𝑠dim∞(𝜈),

we have

𝛽𝐷

𝜈(𝑞)=𝛽𝑁

𝜈(𝑞).

Note that we always have

𝛽𝑁

𝜈(𝑞) ≥

0 for all

𝑞∈ [

]

, implying

𝛼>

1. If

𝛼=∞

then we are ﬁnished. Otherwise, the convexity of

𝛽𝑁

𝜈

and

𝛽𝑁

𝜈(𝑞) ≥ −𝑞dim∞(𝜈)

force that

𝛽𝑁

𝜈(𝑞)=−𝑞dim∞(𝜈)

for all

𝑞≥𝛼

. Indeed, if

𝛽𝑁

𝜈(𝑞)=−𝑞dim∞(𝜈)

, then

by Theorem A.5 and (2.3.1), for all 𝑞′≥𝑞, we have

−𝑞′dim∞(𝜈) ≤ 𝛽𝑁

𝜈(𝑞′)=𝛽𝑁

𝜈(𝑞)+(𝑞′−𝑞)𝛽𝑁

𝜈(𝑞′)−𝛽𝑁

𝜈(𝑞)

𝑞′−𝑞

≤𝛽𝑁

𝜈(𝑞)−(𝑞′−𝑞)dim∞(𝜈)=−𝑞′dim∞(𝜈).

2.4. The spectral partition function

Therefore, Proposition 2.40 yields

−𝑞dim𝐷

∞(𝜈)=−𝑞dim∞(𝜈) ≤ 𝛽𝐷

𝜈(𝑞) ≤ 𝛽𝑁

𝜈(𝑞)=−𝑞dim∞(𝜈).□

Corollary 2.42. Let

𝜈1, . . . ,𝜈𝑑

be non-zero Borel measures on

(

)

and deﬁne

𝜈≔𝜈1⊗...⊗𝜈𝑑

. For

𝑞≥

0, we assume that

𝛽𝐷/𝑁

𝜈𝑖(𝑞)

exists as limit for each

𝑖=1, . . ., 𝑑 −1. Then,

𝛽𝐷

𝜈(𝑞)=𝛽𝑁

𝜈(𝑞)=

𝑑



𝑖=1

𝛽𝐷/𝑁

𝜈𝑖(𝑞).

Proof.

The second equality follows from the simple fact that, for all

𝑞≥

0 and all

𝑛∈N, we have



𝑄∈D𝑁

𝑛

𝜈(𝑄)𝑞=

𝑄∈D𝑁

𝑛

𝑑



𝑖=1

𝜈𝑖(𝜋𝑖(𝑄))𝑞=

𝑑



𝑖=1

𝑄∈D𝑁

𝑛

𝜈𝑖(𝜋𝑖(𝑄))𝑞,

where 𝜋𝑖denotes the projection in the 𝑖-th component. Further, set

D1,𝐷

𝑛≔{(𝑘2−𝑛,(𝑘+1)2−𝑛]:𝑘=1,2𝑛−2},𝑛 ∈N.

Then for the 𝑑-folded product D𝐷

𝑛=D1,𝐷

𝑛×...×D1,𝐷

𝑛, we have



𝑄∈D𝐷

𝑛

𝜈(𝑄)𝑞=

(𝑄1,...,𝑄𝑑)∈(D𝐷

𝑛)𝑑

𝑑



𝑖=1

𝜈𝑖(𝑄𝑖)𝑞=

𝑑



𝑖=1

𝑄𝑖∈D𝐷 ,1

𝑛

𝜈𝑖(𝑄𝑖)𝑞.

Thus, the ﬁrst equality follows from Proposition 2.41 and our assumption that the

𝛽𝐷/𝑁

𝜈𝑖(𝑞)’s exist as limit for each 𝑖=1, . . .,𝑑 −1. □

2.4.3 Examples

In this section, assuming

dim∞(𝜈)>𝑑−

2, we show that for some particular cases

(absolutely continuous measures, product measures, Ahlfors–David regular mea-

sures, and self-conformal measures) the spectral partition function is completely

determined by the

𝐿𝑞

-spectrum. Furthermore, for these classes of measures we

investigate under which conditions the Dirichlet and the Neumann

𝐿𝑞

-spectra coin-

cide. Later, we will use these results to calculate the spectral dimension for these

classes of measures.

2.4. The spectral partition function

2.4.3.1 Absolutely continuous measures

Lemma 2.43. Let

𝜈

be a non-zero absolutely continuous measure with Lebesgue

density 𝑓∈𝐿𝑟

Λ(Q)for some 𝑟≥1. Then, for all 𝑞∈[0,𝑟 ], we have

liminf

𝑛→∞ 𝛽𝐷/𝑁

𝜈,𝑛 (𝑞)=𝛽𝐷/𝑁

𝜈(𝑞)=𝑑(1−𝑞).

Proof.

First, we remark that, since

𝜈(𝜕Q)=

0, there exists an open set

𝑂⊂Q

with

𝜈(𝑂)>

0. Moreover, we have

𝛽𝑁

𝜈(

0 and

𝛽𝑁

𝜈(

) ≤𝑑

. Hence, the convexity of

𝛽𝑁

𝜈implies

𝛽𝑁

𝜈(𝑞) ≤𝑑(1−𝑞)for all 𝑞∈ [0,1].

Furthermore, for

𝑛

large, we have

𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)(

0 and

𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)(

) ≤𝑑

. Conse-

quently, for all 𝑞∈ (1,∞), the convexity of 𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)and Theorem A.5 give

𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)(𝑞)−𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)(1)

𝑞−1≥

𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)(1) −𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)(0)

1−0≥ −𝑑.

Hence,

𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)(𝑞) ≥ 𝑑(1−𝑞).

This implies

𝑑(1−𝑞) ≤ lim inf

𝑛→∞ 𝛽𝐷

𝑛,𝜈 |𝑂/𝜈(𝑂)(𝑞)=lim inf

𝑛→∞ 𝛽𝐷

𝑛,𝜈 |𝑂(𝑞) ≤ lim inf

𝑛→∞ 𝛽𝐷/𝑁

𝑛,𝜈 (𝑞).

Moreover, by Jensen’s inequality, for all 𝑞∈ [0,1]and 𝑛large, we have



𝑄∈D𝐷/𝑁

𝑛

𝜈(𝑄)𝑞=

𝑄∈D𝐷

𝑛𝑄

𝑓dΛ/Λ(𝑄)𝑞

Λ(𝑄)𝑞

≥

𝑄∈D𝐷/𝑁

𝑛

Λ(𝑄)𝑞−1𝑄

𝑓𝑞dΛ

≥Λ(𝑄)𝑞−1𝑂

𝑓𝑞dΛ,

implying

liminf

𝑛→∞ 𝛽𝐷/𝑁

𝜈,𝑛 (𝑞) ≥ 𝑑(1−𝑞).

2.4. The spectral partition function

Further, Jensen’s inequality, for all 𝑞∈ [1,𝑟], yields



𝑄∈D𝐷/𝑁

𝑛

𝜈(𝑄)𝑞=

𝑄∈D𝐷/𝑁

𝑛𝑄

𝑓dΛ/Λ(𝑄)𝑞

Λ(𝑄)𝑞

≤Λ(𝑄)𝑞−1

𝑄∈D𝐷/𝑁

𝑛𝑄

𝑓𝑞dΛ

≤Λ(𝑄)𝑞−1Q

𝑓𝑞dΛ.

Hence, we obtain

limsup

𝑛→∞

𝛽𝐷/𝑁

𝜈,𝑛 (𝑞) ≤ 𝑑(1−𝑞).□

Proposition 2.44. Let

𝑑>

2and

𝜈

be a non-zero absolutely continuous measure

with Lebesgue density 𝑓∈𝐿𝑟

Λ(Q)for some 𝑟≥𝑑/2. Then, for all 𝑞∈[0,𝑟 ],

liminf

𝑛→∞ 𝜏𝐷/𝑁

𝔍𝜈,𝑛 (𝑞)=𝜏𝐷/𝑁

𝔍𝜈(𝑞)=𝛽𝐷/𝑁

𝜈(𝑞)−(2−𝑑)𝑞=𝑑−2𝑞.

Proof. By Jensen’s Inequality, for 𝑑/2≤𝑞≤𝑟and 𝑄∈ D𝐷/𝑁, we have

𝜈(𝑄)𝑞=𝑄

𝑓Λ(𝑄)−1dΛ𝑞

Λ(𝑄)𝑞≤𝑄

𝑓𝑞dΛΛ(𝑄)𝑞−1.

This shows that

𝜈(𝑄)𝑞Λ(𝑄)2𝑞/𝑑−𝑞≤𝑄𝑓𝑞dΛΛ(𝑄)2𝑞/𝑑−1

, and since we have

0≤2𝑞/𝑑−1, we observe that the right-hand side is monotonic in 𝑄. Therefore we

get the following upper bound





𝑄∈D𝐷/𝑁

𝑛sup

𝑄∈D𝑛(

𝑄)

𝜈(𝑄)𝑞Λ(𝑄)2𝑞/𝑑−𝑞≤



𝑄∈D𝐷/𝑁

𝑛

𝑄

𝑓𝑞dΛΛ

𝑄2𝑞/𝑑−1

≤2−𝑛(2𝑞−𝑑)∥𝑓∥𝑞

𝐿𝑞

Λ(Q).

Combining this with Lemma 2.43, we obtain

𝑑−2𝑞=liminf

𝑛→∞ 𝛽𝐷/𝑁

𝜈,𝑛 (𝑞)+(2−𝑑)𝑞

≤liminf

𝑛→∞ 𝜏𝐷/𝑁

𝔍𝜈,𝑛 (𝑞)

≤𝜏𝐷/𝑁

𝔍𝜈(𝑞)≤𝑑−2𝑞.

For the remaining case, we use the convexity of

𝜏𝐷/𝑁

𝔍𝜈

, the lower bound obtained

2.4. The spectral partition function

above, and the fact that

𝜏𝐷/𝑁

𝔍𝜈(

) ≤𝑑

and

𝜏𝐷/𝑁

𝔍𝜈(𝑟)=𝑑−

𝑟

, to obtain for all

𝑞∈ [

,𝑟 ]

𝑑−2𝑞≥𝜏𝐷/𝑁

𝔍𝜈(𝑞) ≥ lim inf

𝑛→∞ 𝜏𝐷/𝑁

𝔍𝜈,𝑛 (𝑞) ≥ lim inf

𝑛→∞ 𝛽𝐷/𝑁

𝜈,𝑛 (𝑞)+(2−𝑑)𝑞=𝑑−2𝑞. □

2.4.3.2 Product measures

The following example will be used to give an example for the non-existence of

the spectral dimension (see Section 5.4.4). Let

𝑑≥

3 and

𝜈𝑑

denotes a non-zero

Borel measure on

(0,1)

and let

Λ1

denote the one-dimensional Lebesgue measure

on (0,1). Here, we consider 𝜈≔Λ1...Λ1

ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ

(𝑑−1)-times

𝜈𝑑. Then, for 𝑄∈ D we have

𝔍𝜈(𝑄)=sup

𝑄′∈D(𝑄)

𝜈(𝑄′)Λ(𝑄′)(2−𝑑)/𝑑=2−𝑛𝜈𝑑(𝜋𝑑(𝑄)).

Hence, for all 𝑞≥0, we have

𝜏𝑁

𝑛,𝔍𝜈(𝑞)=2(𝑑−1)𝑛2−𝑛𝑞 

𝑄∈𝜋𝑑D𝑁

𝑛

𝜈𝑑(𝑄)𝑞

and

𝜏𝐷

𝑛,𝔍𝜈(𝑞)=(2𝑛−2)𝑑−12−𝑛𝑞 

𝑄∈𝜋𝑑D𝐷

𝑛

𝜈𝑑(𝑄)𝑞.

It follows from Proposition 2.41 that

𝜏𝑁

𝔍𝜈(𝑞)=𝑑−1−𝑞+𝛽𝑁

𝜈𝑑(𝑞)=𝑑−1−𝑞+𝛽𝐷

𝜈𝑑(𝑞)=𝜏𝐷

𝔍𝜈(𝑞).

2.4.3.3 Ahlfors–David regular measures

In this example we assume that

𝜈

is an

𝛼

-Ahlfors–David regular probability measure

with

𝛼∈(𝑑−2,𝑑]

, that is, there exists a constant

𝐾>

0 such that for every

𝑥∈supp (𝜈)and 𝑟∈(0,diam(supp(𝜈))]we have

𝐾−1𝑟𝛼≤𝜈(𝐵𝑟(𝑥)) ≤𝐾𝑟 𝛼,

where the diameter of a set 𝐴⊂R𝑑is deﬁned by

diam(𝐴)≔sup{|𝑥−𝑦|:𝑥,𝑦 ∈𝐴}.

Then for appropriate 𝐶>0 and every 𝑄∈ D with 𝜈(𝑄)>0 we have

𝐶−1Λ(𝑄)𝛼/𝑑≤𝜈𝑄

˚2and 𝜈(𝑄)≤𝐶Λ(𝑄)𝛼/𝑑.(2.4.1)

2.4. The spectral partition function

This implies

dim𝑀(𝜈)=lim

𝑛→∞

log card 𝑄∈ D𝑁

𝑛:𝜈(𝑄)>0

log(2𝑛)

=lim

𝑛→∞

log card 𝑄∈ D𝐷

𝑛:𝜈(𝑄)>0

log(2𝑛)=𝛼=dim∞(𝜈).

Indeed, since

𝜈(Q

˚)>

0 we ﬁnd an element

𝐸∈ D

with

𝐸⊂Q

and

𝜈(𝐸)=𝜀>

Then, on the one hand, for 𝑛large we have

card 𝑄′∈ D𝐷

𝑛:𝜈(𝑄′)>0𝐶2−𝑛𝛼 ≥

𝑄∈D𝐷

𝑛

𝜈(𝑄) ≥ 𝜀,

showing

liminf

𝑛→∞

log card 𝑄′∈ D𝐷

𝑛:𝜈(𝑄′)>0

log(2𝑛)≥𝛼 .

On the other hand,

card 𝑄′∈ D𝑁

𝑛:𝜈(𝑄′)>0𝐶−12−𝑛𝛼 3−𝑑≤3−𝑑

𝑄∈D𝑁

𝑛

𝜈𝑄

˚2

≤

𝑄∈D𝑁

𝑛

𝜈(𝑄)=1,

implying

limsup

𝑛→∞

log card 𝑄′∈ D𝑁

𝑛:𝜈(𝑄′)>0

log(2𝑛)≤𝛼 .

Now we prove that for all 𝑞≥0

𝜏𝐷/𝑁

𝔍𝜈(𝑞)=𝛽𝐷/𝑁

𝜈(𝑞)+(2−𝑑)𝑞=(𝛼+2−𝑑)𝑞−𝛼

exists as a limit. Indeed,



𝑄∈D𝑁

𝑛

𝔍𝜈(𝑄)𝑞≤𝐶

𝑄∈D𝑁

𝑛

2−𝑛(𝛼+2−𝑑)𝑞

≤𝐶card 𝑄∈ D𝑁

𝑛:𝜈(𝑄)>02−𝑛(𝛼+2−𝑑)𝑞.

2.4. The spectral partition function

Further, we have



𝑄∈D𝐷

𝑛

𝜈𝑄

˚2𝑞≤

𝑄∈D𝐷

𝑛

𝑄′∈D𝐷

𝑛+1,

𝑄∩𝑄′≠∅

𝜈(𝑄′)𝑞

≤

𝑄∈D𝐷

𝑛

4𝑑𝑞 max

𝑄′∈D𝐷

𝑛+1,

𝑄∩𝑄′≠∅

𝜈(𝑄′)𝑞

≤4𝑑𝑞+𝑑

𝑄∈D𝐷

𝑛+1

𝜈(𝑄)𝑞,

implying



𝑄∈D𝐷

𝑛+1

𝔍𝜈(𝑄)𝑞≥

𝑄∈D𝐷

𝑛+1

𝜈(𝑄)𝑞2(𝑑−2)𝑛𝑞

≥2(𝑑−2)𝑛𝑞4−𝑑𝑞−𝑑

𝑄∈D𝐷

𝑛,𝜈 (𝑄)>0

𝜈𝑄

˚2𝑞

≥𝐶−14−𝑑𝑞−𝑑card 𝑄∈ D𝐷

𝑛:𝜈(𝑄)>02−𝑛(𝛼+2−𝑑)𝑞,

proving the claim.

2.4.3.4 Self-conformal measures

We start with some basic deﬁnitions.

Deﬁnition 2.45. Let

𝑈⊂R𝑑

be an open set. We say a

𝐶1

-map

𝑆

𝑈→R𝑑

is confor-

mal if for every

𝑥∈𝑈

the matrix

𝑆′(𝑥)

, giving the total derivative of

𝑆

𝑥

, satisﬁes

|𝑆′(𝑥)·𝑦|=∥𝑆′(𝑥)∥|𝑦|≠0 for all 𝑦∈R𝑑\{0}with ∥𝑆′(𝑥)∥≔sup |𝑧|=1|𝑆(𝑥) ·𝑧|.

Deﬁnition 2.46. A family of conformal mappings

{𝑆𝑖:𝑋→𝑋}𝑖∈𝐼

on a compact

set

𝑋⊂Q

with

𝐼≔{

, . . . ,ℓ }

ℓ≥

2, is a

-conformal iterated function system

(C1-cIFS) if

Each

𝑆𝑖

extends to an injective conformal map

𝑆𝑖

𝑈→𝑈

on an open set

𝑋⊂𝑈,

2. We have uniform contraction, i.e. sup 𝑆′

𝑖(𝑥):𝑥∈𝑈<1, 𝑖∈𝐼.

3. The contractions (𝑆𝑖)𝑖∈𝐼do not share the same ﬁxed point.

For a conformal iterated function system

{𝑆𝑖:𝑋→𝑋}𝑖∈𝐼

there exists a unique

compact set K ⊂ 𝑋such that

K=

𝑖∈𝐼

𝑆𝑖(K).

2.4. The spectral partition function

Let

(𝑝𝑖)𝑖∈𝐼

be a positive probability vector and deﬁne

𝑝𝑢≔|𝑢|

𝑖=1𝑝𝑢𝑖

. Then there is

a unique Borel probability measure 𝜈with support Ksuch that

𝜈(𝐴)=

ℓ



𝑖=1

𝑝𝑖𝜈𝑆−1

𝑖(𝐴)(2.4.2)

for all 𝐴∈𝔅R𝑑(see [Hut81]). We refer to 𝜈as the self-conformal measure.

In following we provide some standard notations. We call

𝐼={1, . . ., ℓ }

alphabet and

𝐼𝑚

is the set of words of length

𝑚∈N

over

𝐼

and by

𝐼∗=𝑚∈N𝐼𝑚∪{∅}

we refer

to the set of all words with ﬁnite length including the empty word ∅. Furthermore,

the set of words with inﬁnite length will be denoted by

𝐼N

equipped with the metric

𝑑(𝑥, 𝑦)≔2−min {𝑖∈N:𝑥𝑖=𝑦𝑖},if 𝑥≠𝑦,

0,x=y.

The length of a ﬁnite word

𝜔∈𝐼∗

will be denoted by

|𝜔|

and for the concatenation

𝜔∈𝐼∗

with

𝑥∈𝐼∗∪𝐼N

we write

𝜔𝑥

. The shift-map

𝜎

𝐼N∪𝐼∗→𝐼N∪𝐼∗

is deﬁned

𝜎(𝜔)=∅

for

𝜔∈𝐼∪{∅}

𝜎(𝜔1. . . 𝜔𝑚)=𝜔2. . .𝜔𝑚

for

𝜔1. . . 𝜔𝑚∈𝐼𝑚

with

𝑚>

and

𝜎(𝜔1𝜔2, . . .)=(𝜔2𝜔3...)

for

(𝜔1𝜔2...)∈𝐼N

. The cylinder set generated by

𝜔∈𝐼∗

is deﬁned by

[𝜔]≔𝜔𝑥 :𝑥∈𝐼N⊂𝐼N

. Further, for

𝑢=𝑢1. . .𝑢𝑛∈𝐼𝑛

𝑛∈N

we set 𝑢−=𝑢1. . . 𝑢𝑛−1. We say 𝑃⊂𝐼∗is a partition of 𝐼Nif



𝜔∈𝑃[𝜔]=𝐼Nand [𝜔]∩[𝜔′]=∅,for all 𝜔 , 𝜔′∈𝑃with 𝜔≠𝜔′.

Now, we are able to give a coding of the self-conformal set in terms of

𝐼N

. For

𝜔∈𝐼∗

we put

𝑇𝜔≔𝑇𝜔1◦...◦𝑇𝜔𝑛

and deﬁne

𝑇∅≔id[0,1]

to be the identity map on

[

]

For

(𝜔1𝜔2...)∈𝐼N

and

𝑚∈N

we deﬁne the initial word by

𝜔|𝑛≔𝜔1. . . 𝜔𝑛

. For

every

𝜔∈𝐼N

the intersection

𝑛∈N𝑇𝜔|𝑛([

])

contains exactly one point

𝑥𝜔∈ K

and gives rise to a surjection

𝜋

𝐼N→K

𝜔↦→𝑥𝜔

, which we call the natural coding

map.

It is worth pointing out that we have the following remarkable bounded distortion

property in the case 𝑑≥2.

Proposition 2.47 ([MU03][Theorem 4.1.3]).Assume

𝑑≥

2. Then there exists

𝐷≥

such that for all 𝑛∈Nand 𝑢∈𝐼𝑛

𝐷−1≤𝑆′

𝑢(𝑥)

∥𝑆′

𝑢(𝑦)∥≤𝐷for all 𝑥,𝑦 ∈𝑈

with 𝑆𝑢=𝑆𝑢1◦...◦𝑆𝑢|𝑢|.

Proposition 2.48. Any self-conformal measure 𝜈is atomless.

2.4. The spectral partition function

Proof. Fix 𝑥∈𝑋such that 𝜈({𝑥}) =max {𝜈({𝑦}) :𝑦∈𝑋}≕𝑚. Using

𝜈({𝑥}) =

𝜔∈𝐼𝑛

𝑝𝜔𝜈𝑆−1

𝜔{𝑥}, 𝑛 ∈N,

we ﬁnd

𝜈𝑆−1

𝜔({𝑥})=𝑚

for every

𝜔∈𝐼𝑛

and

𝑛∈N

. Hence, if

𝑚>

0, we have

𝑥∈

𝑆𝜔(𝑋)

for all

𝜔∈𝐼𝑛

and

𝑛∈N

and

𝑥

is the common ﬁxed point of all contractions.

This contradicts our assumption that the contractions do not share the same ﬁxed

point. □

We need the following result from [PS00, Theorem 1.1] and [Fen07, Corollary 4.5]:

Theorem 2.49. For a self-conformal measure

𝜈

, the

𝐿𝑞

-spectrum

𝛽𝑁

𝜈

exists as a

limit on R>0.

Example 2.50. In this example, we additionally assume that the

𝑆𝑖

’s are contractive

similitudes 𝑆1, . . . ,𝑆 ℓwith corresponding contraction ratios ℎ𝑖, i.e.

|𝑆𝑖(𝑥)−𝑆𝑖(𝑦)| =ℎ𝑖|𝑥−𝑦|, 𝑥,𝑦 ∈R𝑑.

Furthermore, we assume the OSC, i.e. there exists a bounded open set

𝑂⊂R𝑑

such

that

∀𝑖≠𝑗:𝑆𝑖(𝑂)∩𝑆𝑗(𝑂)=∅and

ℓ



𝑖=1

𝑆𝑖(𝑂) ⊂𝑂 .

Then the 𝐿𝑞-spectrum of 𝜈is implicitly given by

ℓ



𝑖=1

𝑝𝑞

𝑖ℎ𝛽𝑁

𝜈(𝑞)

𝑖=1

(see for instance [Rie95]).

Proposition 2.51. Let

𝜈

denote a self-conformal measure on

such that

𝜈(𝜕Q)=

and dim∞(𝜈)>𝑑−2. Then for all 𝑞≥0,

𝛽𝑁

𝜈(𝑞)+(𝑑−2)𝑞=lim inf

𝑛→∞ 𝛽𝑁

𝜈,𝑛 (𝑞)+(𝑑−2)𝑞=𝜏𝑁

𝔍𝜈(𝑞)=liminf

𝑛→∞ 𝜏𝑁

𝔍𝜈,𝑛 (𝑞).

Proof.

We only have to check the case

𝑑>

2. Note that

𝑎≔

−𝑑>−dim∞(𝜈)

implies

sup

𝑄∈D

𝜈(𝑄)Λ(𝑄)𝑎/𝑑≕𝐾<∞.

Let

𝐼∗

denote the set of all ﬁnite words generated by the alphabet

𝐼

. For

𝑛∈N

, as in

[PS00], we let

𝑊𝑛≔{𝜔∈𝐼∗: diam(𝑆𝜔(K)) ≤ 2−𝑛<diam(𝑆𝜔−(K))},

2.4. The spectral partition function

which deﬁnes a partition of 𝐼N. Now ﬁx 𝑄∈ D𝑁

𝑛. For any 𝑄′⊂ D (𝑄)we set

𝐼𝑄′≔{𝑢∈𝑊𝑛:𝑆𝑢(K)∩𝑄′≠∅}.

𝑄′∈ D𝑁

𝑛+𝑚∩D (𝑄)

𝑚∈N

, and

𝑢∈𝐼𝑄′

, then we have

diam 𝑆−1

𝑢(𝑄′)≤𝐿

−𝑚

for some

𝐿>

0 (see also [PS00, Lemma 2.4]) and hence it is contained in at most 3

𝑑

cubes from

D𝑁

𝑚−𝑘

with

𝑘≔⌈log(𝐿)/log(2)⌉

(this gives

diam 𝑆−1

𝑢(𝑄′)≤

−𝑚+𝑘

Also, by deﬁnition of 𝐼𝑄′and 𝑊𝑛, we have



𝑢∈𝐼𝑄′

𝑆𝑢(K) ⊂ 

𝑄′′∈D𝑁

𝑛,𝑄′′∩𝑄′≠∅

𝑄′′ ⊂

𝑄′′∈D𝑁

𝑛,𝑄′′∩𝑄≠∅

𝑄′′ ≕𝑄′

Then we have

𝜈(𝑄′)Λ(𝑄′)𝑎/𝑑=2−𝑎(𝑛+𝑚)

𝑢∈𝑊𝑛

𝑝𝑢𝜈𝑆−1

𝑢(𝑄′)

=2−𝑎𝑛 

𝑢∈𝐼𝑄′

𝑝𝑢2−𝑎𝑚𝜈𝑆−1

𝑢(𝑄′)

≤2−𝑎𝑛 

𝑢∈𝐼𝑄′

𝑝𝑢2−𝑎𝑘 

𝐶∈D𝑁

𝑚−𝑘,

𝑆−1

𝑢(𝑄′)∩𝐶≠∅

2−𝑎(𝑚−𝑘)𝜈(𝐶)

≤2−𝑎𝑘 3𝑑max

𝐶∈D𝑁

𝑚−𝑘

𝜈(𝐶)Λ(𝐶)𝑎/𝑑2−𝑎𝑛 

𝑢∈𝐼𝑄′

𝑝𝑢

≤2−𝑎𝑘 3𝑑max

𝐶∈D𝑁

𝑚−𝑘

𝜈(𝐶)Λ(𝐶)𝑎/𝑑2−𝑎𝑛𝜈

𝑢∈𝐼𝑄′

𝑆𝑢(K)

≤2−𝑎𝑘 3𝑑𝐾𝜈 𝑄′

32−𝑎𝑛.

Since in the above inequality 𝑄′∈ D (𝑄)was arbitrary, we deduce for 𝑞>0,



𝑄∈D𝑁

𝑛

𝔍𝜈(𝑄)𝑞≤2(𝑑−2)𝑘𝑞 3𝑑𝑞𝐾𝑞2−𝑛𝑎𝑞 

𝑄∈D𝑁

𝑛

𝜈𝑄′

3𝑞

≤2(𝑑−2)𝑘𝑞 3𝑑𝑞𝐾𝑞2−𝑛𝑎𝑞 

𝑄∈D𝑁

𝑛

𝑄′∈D𝑁

𝑛,𝑄′∩𝑄≠∅

𝜈(𝑄′)

𝑞

≤2(𝑑−2)𝑘𝑞 3𝑑𝑞𝐾𝑞2−𝑛𝑎𝑞3𝑑𝑞 

𝑄∈D𝑁

𝑛

max

𝑄′∈D𝑁

𝑛,𝑄′∩𝑄≠∅

𝜈(𝑄′)𝑞

≤2(𝑑−2)𝑘𝑞 3𝑑𝑞𝐾𝑞2−𝑛𝑎𝑞3𝑑𝑞+𝑑

𝑄∈D𝑁

𝑛

𝜈(𝑄)𝑞.

2.4. The spectral partition function

This gives 𝛽𝑁

𝜈(𝑞)−𝑎𝑞 ≥𝜏𝑁

𝔍𝜈(𝑞). Furthermore, we observe that

𝛽𝑁

𝜈(0)=dim𝑀(𝜈)=𝜏𝑁

𝔍𝜈(0).

To complete the proof, notice that



𝑄∈D𝑁

𝑛

𝜈(𝑄)𝑞Λ(𝑄)𝑎≤

𝑄∈D𝑁

𝑛

𝔍𝜈(𝑄)𝑞.

Finally, Theorem 2.49 gives 𝛽𝑁

𝜈(𝑞)−𝑎𝑞 ≤liminf𝑛→∞ 𝜏𝑁

𝔍𝜈,𝑛 (𝑞)for 𝑞>0. □

Proposition 2.52. Let

𝜈

denote a self-conformal measure on

with

𝜈(𝜕Q)=

0and

dim∞(𝜈)>𝑑−2. Then

𝛽𝑁

𝜈(𝑞)=𝛽𝐷

𝜈(𝑞)=liminf

𝑛→∞ 𝛽𝐷

𝜈,𝑛 (𝑞)=lim inf

𝑛→∞ 𝛽𝑁

𝜈,𝑛 (𝑞)

for all 𝑞>0.

Proof.

We use the same notation as in the proof of Proposition 2.51. By our

assumption there exists

𝑛∈N

such that

𝑆𝑢(K)⊂Q

for some

𝑢∈𝑊𝑛

. Indeed

assume for all 𝑛∈Nand 𝑢∈𝑊𝑛, we have

𝑆𝑢(K)∩𝜕Q≠∅.

Further, using

sup𝑢∈𝑊𝑛diam (𝑆𝑢(K)) ≤

−𝑛→

0 for

𝑛→∞

and

K=𝑢∈𝑊𝑛𝑆𝑢(K)

we deduce that K ⊂ 𝜕Q. This gives 𝜈(𝜕Q)>0 contradicting our assumption.

Let us assume that the distance of

𝑆𝑢(K)

to the boundary of

is at least 2

−𝑛−𝑚0+2√𝑑

for some

𝑚0∈N

. Then all cubes

𝑄∈ D𝑁

𝑛+𝑚

intersecting

𝑆𝑢(K)

lie in

D𝐷

𝑛+𝑚

for all

𝑚>𝑚0

. Therefore, using the self-similarity and [PS00, Lemma 2.2 and Lemma

2.4] (with constant 𝐶1from there) we have for 𝑞>0



𝑄∈D𝐷

𝑛+𝑚

𝜈(𝑄)𝑞=

𝑄∈D𝐷

𝑛+𝑚

𝑣∈𝑊𝑛

𝑝𝑣𝜈𝑆−1

𝑣𝑄𝑞

≥𝑝𝑞

𝑢

𝑄∈D𝑁

𝑛+𝑚

𝜈𝑆−1

𝑢𝑄𝑞≥𝐶−1

1𝑝𝑞

𝑢

𝑄∈D𝑁

𝑚

𝜈(𝑄)𝑞.

This gives

𝛽𝑁

𝜈(𝑞)≤𝛽𝐷

𝜈(𝑞)

and

liminf𝑛→∞ 𝛽𝑁

𝑛(𝑞) ≤ lim inf𝑛→∞ 𝛽𝐷

𝑛(𝑞)

. The reverse

inequalities are obvious. Hence, the claim follows from Theorem 2.49. □

Corollary 2.53. Let

𝜈

denote a self-conformal measure on

with

𝜈(𝜕Q)=

0and

dim∞(𝜈)>𝑑−2. Then, for all 𝑞>0, we have

𝛽𝑁

𝜈(𝑞)+(𝑑−2)𝑞=𝜏𝑁

𝔍𝜈(𝑞)=liminf

𝑛→∞ 𝜏𝑁

𝔍𝜈,𝑛 (𝑞)=𝜏𝐷

𝔍𝜈(𝑞)=liminf

𝑛→∞ 𝜏𝐷

𝔍𝜈,𝑛 (𝑞).

2.4. The spectral partition function

Proof.

The cases

𝑑=

2 follow immediately from Proposition 2.35 and Proposition

2.52. For

𝑑>

2, we obtain from Proposition 2.51 and Proposition 2.52 the following

chain of inequalities

𝛽𝑁

𝜈(𝑞)+(𝑑−2)𝑞=lim inf

𝑛→∞ 𝛽𝐷

𝑛(𝑞)+(𝑑−2)𝑞

≤liminf

𝑛→∞ 𝜏𝐷

𝔍𝜈,𝑛 (𝑞) ≤ lim inf

𝑛→∞ 𝜏𝑁

𝔍𝜈,𝑛 (𝑞)

=𝜏𝑁

𝔍𝜈(𝑞)=𝛽𝑁

𝜈(𝑞)+(𝑑−2)𝑞. □

Chapter 3

Partition entropy and optimized

coarse multifractal dimension

Throughout this chapter let

𝔍

be a non-negative set function deﬁned on

which

is monotone, locally non-vanishing and uniformly vanishing (for the deﬁnitions

we refer to Section 2.3). This chapter is devoted to the study of the lower and

upper optimized coarse multifractal dimension with respect to

𝔍

and the lower and

upper

𝔍

-partition entropy. If

𝔍

is equal to the spectral partition function

𝔍𝜈,𝑎,𝑏

for

a certain choice of

𝑎,𝑏

, then these quantities will be important in estimating the

lower and upper spectral dimension of Kre

ın–Feller operators and the lower and

upper quantization dimension. First, we brieﬂy recall the basic deﬁnitions from the

introduction. The upper, resp. lower 𝔍-partition entropy is given by

ℎ𝔍=limsup

𝑥→∞

log M𝔍(𝑥)

log(𝑥), ℎ𝔍=lim inf

𝑥→∞

log M𝔍(𝑥)

log(𝑥),

with

M𝔍(𝑥)=inf card (𝑃):𝑃∈Π𝔍|max

𝐶∈𝑃𝔍(𝐶)<1/𝑥,1/𝔍(Q)<𝑥.

Here

Π𝔍

denotes the set of ﬁnite collections of dyadic cubes such that for all

𝑃∈Π𝔍

there exists a partition



𝑃

by dyadic cubes from

with

𝑃=𝑄∈

𝑃:𝔍(𝑄)>0

The lower and upper optimized (Dirichlet/Neumann) coarse multifractal dimension

with respect to 𝔍is given by

𝐹𝐷/𝑁

𝔍=sup

𝛼>0

𝐹𝐷/𝑁

𝔍(𝛼)

𝛼and 𝐹𝐷/𝑁

𝔍=sup

𝛼>0

𝐹𝐷/𝑁

𝔍(𝛼)

𝛼

3.1. Bounds for the partition entropy and optimized coarse multifractal dimension

with

𝐹𝐷/𝑁

𝔍(𝛼)=liminf

𝑛→∞

log+N𝐷/𝑁

𝔍,𝛼 (𝑛)

log(2𝑛)and 𝐹𝐷/𝑁

𝔍(𝛼)=limsup

𝑛→∞

log+N𝐷/𝑁

𝔍,𝛼 (𝑛)

log(2𝑛)

and

N𝐷/𝑁

𝔍,𝛼 (𝑛)=card 𝑀𝐷/𝑁

𝑛(𝛼), 𝑀𝐷/𝑁

𝔍,𝑛 (𝛼)=𝑄∈ D𝐷/𝑁

𝑛:𝔍(𝑄)≥2−𝛼𝑛 .

This chapter is divided into four sections. In Section 3.1, we present an adaptive

approximation partition algorithm (see Proposition 3.1) which yields an upper

bound for the upper

𝔍

-partition entropy in terms of the zero of the

𝔍

-partition

function. Further, we show that the lower and upper

𝔍

-partition entropy is always

bounded from below by the lower and upper optimized coarse multifractal dimen-

sion. In Section 3.2, we prove that the

𝔍𝜈,𝑎,𝑏

-partition entropy, under the assumption

𝑏dim∞(𝜈)+𝑎𝑑 >0, is bounded from above by

𝑞𝑁

𝔍𝜈,𝑎,𝑏 =inf{𝑞≥0 : 𝜏𝑁

𝔍𝜈,𝑎𝑏 (𝑞)<0}.

In Section 3.3, we establish a connection to the classical works of Solomjak and

Birman [BS66; BS74], Borzov [Bor71], and the partition entropy. This enables us

to partially improve [Bor71, Theorem 1] for a wide class of singular set functions on

. In the last chapter we study the lower and upper optimized coarse multifractal

dimension under mild conditions on

𝔍

. One of the main result of this section is

the identiﬁcation of the upper optimized coarse multifractal dimension by

𝑞𝐷/𝑁

𝔍

Further, we establish regularity conditions (see Deﬁnition 3.22) for which we can

guarantee that

𝐹𝐷/𝑁

𝔍=𝐹𝐷/𝑁

𝔍

. Later, we will use this regularity result to give criteria

that allow us to ensure, depending on the setting, the existence of the spectral

dimension or quantization dimension, respectively.

3.1

Bounds for the partition entropy and optimized coarse

multifractal dimension

We begin with a motivation for the upper estimate of

ℎ𝔍

: For a given threshold

<𝑡<𝔍(Q)

, we will construct partitions by dyadic cubes of

as a function

𝑡

via an adaptive approximation algorithm in the sense of [DeV87] (see also

[HKY00]) as follows. We say

𝑄∈ D

is bad if

𝔍(𝑄) ≥ 𝑡

, otherwise we call

𝑄

good. We generate a partition of

of elements of

into good intervals which

will be denoted by

𝑃𝑡

. By the choice of

𝑡

, we see that

is bad. Hence, we put

B≔{Q}

. Now, we divide each element of

into 2

𝑑

cubes of

of equal size and

check whether they are good, in which case we move these cubes to

𝑃𝑡

, or they are

3.1. Bounds for the partition entropy and optimized coarse multifractal dimension

bad, in which case they are put into

. We repeat this procedure until the set of

bad cubes is empty. The process terminates, which is ensured by the assumption

that

𝔍

decreases uniformly. Using the deﬁnition of

and

M𝔍

, it follows that the

resulting ﬁnite partition

𝑃𝑡

is optimal (in the sense of minimizing the cardinality,

i.e.

M𝔍(

/𝑡)=card(𝑃𝑡))

among all partitions

𝑃

by dyadic cubes of

fulﬁlling

max𝑄∈𝑃𝔍(𝑄)<𝑡

. We provide a two-dimensional illustration (Figure 3.1.1) of

these partitions

𝑃𝑡

for three diﬀerent values of

𝑡∈(0,1)

for the particular choice

𝔍(𝑄)=(𝜈𝜈)(𝑄)Λ(𝑄)2

𝑄∈ D

, where

𝜈

denotes the

(𝑝, 1−𝑝)

-Cantor measure

supported on the triadic Cantor set.

Figure 3.1.1 Illustration of the adaptive approximation algorithm for

𝔍(𝑄)=(𝜈

𝜈)(𝑄)Λ(𝑄)2

𝑄∈ D

𝑑=

2, where

𝜈

is the

(

)

-cantor measure. Here, the light

gray cubes belong to

𝑃10−3

, the gray cubes to

𝑃10−4

, and the black cubes to

𝑃10−7

. In this

ﬁgure we neglected all cubes with 𝜈-measure zero.

Now, the remaining task is to connect the asymptotic behavior of

card(𝑃𝑡)

with

the partition function

𝜏𝑁

𝔍

. Motivated by ideas from large derivation theory and the

thermodynamic formalism [Rue04], we are able to bound

ℎ𝔍

from above by

𝑞𝑁

𝔍

namely, by comparing the cardinality of

𝑃𝑡

and

𝑄𝑡≔{𝑄∈ D :𝔍(𝑄) ≥ 𝑡}

. This

will be the key idea in the proof of Proposition 3.1.

3.1. Bounds for the partition entropy and optimized coarse multifractal dimension

Proposition 3.1. For 0<𝑡<𝔍(Q), we have that

𝑃𝑡≔𝑄∈ D :𝔍(𝑄)<𝑡&∃𝑄′∈ D𝑁

|log2(Λ(𝑄)) |/𝑑−1:𝑄′⊃𝑄&𝔍(𝑄′) ≥ 𝑡

is a ﬁnite partition of dyadic cubes of

, and the growth rate of

card(𝑃𝑡)

gives rise

to the following inequalities:

𝐹𝑁

𝔍≤ℎ𝔍≤limsup

𝑡↓0

log (card(𝑃𝑡))

−log(𝑡)≤𝜅𝔍≤𝑞𝑁

𝔍,(3.1.1)

𝐹𝐷

𝔍≤𝑞𝐷

𝔍, and

𝐹𝑁

𝔍≤ℎ𝔍≤liminf

𝑡↓0

log (card(𝑃𝑡))

−log(𝑡).

Remark 3.2.At this stage we would like to point out that in the next section (see

Proposition 3.20) , we will show equality in the above chain of inequalities

(3.1.1)

using the coarse multifractal formalism under some mild additional assumptions on

𝔍.

Proof.

We only have to consider the case

𝜅𝔍<∞

. The ﬁrst statement follows from

the monotonicity of 𝔍,

lim

𝑛→∞ sup

𝑖≥𝑛,𝐶 ∈D𝑁

𝑖

𝔍(𝐶)=0,

the deﬁnition of

and Lemma 2.1. Further, Lemma 2.25 gives

𝜅𝔍≤𝑞𝑁

𝔍

(where

equality holds if dim∞(𝔍)>0, otherwise 𝑞𝑁

𝔍=∞). Let 0 <𝑡<𝔍(Q). Setting

𝑅𝑡≔{𝑄∈ D :𝔍(𝑄) ≥ 𝑡},

we note that for

𝑄∈𝑃𝑡

there is exactly one

𝑄′∈𝑅𝑡∩D𝑁

|log2(Λ(𝑄))|/𝑑−1

with

𝑄⊂𝑄′

and for each

𝑄′∈𝑅𝑡∩ D𝑁

|log2(Λ(𝑄))|/𝑑−1

there are at most 2

𝑑

elements of

𝑃𝑡∩

D𝑁

|log2(Λ(𝑄))|/𝑑such that they are subsets of 𝑄′. Hence,

card(𝑃𝑡) ≤ 2𝑑card(𝑅𝑡).

3.1. Bounds for the partition entropy and optimized coarse multifractal dimension

For 𝑞>𝜅𝔍we obtain

𝑡𝑞card (𝑃𝑡)=∞



𝑛=1

𝑡𝑞

𝑄∈𝑃𝑡∩D𝑁

𝑛

≤2𝑑∞



𝑛=0

𝑄∈𝑅𝑡∩D𝑁

𝑛

𝑡𝑞

≤2𝑑∞



𝑛=0

𝑄∈𝑅𝑡∩D𝑁

𝑛

𝔍(𝑄)𝑞

≤2𝑑∞



𝑛=0

𝑄∈D𝑁

𝑛

𝔍(𝑄)𝑞<∞.

This implies

limsup

𝑡↓0

log (card (𝑃𝑡))

−log(𝑡)≤𝑞.

Now,

𝑞

tending to

𝜅𝔍

proves the third inequality. The second and sixth inequality

follow immediately from the observation that

M𝔍(𝑥)≤card 𝑃1/𝑥

. For

𝛼>

𝑛∈N, and 𝑃∈Π𝔍such that max𝐶∈𝑃𝔍(𝐶)<2−𝑛𝛼 , we have

N𝑁

𝛼,𝔍(𝑛)=card 𝑄∈ D𝑁

𝑛:𝔍(𝑄) ≥ 2−𝛼𝑛 ≤card (𝑃),

where we used the fact that for each

𝑄∈ D𝑁

𝑛

with

𝔍(𝑄) ≥

−𝛼𝑛

there exists at least

one

𝑄′∈ D (𝑄)∩𝑃

and this assignment is injective. Indeed, since

𝑃∈Π𝔍

𝔍

locally non-vanishing, and

𝔍(𝑄) ≥

/𝑥

, it follows

𝑄∩𝑃≠∅

. Therefore, using that

𝔍

is monotone, we deduce that there at least one exists

𝑄′∈ D (𝑄)∩𝑃

. Thus, we

obtain (for an illustration see Figure 3.1.2)

N𝑁

𝛼,𝔍(𝑛)≤ M𝔍(2𝛼𝑛 ).

To prove

𝐹𝑁

𝔍≤ℎ𝔍

, ﬁx

𝛼>

, 𝑥 >max{

,𝔍(Q)}

, and

𝑃∈Π𝔍

such that

M𝔍(𝑥)=

card(𝑃). Then there exists 𝑛𝑥∈Nsuch that

2−(𝑛𝑥+1)𝛼<1

𝑥≤2−𝑛𝑥𝛼.

It follows

N𝑁

𝛼,𝔍(𝑛𝑥)≤ M𝔍(2𝛼𝑛𝑥)≤ M𝔍(𝑥).

Therefore, we obtain

N𝑁

𝛼,𝔍(𝑛𝑥)≤card(𝑃).

3.1. Bounds for the partition entropy and optimized coarse multifractal dimension

Hence,

log N𝑁

𝛼,𝔍(𝑛𝑥)

log(2𝑛𝑥𝛼) +log(2)𝛼≤

log N𝑁

𝛼,𝔍(𝑛𝑥)

log(𝑥)≤log M𝔍(𝑥)

log(𝑥),

implying

liminf

𝑛→∞

log N𝑁

𝛼,𝔍(𝑛)

log(2𝑛𝛼 )≤lim inf

𝑥→∞

log N𝑁

𝛼,𝔍(𝑛𝑥)

log(2𝑛𝑥𝛼)≤ℎ𝔍.

Therefore, taking the supremum over

𝛼>

0 gives

𝐹𝑁

𝔍≤ℎ𝔍

. The last claim follows

from the fact that for every 𝛼>0 and 𝑞=𝑞𝐷

𝔍, we have

N𝐷

𝛼,𝔍(𝑛)≤2𝛼𝑛𝑞 

𝑄∈D𝐷

𝑛

𝔍(𝑄)𝑞.□

Figure 3.1.2 Illustration of the cubes of

𝑀𝑁

𝛼,𝔍(𝑛)

(gray) and

𝑃2−𝛼𝑛

(underneath of gray cubes

in black) for 𝑛=4 and 𝛼=5.734 with 𝔍as deﬁned in Figure 3.1.1.

In Section 3.4 (see Proposition 3.20), we will show equality in the above chain

of inequalities

(3.1.1)

using the coarse multifractal formalism under some mild

additional assumptions on 𝔍.

3.2. Upper bounds for the 𝔍𝜈,𝑎,𝑏-partition entropy

Proposition 3.3. Assume there exists a sequence

(𝑛𝑘)𝑘∈NN

and

𝐾>

0such that

for all 𝑘∈N,

max

𝑄∈D𝑁

𝑛𝑘

𝔍(𝑄)𝑞𝑛𝑘≤𝐾

2𝑛𝑘𝜏𝑁

𝔍,𝑛𝑘(0)

𝑄∈D𝑁

𝑛𝑘

𝔍(𝑄)𝑞𝑛𝑘,

where

𝑞𝑛𝑘

is the unique zero of

𝜏𝑁

𝑛𝑘

. Further, suppose

liminf𝑘→∞ 𝑞𝑛𝑘>

0. Then we

have

ℎ𝔍≤liminf

𝑘→∞ 𝑞𝑛𝑘.

Proof.

First of all, note that we have

𝑄∈D𝑁

𝑛𝑘

𝔍(𝑄)𝑞𝑛𝑘=

1. Further, since

𝔍

uniformly decreasing, we choose

𝑘

large enough such that

𝔍(𝑄)<

1 for all

𝑄∈ D𝑁

𝑛𝑘

Ensuring that 𝜏𝑁

𝑛𝑘has a unique zero. Hence, we obtain

max

𝑄∈D𝑁

𝑛𝑘

𝔍(𝑄) ≤ 𝐾1/𝑞𝑛𝑘

2𝑛𝑘𝜏𝑁

𝔍,𝑛𝑘(0)/𝑞𝑛𝑘

This implies

log M𝔍2𝜏𝑁

𝔍,𝑛𝑘(0)𝑛𝑘/𝑞𝑛𝑘

2𝐾1/𝑞𝑛𝑘

log 2𝜏𝑁

𝔍,𝑛𝑘(0)𝑛𝑘/𝑞𝑛𝑘

2𝐾1/𝑞𝑛𝑘≤

log 2𝑛𝑘𝜏𝑁

𝔍,𝑛𝑘(0)

log 2𝜏𝑁

𝔍,𝑛𝑘(0)𝑛𝑘/𝑞𝑛𝑘

2𝐾1/𝑞𝑛𝑘,

which proves the claim. □

3.2 Upper bounds for the 𝔍𝜈,𝑎,𝑏-partition entropy

This section is devoted to the study of the

𝔍𝜈,𝑎,𝑏

-partition entropy, which is ultimately

associated with the spectral dimension for a certain choice of parameters

𝑎∈R, 𝑏 >

Let us introduce the following notation: M𝑎,𝑏 (𝑥)≔M𝔍𝜈,𝑎,𝑏 (𝑥),𝑥>0, as well as

ℎ𝑎,𝑏 ≔ℎ𝔍𝜈,𝑎,𝑏, ℎ𝑎,𝑏 ≔ℎ𝔍𝜈,𝑎,𝑏 and ℎ𝑎≔ℎ𝑎,1, ℎ𝑎≔ℎ𝑎,1.

The following theorem treats an upper estimate of ℎ𝑎,𝑏 for the case 𝑎=0.

Proposition 3.4. If dim∞(𝜈)>0, then

ℎ0,𝑏 ≤𝑞𝑁

𝔍𝜈,0,𝑏 =inf 𝑞≥0 : 

𝐶∈D

𝔍𝜈,0,𝑏 (𝐶)𝑞<∞=1

𝑏.

Proof.

First, note that Proposition 2.35 implies

dim∞(𝔍𝜈,0,𝑏)=𝑏dim∞(𝜈)>

Therefore, we deduce that

𝔍𝜈,0,𝑏

is uniformly vanishing by Lemma 2.23. An

3.2. Upper bounds for the 𝔍𝜈,𝑎,𝑏-partition entropy

application of Proposition 3.1 with

𝔍 = 𝔍𝜈,0,𝑏

gives

ℎ0,𝑏 ≤𝑞𝑁

𝔍𝜈,0,𝑏

. Furthermore, we

have by Proposition 2.35 that

𝛽𝑁

𝜈(𝑞𝑏)=𝜏𝑁

𝔍𝜈,0,𝑏 (𝑞)

. Thus, by Lemma 2.25, we obtain

inf 𝑞≥0 : 

𝐶∈D

𝔍𝜈,0,𝑏 (𝐶)𝑞<∞=𝑞𝑁

𝔍𝜈,0,𝑏 =1

𝑏.□

The rest of this section deals with the case

𝑎≠

0. Recalling the deﬁnition of

𝑞𝑁

𝔍

, we

ﬁnd

𝑞𝑁

𝜈Λ𝑎≤𝑞𝑁

𝔍𝜈,𝑎,1

with equality for the case

𝑎>

0. We need the following elementary

lemma.

Lemma 3.5. For

𝑐,𝑑 ∈R

with

𝑐<𝑑

, let

(𝑓𝑛

[𝑐,𝑑]→R)𝑛∈N

be a sequence of

decreasing functions converging pointwise to a function

𝑓

. We assume that

𝑓𝑛

has a

unique zero in 𝑥𝑛,𝑛∈N, and 𝑓has a unique zero in 𝑥. Then 𝑥=lim𝑛→∞𝑥𝑛.

Proof.

Assume that

(𝑥𝑛)𝑛

does not converge to

𝑥

. Then, because

[𝑐,𝑑]

is compact,

there exists a subsequence

(𝑛𝑘)𝑘

such that

𝑥𝑛𝑘→𝑥∗≠𝑥

for

𝑘→ ∞

and

𝑥∗∈ [𝑐,𝑑]

We only consider

𝑥∗<𝑥

, the case

𝑥∗>𝑥

follows analogously. Then, for

𝑘

large, we

have 𝑥𝑛𝑘≤ (𝑥∗+𝑥)/2. Thus, for each 𝑦∈((𝑥∗+𝑥)/2,𝑥), we have

0=𝑓𝑛𝑘(𝑥𝑛𝑘)>𝑓𝑛𝑘(𝑦) ≥ 𝑓𝑛𝑘(𝑥) → 𝑓(𝑥)=0,for 𝑘→ ∞.

Consequently,

𝑓(𝑦)=

0 for all

𝑦∈((𝑥∗+𝑥)/2,𝑥 )

, contradicting the uniqueness of

the zero of 𝑓.□

Proposition 3.6. Suppose 𝑏dim∞(𝜈) +𝑎𝑑 >0. If 𝑎<0, then

ℎ𝑎,𝑏 =ℎ𝑎/𝑏,1

𝑏≤

𝑞𝑁

𝔍𝜈,𝑎/𝑏,1

𝑏≤dim∞(𝜈)

𝑏dim∞(𝜈)+𝑎𝑑 .

If 𝑎>0, then

ℎ𝑎,𝑏 ≤𝑞𝑁

𝔍𝜈,𝑎,𝑏 =inf 𝑞>0 : 𝛽𝑁

𝜈(𝑏𝑞)<𝑎𝑑𝑞 ≤dim𝑀(𝜈)

𝑏dim𝑀(𝜈)+𝑎𝑑 ≤1

𝑏+𝑎.

In particular, if

dim∞(𝜈)>𝑑−

2and

𝑑>

2, then for all

𝑡∈(0,2dim∞(𝜈)/(𝑑−2))

we have

ℎ2/𝑑−1,2/𝑡=𝑡

2ℎ𝑡(2/𝑑−1)/2,1≤𝑡

2𝑞𝑁

𝔍𝜈,𝑡 (2/𝑑−1)/2,1≤dim∞(𝜈)

2dim∞(𝜈)/𝑡+2−𝑑.

Moreover, lim𝑡↓2𝑞𝑁

𝔍𝜈,𝑡 (2/𝑑−1)/2,1=𝑞𝑁

𝔍𝜈.

3.2. Upper bounds for the 𝔍𝜈,𝑎,𝑏-partition entropy

Proof. Since 𝑏dim∞(𝜈)+𝑎𝑑 >0, we obtain from Fact 2.37 that

dim∞(𝔍𝜈,𝑎/𝑏,1)=dim∞(𝜈)+𝑎𝑑/𝑏>0.

Using the deﬁnition of

M𝑎,𝑏 (𝑥)

and Proposition 3.1 applied to

𝔍 = 𝔍𝜈,𝑎/𝑏,1

, we

obtain

ℎ𝑎,𝑏 =limsup

𝑥→∞

log M𝑎,𝑏 (𝑥)

log(𝑥)=lim sup

𝑥→∞

log M𝑎/𝑏,1𝑥1/𝑏

𝑏log(𝑥1/𝑏)=1

𝑏ℎ𝑎/𝑏,1≤

𝑞𝑁

𝔍𝜈,𝑎/𝑏,1

𝑏,

where for the third equality we used the bijectivity of

𝑥↦→𝑥1/𝑏

𝑥>

0. The estimate

of 𝑞𝑁

𝔍𝜈,𝑎,𝑏 for the case 𝑎>0 follows from

𝛽𝑁

𝜈(𝑏𝑞) ≤ dim𝑀(𝜈)(1−𝑞𝑏)

for all 0 ≤𝑞≤1/𝑏. For the case 𝑎<0, Fact 2.37 implies

𝑞𝑁

𝔍𝜈,𝑎/𝑏,1≤dim∞(𝜈)/(dim∞(𝜈)+𝑎𝑑/𝑏).

Now, let dim∞(𝜈)>𝑑−2, 𝑑>2, and 𝑡∈𝐼≔(0,2 dim∞(𝜈)/(𝑑−2)). we obtain

dim∞𝔍𝜈,𝑡 (2/𝑑−1)/2,1=dim∞(𝜈)+𝑑𝑡 (2/𝑑−1)/2>0.

Hence, the third claim follows from the ﬁrst part. The rest of the proof is devoted to

the proof of lim𝑡↓2𝑞𝑁

𝔍𝜈,𝑡 (2/𝑑−1)/2,1=𝑞𝑁

𝔍𝜈, (2/𝑑−1),1. First, observe that, using

0<𝑡(𝑑−2)/2<𝑠<dim∞(𝜈),

we have for 𝑛large

𝜈(𝐶) ≤ 2−𝑠𝑛, 𝐶 ∈ D𝑁

𝑛.

Set 𝑎≔2/𝑑−1 and for ﬁxed 𝑞≥0, consider

𝑡↦→𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1(𝑞)=limsup

𝑛→∞

log 𝑄∈D𝑁

𝑛max𝑄′∈D(𝑄)𝜈(𝑄′)𝑞(Λ(𝑄′)𝑞𝑎)𝑡/2

log(2𝑛).

Since

𝑓𝑄

𝑡↦→𝜈(𝑄)𝑞(Λ(𝑄)𝑞𝑎 )𝑡/2

𝑄∈ D𝑁

𝑛

with

𝜈(𝑄)>

0 and

𝑡∈𝐼

, is log-convex,

i.e. for all 𝜃∈ (0,1)and 𝑠, 𝑡 ∈R>0, we have

log 𝑓𝑄(𝜃𝑡 + (1−𝜃)𝑠)≤𝜃log 𝑓𝑄(𝑡)+(1−𝜃)log 𝑓𝑄(𝑠),

it follows that

𝑡↦→ max

𝑄′∈D(𝑄)𝜈(𝑄′)𝑞(Λ(𝑄′)𝑞𝑎 )𝑡/2

3.3. The dual problem

is also log-convex (the existence of the maximum is ensured by

𝔍𝜈,𝑎𝑡/2,1(𝐶) ≤ 2𝑛(−𝑠+(𝑑−2)𝑡/2)

for all

𝐶∈ D𝑁

𝑛

). Therefore, we get with the H

older inequality that

𝑡↦→𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1,𝑛 (𝑞)

𝑡∈𝐼

is convex, which carries over to the limit superior

𝑡↦→𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1(𝑞)

𝑡∈𝐼

, of

convex functions. Further, we have

𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1(𝑞)<∞

for all

𝑡∈𝐼

. Hence, by Theorem

A.5, it follows that

𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1

is continuous on

𝐼

. In particular, for each

𝑞≥

0, we have

lim

𝑡→2𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1(𝑞)=𝜏𝑁

𝔍𝜈,𝑎,1(𝑞).

By Lemma 2.27 we deduce that

𝑞𝑁

𝔍𝜈,𝑎𝑡 /2,1

is the unique zero of

𝑞↦→ 𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1(𝑞)

Hence, for ﬁxed

𝑡∈𝐼

, we have that

𝑞↦→𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1(𝑞)

is decreasing and has a unique

zero given by 𝑞𝑁

𝔍𝜈,𝑎𝑡 /2,1. Further, for all 𝑡∈ [2,(2+2dim∞(𝜈)/(𝑑−2))/2], we have

0≤𝑞𝑁

𝔍𝜈,𝑎𝑡 /2,1≤dim∞(𝜈)

dim∞(𝜈)+(2−𝑑)𝑡/2

≤dim∞(𝜈)

dim∞(𝜈)+(2−𝑑)(1+dim∞(𝜈)/(𝑑−2))/2≕𝑔.

Now, Lemma 3.5, applied to

𝑞↦→𝜏𝑁

𝔍𝜈,𝑎𝑡 /2,1|[0,𝑔], 𝑡 ∈ [2,(2+dim∞(𝜈)/(𝑑−2))/2],

implies

lim

𝑡↓2𝑞𝑁

𝔍𝜈,𝑎𝑡 /2,1=𝑞𝑁

𝔍𝜈,𝑎,1.□

3.3 The dual problem

This section is devoted to study the dual problem of

M𝔍

. Recall from Section 1.1.3

that the dual problem is concerned with the control of the asymptotic behavior of

𝛾𝔍,𝑛 =inf

𝑃∈Π𝔍,

card(𝑃) ≤𝑛

max

𝑄∈𝑃𝔍(𝑄).

In particular, we are interested in the special choice

𝔍𝐽 ,𝑎 (𝑄)≔𝐽(𝑄)Λ(𝑄)𝑎

𝑎>

𝑄∈ D

, where

𝐽

is a non-negative, ﬁnite, locally non-vanishing, and superadditive

function on

, that is, if

𝑄∈ D

is decomposed into a ﬁnite number of disjoint cubes

𝑄𝑗𝑗=1,...,𝑁

, then

𝑁

𝑗=1𝐽(𝑄𝑗) ≤ 𝐽(𝑄)

. We are now interested in the growth

properties of

𝛾𝔍𝐽 ,𝑎,𝑛

. Upper estimates for

𝛾𝔍𝐽 ,𝑎,𝑛

have been ﬁrst obtained in [BS67;

Bor71]. Here, we proceed as follows: First we present an adaptive approximation

3.3. The dual problem

algorithm going back to Birman/Solomjak [BS67; Bor71] to obtain well-known

upper bounds on

𝛾𝔍𝐽 ,𝑎,𝑛

. After that we employ the estimates of Proposition 3.1 to

partially improve and extend the results in [BS67; Bor71].

In the following we use the terminology as in [DKS20]. Let

Ξ0

be a ﬁnite partition

of dyadic cubes from

. We say a partition

Ξ′

is an elementary extension

Ξ0

if it can be obtained by uniformly splitting some of its cubes into 2

𝑑

equal

sized disjoint cubes lying in

with half side length. We call a partition

dyadic

subdivision of an initial partition

Ξ0

if it is obtained from the partition

Ξ0

with the

help of a ﬁnite number of elementary extensions.

Proposition 3.7 ([BS67], [Bor71]).Let

Ξ0

be a ﬁnite partition of

with dyadic

cubes from Dand suppose there exists 𝜀>0and a subset Ξ′

0⊂Ξ0such that



𝑄∈Ξ0\Ξ′

Λ(𝑄) ≤ 𝜀and 

𝑄∈Ξ′

𝐽(𝑄) ≤ 𝜀.

Let

(𝑃𝑘)𝑘∈N

denote a sequence of dyadic partitions obtained recursively as follows:

We set

𝑃0≔Ξ0

and, for

𝑘∈N

, we construct an elementary extension

𝑃𝑘

𝑃𝑘−1

subdividing all cubes 𝑄∈𝑃𝑘−1for which

𝔍𝐽 ,𝑎 (𝑄) ≥ 2−𝑑𝑎𝐺𝑎(𝑃𝑘−1)

with

𝐺𝑎(𝑃𝑘−1)≔max𝑄∈𝑃𝑘−1𝔍𝐽 ,𝑎 (𝑄)

, into 2

𝑑

equal sized cubes. Then, for all

𝑘∈N,

we have

𝐺𝑎(𝑃𝑘)=max

𝑄∈𝑃𝑘

𝔍𝐽 ,𝑎 (𝑄) ≤𝐶𝜀 min (1,𝑎)(𝑁𝑘−𝑁0)−(1+𝑎)𝐽(Q)

with

𝑁𝑘≔card(𝑃𝑘)

𝑘∈N0

, and the constant

𝐶>

0depends only on

𝑎

and

𝑑

. In

particular, there exists 𝐶′>0such that for all 𝑛>𝑁0,

𝛾𝔍𝐽 ,𝑎,𝑛 ≤𝐶′𝜀min(1,𝑎)𝑛−(1+𝑎)𝐽(Q).

Proof.

Here, we follow closely [DKS20]. Without loss of generality we may

assume

𝐽(Q) ≤

1. Fix

𝑘∈N

and let

𝑆𝑘

denote the set of all cubes from

𝑃𝑘−1

that

are subdivided to obtain 𝑃𝑘. Further, let 𝑆1

𝑘,𝑆2

𝑘⊂𝑆𝑘with 𝑆𝑘=𝑆1

𝑘∪𝑆2

𝑘,



𝑄∈𝑆1

𝑘

𝑄⊂

𝑄∈Ξ0\Ξ′

𝑄,

and 

𝑄∈𝑆2

𝑘

𝑄⊂

𝑄∈Ξ′

𝑄.

We deﬁne

𝑡𝑘≔card(𝑆𝑘)

𝑡1,𝑘 ≔card(𝑆1

𝑘)

, and

𝑡2,𝑘 ≔card(𝑆2

𝑘)

. By the deﬁnition of

3.3. The dual problem

𝑃𝑘,we have min𝑄∈𝑆𝑖

𝑘𝔍𝐽 ,𝑎 (𝑄) ≥ 2−𝑑𝑎𝐺𝑎(𝑃𝑘−1)and we obtain

2−𝑑𝑎𝐺𝑎(𝑃𝑘−1)1

1+𝑎≤min

𝑄∈𝑆𝑖

𝑘

𝔍𝐽 ,𝑎 (𝑄)1

1+𝑎≤1

𝑡𝑖,𝑘 

𝑄∈𝑆𝑖

𝑘

Λ(𝑄)𝑎

1+𝑎𝐽(𝑄)1

1+𝑎.

By the H¨

older inequality and the superadditivity of 𝐽, we obtain for 𝑖=1,2,

2−𝑑𝑎𝐺𝑎(𝑃𝑘−1)1

1+𝑎≤1

𝑡𝑖,𝑘 

𝑄∈𝑆𝑖

𝑘

Λ(𝑄)

𝑎

1+𝑎

𝑄∈𝑆𝑖

𝑘

𝐽(𝑄)

1+𝑎

≤1

𝑡𝑖,𝑘

𝜀min(1,𝑎)

1+𝑎.

This is equivalent to

𝑡𝑖,𝑘 ≤2𝑎𝑑

1+𝑎𝜀min(1,𝑎)

1+𝑎(𝐺𝑎(𝑃𝑘−1))−1

1+𝑎.

Since

𝑡𝑘=𝑡1,𝑘 +𝑡2,𝑘

, we have

𝑡𝑘≤

1+𝑎𝑑

1+𝑎𝜀min(1,𝑎)

1+𝑎(𝐺𝑎(𝑃𝑘−1))−1

1+𝑎

. By the deﬁnition

of the dyadic subdivision 𝑃𝑗,𝑗∈N,

𝐺𝑎(𝑃𝑗) ≤ max max

𝑄∈𝑃𝑗∩𝑃𝑗−1

𝔍𝐽 ,𝑎 (𝑄),max

𝑄∈𝑃𝑗\𝑃𝑗−1

𝔍𝐽 ,𝑎 (𝑄)

≤max 2−𝑎𝑑𝐺𝑎(𝑃𝑗−1),max

𝑄∈𝑆𝑗

2−𝑎𝑑 𝔍𝐽 ,𝑎 (𝑄)

≤2−𝑎𝑑𝐺𝑎(𝑃𝑗−1).

(3.3.1)

Now, applying (3.3.1) recursively, for all integers 𝑗≤𝑘, we obtain

𝐺𝑎(𝑃𝑘−1) ≤ 2−𝑎𝑑 (𝑘−𝑗)𝐺𝑎(𝑃𝑗−1).

Since for all

𝑗∈N

we have

𝑁𝑗−𝑁𝑗−1=(

𝑑−

)𝑡𝑗

. Hence, for all

𝑘∈N

, we deduce

𝑁𝑘−𝑁0=(2𝑑−1)

𝑘



𝑗=1

𝑡𝑗

≤21+𝑎𝑑

1+𝑎𝜀min(1,𝑎)

1+𝑎(2𝑑−1)

𝑘



𝑗=1

𝐺𝑎(𝑃𝑗−1)−1

1+𝑎

≤21+𝑎𝑑

1+𝑎𝜀min(1,𝑎)

1+𝑎(2𝑑−1)𝐺𝑎(𝑃𝑘−1)−1

1+𝑎

𝑘



𝑗=1

2−𝑎𝑑

1+𝑎(𝑘−𝑗)

≤1−2−𝑎𝑑

1+𝑎−1

21+𝑎𝑑

1+𝑎𝜀min(1,𝑎)

1+𝑎(2𝑑−1)𝐺𝑎(𝑃𝑘−1)−1

1+𝑎

≤1−2−𝑎𝑑

1+𝑎−1

21+𝑎𝑑

1+𝑎𝜀min(1,𝑎)

1+𝑎(2𝑑−1)2−𝑎𝑑

1+𝑎𝐺𝑎(𝑃𝑘)−1

1+𝑎.

3.3. The dual problem

This proves our ﬁrst claim. For the second claim note that, since

lim𝑘→∞ 𝑁𝑘=∞

for each

𝑛∈N

there exists

𝑘∈N

such that

𝑁𝑘−1≤𝑛<𝑁𝑘

. Furthermore, we always

have 𝑁𝑘−1≤𝑁𝑘≤2𝑑𝑁𝑘−1. Thus, by combining both inequalities, we obtain

𝛾𝔍𝐽 ,𝑎,𝑛 ≤𝛾𝔍𝐽 ,𝑎,𝑁𝑘−1

≤𝐶𝜀min (1,𝑎)(𝑁𝑘−1−𝑁0)−(1+𝑎)

≤𝐶𝜀min (1,𝑎)2−𝑑𝑛−𝑁0−(1+𝑎).□

Deﬁnition 3.8. We call

𝐽

a singular function with respect to

if for every

𝜀>

there exist two partitions Ξ′

0⊂Ξ0⊂ D of Qsuch that



𝑄∈Ξ0\Ξ′

Λ(𝑄) ≤ 𝜀and 

𝑄∈Ξ′

𝐽(𝑄) ≤ 𝜀.

Remark 3.9.Since

is a semiring of sets, it follows that a measure

𝜈

which is

singular with respect to the Lebesgue measure, is also singular as a function

𝐽=𝜈

in the sense of Deﬁnition 3.8.

As an immediate corollary of Proposition 3.7, we obtain the following statement by

[Bor71].

Corollary 3.10. We always have

𝛾𝔍𝐽 ,𝑎,𝑛 =𝑂𝑛−(1+𝑎)and M𝔍𝐽 ,𝑎 (𝑥)=𝑂𝑥1/(1+𝑎).

If additionally 𝐽is singular, then

𝛾𝔍𝐽 ,𝑎,𝑛 =𝑜𝑛−(1+𝑎)and M𝔍𝐽 ,𝑎 (𝑥)=𝑜𝑥1/(1+𝑎).

Using Proposition 3.1, we are able to extend the class of set functions considered in

[BS67, Theorem 2.1] (i.e. we allow set functions

𝔍

for which

𝔍

is only assumed to

be non-negative, monotone, and

dim∞(𝔍)>

0). We obtain the following estimate

for the upper exponent of divergence of 𝛾𝔍,𝑛 given by

𝛼𝔍≔limsup

𝑛→∞

log 𝛾𝔍,𝑛 

log(𝑛)and 𝛼𝔍≔lim inf

𝑛→∞

log 𝛾𝔍,𝑛 

log(𝑛).

Proposition 3.11. If dim∞(𝔍)>0, then

−1

ℎ𝔍

=𝛼𝔍≤ − 1

𝑞𝑁

𝔍≤ − dim∞(𝔍)

dim𝑀(supp (𝔍))and −1

ℎ𝔍

=𝛼𝔍.

3.3. The dual problem

In particular, for 𝔍 = 𝔍𝐽 ,𝑎 , we have dim∞(𝔍)=dim∞𝐽+𝑎𝑑 >0and

−1

ℎ𝔍𝐽 ,𝑎

=𝛼𝔍𝐽 ,𝑎 ≤ − 1

𝑞𝑁

𝔍𝐽 ,𝑎 ≤ − dim𝑀((𝐽))+𝑎𝑑

dim𝑀(supp (𝐽))≤ −(1+𝑎).

Remark 3.12.If

𝜏𝑁

𝔍𝐽 ,𝑎 (𝑞)<𝑑(

−𝑞(

+𝑎))

for some

𝑞∈ (

)

, then this estimate

improves the corresponding results of [Bor71; BS67, Theorem 2.1], where only

𝛼𝔍𝐽 ,𝑎 ≤ −(1+𝑎)has been shown.

Proof. For all 𝜀>0, we have for 𝑛large

M𝔍𝑛1/(ℎ𝔍+𝜀)≤𝑛,

this gives

min𝑃∈Π𝔍,card(𝑃)≤𝑛max𝑄∈𝑃𝔍(𝑄) ≤ 𝑛−1/(ℎ𝔍+𝜀)

. Thus, in tandem with Lemma

2.25 and Proposition 3.1, we see that

𝛼𝔍≤ −1/ℎ𝔍≤ −1/𝑞𝑁

𝔍≤ −dim∞(𝔍)/dim𝑀(supp (𝔍)).

In particular, since

𝑞𝑁

𝔍≥

0, we have

𝛼𝔍<

0. To prove the equality, it is left to show

𝛼𝔍≥ −

/ℎ𝔍

. First, assume

𝛼𝔍>−∞

, then for

𝜀>

0 with

𝛼𝔍+𝜀<

0 and

𝑛

large, we

have

inf

𝑃∈Π𝔍,

card(𝑃) ≤𝑛

max

𝑄∈𝑃𝔍(𝑄) ≤ 𝑛𝛼𝔍+𝜀.

By the deﬁnition of inﬁmum, there exists 𝑃′∈Π𝔍with card(𝑃′) ≤ 𝑛such that

max

𝑄∈𝑃′𝔍(𝑄) ≤ 1+2

3𝑛𝛼𝔍+𝜀<2𝑛𝛼𝔍+𝜀,

implying

M𝔍𝑛−(𝛼𝔍+𝜀)/2≤𝑛

. Moreover, for each

𝑥≥

1 there exists

𝑚∈N

such

that

𝑚−(𝛼𝔍+𝜀)/2≤𝑥<(𝑚+1)−(𝛼𝔍+𝜀)/2.

Consequently, for 𝑥large, we have

log M𝔍(𝑥)

log(𝑥)≤log M𝔍(𝑚+1)−(𝛼𝔍+𝜀)/2

log(𝑚−(𝛼𝔍+𝜀)/2)

≤log(𝑚+1)

log(𝑚−(𝛼𝔍+𝜀)/2).

Therefore, we infer that

ℎ𝔍≤ −

/𝛼𝔍

or equivalently

𝛼𝔍≥ −

/ℎ𝔍

. In the case

𝛼𝔍=−∞

it follows in a similar way that

ℎ𝔍=

0. Now, we consider the special case

𝔍 = 𝔍𝐽 ,𝑎

. Observe that

𝜏𝑁

𝔍𝐽 ,𝑎 (𝑞)=𝜏𝑁

𝐽(𝑞)−𝑎𝑑𝑞

for

𝑞≥

0 and

𝜏𝑁

𝐽(

) ≤ 𝑑

. From the

3.4. Coarse multifractal analysis

fact that

𝐽

is sub-additive, it follows that

𝜏𝑁

𝐽(

) ≤

0. We only have to consider the

case 𝜏𝑁

𝐽(1)>−∞. Since 𝜏𝑁

𝐽is convex, for every 𝑞∈ [0,1], we deduce

𝜏𝑁

𝔍𝐽 ,𝑎 (𝑞)=𝜏𝑁

𝐽(𝑞)−𝑎𝑑𝑞 ≤𝜏𝑁

𝐽(0)(1−𝑞)−𝑎𝑑𝑞 ≤𝑑(1−𝑞)−𝑎𝑑𝑞.

This implies

𝑞𝑁

𝔍𝐽 ,𝑎 ≤𝜏𝑁

𝐽(

)/(𝜏𝑁

𝐽(

) +𝑎𝑑 ) ≤

+𝑎)

. From Proposition 3.1, we

deduce

−1

ℎ𝔍≤ − 1

𝑞𝑁

𝔍𝐽 ,𝑎 ≤ −dim𝑀(supp(𝐽)) +𝑎𝑑

dim𝑀(supp (𝐽))≤ −(1+𝑎).□

The following proposition establishes an upper bound of

ℎ𝔍𝐽 ,𝑎

in terms of the lower

Minkowski dimension of 𝐽and the lower ∞-dimension of 𝐽.

Proposition 3.13. If dim∞(𝐽) ∈ [0,∞), then we have

ℎ𝔍𝐽 ,𝑎 ≤dim𝑀(supp(𝐽))

𝑎𝑑 +dim∞(𝐽).

Proof.

We only consider the case

dim∞(𝐽)>

0. The case

dim∞(𝐽)=

0 follows

along the same lines. Let 0

<𝑠<dim∞(𝐽)

and set

𝑎≔𝑟/𝑑

. Then, for

𝑛

large, we

have

max

𝑄∈D𝑁

𝑛

𝔍(𝑄)Λ(𝑄)𝑎≤2−(𝑠+𝑎𝑑 )𝑛<2−(𝑠+𝑎𝑑)𝑛+1.

This implies

M𝔍𝐽 ,𝑎 2−(𝑠+𝑎𝑑)𝑛+1≤2𝑛𝜏𝐽 ,𝑛 (0).

Therefore, we obtain

ℎ𝔍𝜈,𝑎 ≤lim inf

𝑛→∞

log M𝔍𝜈,𝑎 2(𝑠+𝑎𝑑)𝑛−1

log 2(𝑠+𝑎𝑑)𝑛−1

≤liminf

𝑛→∞

log 2𝑛𝛽𝜈,𝑛 (0)

log 2(𝑠+𝑎𝑑)𝑛−1

=liminf

𝑛→∞

𝜏𝐽 ,𝑛 (0)

(𝑠+𝑎𝑑) −1/𝑛=dim𝑀(supp(𝐽))

𝑎𝑑 +𝑠.

Now, 𝑠↑dim∞(𝐽)gives ℎ𝔍𝐽 ,𝑎 ≤dim𝑀(supp(𝐽))/(𝑎𝑑 +dim∞(𝐽)).□

3.4 Coarse multifractal analysis

Throughout this section let

𝔍

be a non-negative, monotone and locally non-vanishing

set function deﬁned on the set of dyadic cubes D. We additionally assume that

3.4. Coarse multifractal analysis

(A1)

there exists

𝑎>

0 and

𝑏∈R

such that

𝜏𝐷/𝑁

𝔍,𝑛 (𝑎)≥𝑏

for all

𝑛

large enough

(excluding trivial cases),

(A2)

the maximal asymptotic direction of

𝜏𝑁

𝔍

is negative, i.e.

dim∞(𝔍)>

0 (this

generalizes the assumption dim∞(𝜈)−𝑑+2>0).

Lemma 3.14. Under the assumptions (A1) and (A2) with

𝑎

and

𝑏

as determined

there and 𝐿≔(𝑏−𝑑)/𝑎<0, for all 𝑛large enough and 𝑞≥0, we have

𝑏+𝑞𝐿 ≤𝜏𝐷/𝑁

𝔍,𝑛 (𝑞).

In particular, −∞ <liminf𝑛→∞ 𝜏𝐷/𝑁

𝔍,𝑛 (𝑞)and dim∞(𝔍)≤ −𝐿.

Proof.

By our assumptions, we have

dim∞(𝔍)>

0, therefore, for

𝑛

large,

𝜏𝐷/𝑁

𝔍,𝑛

monotone decreasing and also

𝑏≤𝜏𝐷/𝑁

𝔍,𝑛 (𝑎)

. By the deﬁnition of

𝜏𝐷/𝑁

𝔍,𝑛

we have

𝜏𝐷/𝑁

𝔍,𝑛 (0) ≤𝑑for all 𝑛∈Nand the convexity of 𝜏𝐷/𝑁

𝔍,𝑛 implies for all 𝑞∈ [0,𝑎]

𝜏𝐷/𝑁

𝔍,𝑛 (𝑞)≤𝜏𝐷/𝑁

𝔍,𝑛 (0)+

𝑞𝜏𝐷/𝑁

𝔍,𝑛 (𝑎)−𝜏𝐷/𝑁

𝔍,𝑛 (0)

𝑎.

In particular, by Theorem A.5, the convexity of 𝜏𝐷/𝑁

𝔍,𝑛 implies for 𝑞>𝑎

𝜏𝐷/𝑁

𝔍,𝑛 (𝑎)−𝜏𝐷/𝑁

𝔍,𝑛 (0)

𝑎≤

𝜏𝐷/𝑁

𝔍,𝑛 (𝑞)−𝜏𝐷/𝑁

𝔍,𝑛 (0)

𝑞.

Thus, we obtain

𝑏+𝑞(𝑏−𝑑)/𝑎≤𝜏𝐷/𝑁

𝔍,𝑛 (0)+

𝑞𝜏𝐷/𝑁

𝔍,𝑛 (𝑎)−𝜏𝐷/𝑁

𝔍,𝑛 (0)

𝑎

≤𝜏𝐷/𝑁

𝔍,𝑛 (0)+

𝑞𝜏𝐷/𝑁

𝔍,𝑛 (𝑞)−𝜏𝐷/𝑁

𝔍,𝑛 (0)

𝑞=𝜏𝐷/𝑁

𝔍,𝑛 (𝑞).

Since 𝜏𝐷/𝑁

𝔍,𝑛 is decreasing with 0 ≤𝜏𝐷/𝑁

𝔍,𝑛 (0) ≤𝑑and 𝜏𝐷/𝑁

𝔍,𝑛 (𝑎) ≥𝑏, we obtain for all

𝑞∈ [0,𝑎]𝑏+𝑞(𝑏−𝑑)/𝑎≤𝑏≤𝜏𝐷/𝑁

𝔍,𝑛 (𝑎) ≤𝜏𝐷/𝑁

𝔍,𝑛 (𝑞).□

Remark 3.15.If

dim∞(𝜈)>𝑑−

2 and

𝜈(Q

˚)>

0, then the assumptions of Lemma

3.14 are satisﬁed for 𝜏𝐷/𝑁

𝔍𝜈,𝑛 . This follows from

𝜏𝐷/𝑁

𝔍𝜈,𝑛 (1) ≥𝑑−2+𝛽𝐷/𝑁

𝑛(1) ≥ (𝑑−2)−𝛿

3.4. Coarse multifractal analysis

for

𝛿>

0 and

𝑛

suﬃciently large, where we used

𝜈(Q

˚)>

0 for the Dirichlet case.

Consequently, 𝛽𝐷

𝑛(1) → 0, for 𝑛→ ∞.

Lemma 3.16. For 𝛼∈(0,dim∞(𝔍)) and 𝑛large, we have

N𝐷/𝑁

𝛼,𝔍(𝑛)=0.

In particular,

𝐹𝐷/𝑁

𝔍=sup

𝛼≥dim∞(𝔍)

limsup

𝑛→∞

log+N𝐷/𝑁

𝛼,𝔍(𝑛)

𝛼log(2𝑛)

and

𝐹𝐷/𝑁

𝔍=sup

𝛼≥dim∞(𝔍)

liminf

𝑛→∞

log+N𝐷/𝑁

𝛼,𝔍(𝑛)

𝛼log(2𝑛).

Proof.

For ﬁxed

𝛼>

0 with

𝛼<dim∞(𝔍)

, by the deﬁnition of

dim∞(𝔍)

, for

𝑛

large

we have

max𝑄∈D𝐷/𝑁

𝑛𝔍(𝑄)≤

−𝛼𝑛

. Hence, for all

𝑄∈ D𝑁

𝑛

, we have

𝔍(𝑄)≤

−𝑛𝛼

For every 0 <𝛼′<𝛼, it follows that N𝐷/𝑁

𝛼′,𝔍(𝑛)=0. This proves the claim. □

We need the following elementary observation from large deviation theory which

seems not to be standard in the relevant literature. For this purpose, we need some

standard facts about convex functions, which are summarized in Appendix A.1.

Lemma 3.17. Suppose

(𝑋𝑛)𝑛

are real-valued random variables on some probability

spaces

(Ω𝑛,A𝑛, 𝜇𝑛)𝑛

such that the rate function

𝔠(𝑡)≔limsup𝑛→∞𝔠𝑛(𝑡)

is a proper

convex function with

𝔠𝑛(𝑡)≔𝑎−1

𝑛log exp(𝑡𝑋𝑛)d𝜇𝑛

𝑡∈R

𝑎𝑛→ ∞

and such

that 0belongs to the interior of the domain of ﬁniteness

{𝑡∈R:𝔠(𝑡)<∞}

. Let

𝐼=(𝑎,𝑑)

be an open interval containing the subdiﬀerential

𝜕𝔠(0)=[𝑏,𝑐]

𝔠

in 0.

Then there exists 𝑟>0such that for all 𝑛suﬃciently large,

𝜇𝑛𝑎−1

𝑛𝑋𝑛∉𝐼≤2 exp (−𝑟𝑎𝑛).

Proof.

We assume that

𝜕𝔠(0)=[𝑏,𝑐]

and

𝐼=(𝑎,𝑑)

with

𝑎<𝑏≤𝑐<𝑑

. First note

that the assumptions ensure that

−∞ <𝑏≤𝑐<∞

. By the Chebychev inequality for

all 𝑞>0 and 𝑛∈N, we have

𝜇𝑛𝑎−1

𝑛𝑋𝑛≥𝑑=𝜇𝑛(𝑞𝑋𝑛≥𝑞𝑎𝑛𝑑)≤exp (−𝑞𝑎𝑛𝑑)exp (𝑞𝑋𝑛)d𝜇𝑛,

implying

limsup

𝑛→∞

𝑎−1

𝑛log 𝜇𝑛𝑎−1

𝑛𝑋𝑛≥𝑑≤inf

𝑞>0𝔠(𝑞)−𝑞𝑑 =inf

𝑞∈R𝔠(𝑞)−𝑞𝑑 ≤0,

3.4. Coarse multifractal analysis

where the equality follows from the assumption 𝑐<𝑑,

𝔠(𝑞)−𝑞𝑑 ≥𝔠(0)+(𝑞−0)𝑐−𝑞𝑑 =(𝑐−𝑑)𝑞≥0

for all 𝑞≤0, 𝔠(0)=0, and the continuity of 𝔠at 0. Similarly, we ﬁnd

limsup

𝑛→∞

𝑎−1

𝑛log 𝜇𝑛𝑎−1

𝑛𝑋𝑛≤𝑎≤inf

𝑞<0𝔠(𝑞)−𝑞𝑎 =inf

𝑞∈R𝔠(𝑞)−𝑞𝑎.

We are left to show that both upper bounds are negative. We show the ﬁrst case

by contradiction – the other case follows in exactly the same way. Assuming

inf𝑞∈R𝔠(𝑞)−𝑞𝑑 ≥

0, then for all

𝑞∈R

we have

𝔠(𝑞)−𝑞𝑑 ≥

0, or after rearranging,

𝔠(𝑞)−𝔠(0)≥𝑑𝑞

. This means, according to the deﬁnition of the sub-diﬀerential,

that 𝑑∈𝜕𝔠(0), contradicting our assumptions. □

Proposition 3.18. For a subsequence

(𝑛𝑘)𝑘

deﬁne the convex function on

R≥0

𝐵≔limsup𝑘→∞𝜏𝐷/𝑁

𝔍,𝑛𝑘

and for some

𝑞≥

0, we assume

𝐵(𝑞)=lim𝑘→∞𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)

and

set [𝑎′,𝑏 ′]≔−𝜕𝐵 (𝑞). Then we have 𝑎′≥dim∞(𝔍)and

𝑎′𝑞+𝐵(𝑞)

𝑏′≤sup

𝛼>𝑏′

liminf

𝑘→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛𝑘)

𝛼log(2𝑛𝑘)

≤sup

𝛼≥dim∞(𝔍)

liminf

𝑘→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛𝑘)

𝛼log(2𝑛𝑘)=sup

𝛼>0

liminf

𝑘→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛𝑘)

𝛼log(2𝑛𝑘).

Moreover, if

𝐵(𝑞)=𝜏𝐷/𝑁

𝔍(𝑞),

then

[𝑎,𝑏]=−𝜕𝜏 𝐷/𝑁

𝔍(𝑞)⊃ −𝜕𝐵 (𝑞)

. Further, if addi-

tionally 0≤𝑞≤𝑞𝐷/𝑁

𝔍, then

𝑎𝑞 +𝜏𝐷/𝑁

𝔍(𝑞)

𝑏≤𝑎′𝑞+𝐵(𝑞)

𝑏′.

Proof.

Without loss of generality we can assume

𝑏′<∞

. Moreover,

dim∞(𝔍)>

implies

𝑏′≥𝑎′≥dim∞(𝔍)>

0. Indeed, observe that

𝐵

is again a convex function

on R. Thus, by the deﬁnition of the sub-diﬀerential, we have for all 𝑥>0

𝐵(𝑞)−𝑎′(𝑥−𝑞) ≤ 𝐵(𝑥) ≤ 𝜏𝐷/𝑁

𝔍(𝑥) ≤𝜏𝑁

𝔍(𝑥) ≤ −𝑥dim∞(𝔍)+𝑑,

which gives

𝑎′≥dim∞(𝔍)>

0. Let

𝑞≥

0. Now, for all

𝑘∈N

and

𝑠<𝑎′≤𝑏′<𝑡

3.4. Coarse multifractal analysis

we have

N𝐷/𝑁

𝑡,𝔍(𝑛𝑘)≥card(𝐶∈ D𝐷/𝑁

𝑛𝑘: 2−𝑠𝑛𝑘>𝔍(𝐶)>2−𝑡𝑛𝑘

ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ

≕𝐿𝑠,𝑡

𝑛𝑘

)

≥

𝐶∈𝐿𝑠,𝑡

𝑛𝑘

𝔍(𝐶)𝑞2𝑠𝑛𝑘𝑞

=2𝑠𝑛𝑘𝑞+𝑛𝑘𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)

𝐶∈D𝐷/𝑁

𝑛𝑘

𝐿𝑠,𝑡

𝑛𝑘(𝐶)𝔍(𝐶)𝑞2−𝑛𝑘𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)

=2𝑠𝑛𝑘𝑞+𝑛𝑘𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)

1−

𝐶∈D𝐷/𝑁

𝑛𝑘

𝐿𝑠,𝑡

𝑛𝑘∁(𝐶)𝔍(𝐶)𝑞2−𝑛𝑘𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)

We use the lower large deviation principle for the process

𝑋𝑘(𝐶)≔log (𝔍(𝐶))

with

probability measure on

D𝐷/𝑁

𝑛𝑘

given by

𝜇𝑘({𝐶}) ≔𝔍(𝐶)𝑞

−𝑛𝑘𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)

. We ﬁnd for

the free energy function

𝔠(𝑥)≔limsup

𝑘→∞

log(2𝑛𝑘)log E𝜇𝑘(exp (𝑥𝑋𝑘))

=limsup

𝑘→∞

log(2𝑛𝑘)log 

𝐶∈D𝐷/𝑁

𝑛𝑘

𝔍(𝐶)𝑥+𝑞/2𝑛𝑘𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)

=limsup

𝑘→∞

𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞+𝑥)−𝐵(𝑞)=𝐵(𝑥+𝑞) −𝐵(𝑞),

with

−𝜕𝔠(0)=[𝑎′,𝑏 ′]⊂ (𝑠, 𝑡 )

and hence there exists a constant

𝑟>

0 depending on

𝑠, 𝑡 , and 𝑞such that for 𝑘large by Lemma 3.17



𝐶∈D𝐷/𝑁

𝑛𝑘

𝐿𝑠,𝑡

𝑛𝑘∁(𝐶)𝔍(𝐶)𝑞/2𝑛𝑘𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)=𝜇𝑘𝑋𝑘

log (2𝑛𝑘)∉(−𝑡, −𝑠)≤2 exp (−𝑟𝑛𝑘).

Therefore,

liminf

𝑘→∞

log N𝐷/𝑁

𝑡,𝔍(𝑛𝑘)

log(2𝑛𝑘)≥𝑠𝑞 +𝐵(𝑞),

for all 𝑠<𝑎′and 𝑡>𝑏′. Hence, we have

sup

𝑡>𝑏′

liminf

𝑘→∞

log N𝐷/𝑁

𝑡,𝔍(𝑛𝑘)

𝑡log(2𝑛𝑘)≥sup

𝑡>𝑏′

𝑎′𝑞+𝐵(𝑞)

𝑡=𝑎′𝑞+𝐵(𝑞)

𝑏′.

3.4. Coarse multifractal analysis

The fact that

−𝜕𝜏𝐷/𝑁

𝔍(𝑞)⊃ −𝜕𝐵 (𝑞)

𝜏𝐷/𝑁

𝔍(𝑞)=𝐵(𝑞)

follows immediately from

limsup𝑘→∞𝜏𝐷/𝑁

𝔍,𝑛𝑘≤𝜏𝐷/𝑁

𝔍.□

Corollary 3.19. Let

𝔍(𝑄)≔𝜈(𝑄)Λ(𝑄)𝛾

with

𝑄∈ D

and

𝛾>

0. Then

𝜏𝐷/𝑁

𝔍(𝑞)=

𝛽𝐷/𝑁

𝜈(𝑞)−𝛾𝑑𝑞

𝑞≥

0, and

dim∞(𝔍)=dim∞(𝜈)+𝑑𝛾 >

0. Suppose there exists a

subsequence

(𝑛𝑘)𝑘

and

𝑞∈ [

]

such that

𝜏𝐷/𝑁

𝔍(𝑞)=lim𝑘→∞𝜏𝐷/𝑁

𝔍,𝑛𝑘(𝑞)

. Then for

𝐵≔limsup𝑘→∞𝜏𝐷/𝑁

𝔍,𝑛𝑘, we have −𝜕𝐵 (𝑞)≔[𝑎′,𝑏 ′]⊂ −𝜕𝜏𝐷/𝑁

𝔍(𝑞)≔[𝑎,𝑏 ]and

𝑎𝑞 +𝜏𝐷/𝑁

𝔍(𝑞)

𝑏≤

𝑎′𝑞+𝜏𝐷/𝑁

𝔍(𝑞)

𝑏′≤sup

𝛼≥dim∞(𝜈)+𝑑𝛾

liminf

𝑘→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛𝑘)

𝛼log(2𝑛𝑘).

Proof.

The ﬁrst claim is obvious since

𝛾>

0. The second inequality follows im-

mediately from Proposition 3.18 and

dim∞(𝔍)=dim∞(𝜈)+𝑑𝛾

. To prove the ﬁrst

inequality observe that

−𝜕𝜏𝐷/𝑁

𝔍(𝑞)=[𝑎1+𝛾𝑑,𝑏1+𝛾𝑑]

with

−𝜕𝛽𝐷/𝑁

𝜈(𝑞)=[𝑎1,𝑏1]

Using

[𝑎′,𝑏 ′]⊂ [𝑎1+𝛾𝑑,𝑏1+𝛾𝑑]

𝜏𝐷/𝑁

𝔍(𝑞)=𝛽𝐷/𝑁

𝜈(𝑞)−𝑑𝛾𝑞

, and

𝛽𝐷/𝑁

𝜈(𝑞)≥

0, we

obtain

(𝑎1+𝑑𝛾)𝑞+𝜏𝐷/𝑁

𝔍(𝑞)

𝑏1+𝛾𝑑 =𝑎1𝑞+𝛽𝐷/𝑁

𝜈(𝑞)

𝑏1+𝛾𝑑 ≤(𝑎1+𝑑𝛾 )𝑞+𝛽𝐷/𝑁

𝜈(𝑞)−𝛾𝑑

𝑏′

≤

𝑎′𝑞+𝜏𝐷/𝑁

𝔍(𝑞)

𝑏′

≤sup

𝛼≥dim∞(𝜈)+𝑑𝛾

liminf

𝑘→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛𝑘)

𝛼log(2𝑛𝑘).□

Proposition 3.20. We have

𝐹𝐷/𝑁

𝔍=𝑞𝐷/𝑁

𝔍.

Proof. First, note that by Proposition 3.1, we always have

𝐹𝐷/𝑁

𝔍≤𝑞𝐷/𝑁

𝔍.

Hence, we can restrict our attention to the case

𝑞𝐷/𝑁

𝔍>

0. Further, by Lemma 3.14,

we observe that, for

𝑛

large, the family of convex functions

𝜏𝐷/𝑁

𝔍,𝑛 𝑛

restricted to

0,𝑞𝐷/𝑁

𝔍+1

only takes values in

𝑞𝐷/𝑁

𝔍+1𝐿+𝑏,𝑑 

and on any compact interval

[𝑐,𝑒]⊂0,𝑞𝐷/𝑁

𝔍+1, by Theorem A.5, for all 𝑐≤𝑥≤𝑦≤𝑒, we have

3.4. Coarse multifractal analysis

𝜏𝐷/𝑁

𝔍,𝑛 (𝑥)−𝜏𝐷/𝑁

𝔍,𝑛 (0)

𝑥−0≤

𝜏𝐷/𝑁

𝔍,𝑛 (𝑦)−𝜏𝐷/𝑁

𝔍,𝑛 (𝑥)

𝑦−𝑥≤

𝜏𝐷/𝑁

𝔍,𝑛 𝑞𝐷/𝑁

𝔍+1−𝜏𝐷/𝑁

𝔍,𝑛 (𝑦)

𝑞𝐷/𝑁

𝔍+1−𝑦

By Lemma 3.14 and the fact 𝜏𝐷/𝑁

𝔍,𝑛 (0) ≤𝑑, we obtain

𝑞𝐷/𝑁

𝔍+1𝐿+𝑏−𝑑

𝑐≤

𝜏𝐷/𝑁

𝔍,𝑛 (𝑥)−𝜏𝐷/𝑁

𝔍,𝑛 (0)

𝑐≤

𝜏𝐷/𝑁

𝔍,𝑛 (𝑥)−𝜏𝐷/𝑁

𝔍,𝑛 (0)

𝑥−0

and

𝜏𝐷/𝑁

𝔍,𝑛 𝑞𝐷/𝑁

𝔍+1−𝜏𝐷/𝑁

𝔍,𝑛 (𝑦)

𝑞𝐷/𝑁

𝔍+1−𝑦≤

𝑑−𝑞𝐷/𝑁

𝔍+1𝐿−𝑏

𝑞𝐷/𝑁

𝔍+1−𝑒

which implies

𝜏𝐷/𝑁

𝔍,𝑛 (𝑦)−𝜏𝐷/𝑁

𝔍,𝑛 (𝑥)≤max 









|𝑏| − 𝑞𝐷/𝑁

𝔍+1𝐿+𝑑

𝑐,

𝑑−𝑞𝐷/𝑁

𝔍+1𝐿+|𝑏|

𝑞𝐷/𝑁

𝔍+1−𝑒







|𝑥−𝑦|.

Hence,

𝜏𝐷/𝑁

𝔍,𝑛 |[𝑐,𝑒]𝑛

is uniformly bounded and uniformly Lipschitz. Thus, by

Arzel

a–Ascoli relatively compact. Using this fact, we ﬁnd a subsequence

(𝑛𝑘)𝑘

such that

lim

𝑘→∞𝜏𝐷/𝑁

𝔍,𝑛𝑘𝑞𝐷/𝑁

𝔍=limsup

𝑛→∞

𝜏𝐷/𝑁

𝔍,𝑛 𝑞𝐷/𝑁

𝔍=0

and 𝜏𝐷/𝑁

𝔍,𝑛𝑘converges uniformly to the proper convex function 𝐵on

𝑞𝐷/𝑁

𝔍−𝛿, 𝑞𝐷/𝑁

𝔍+𝛿⊂0,𝑞𝐷/𝑁

𝔍+1,

for

𝛿

suﬃciently small. We put

[𝑎,𝑏]≔−𝜕𝐵 𝑞𝐷/𝑁

𝔍

. Since the points where

𝐵

diﬀerentiable are dense and since

𝐵

is convex, we ﬁnd for every

𝛿>𝜀>

0 an element

𝑞∈𝑞𝐷/𝑁

𝔍−𝜀,𝑞𝐷/𝑁

𝔍

such that

𝐵

is diﬀerentiable at

𝑞

with

−𝐵′(𝑞)∈ [𝑏,𝑏 +𝜀]

. This

follows from the fact that

𝜏𝐷/𝑁

𝔍

is a decreasing function and that the left-hand

derivative of the convex function 𝜏𝐷/𝑁

𝔍is left-hand continuous and non-decreasing

(see Theorem A.5). Noting

𝐵≤𝜏𝐷/𝑁

𝔍

, we have

−𝐵′(𝑞)≥dim∞(𝔍)

. Hence, from

3.4. Coarse multifractal analysis

Proposition 3.18 we deduce

sup

𝛼≥dim∞(𝔍)

limsup

𝑛→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛)

𝛼log(2𝑛)≥sup

𝛼>−𝐵′(𝑞)

limsup

𝑘→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛𝑘)

𝛼log(2𝑛𝑘)

≥sup

𝛼>−𝐵′(𝑞)

liminf

𝑘→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛𝑘)

𝛼log(2𝑛𝑘)

≥−𝐵′(𝑞)𝑞+𝐵(𝑞)

−𝐵′(𝑞)≥

𝑏𝑞𝐷/𝑁

𝔍−𝜀

𝑏+𝜀.

Taking the limit 𝜀↓0 gives the assertion. □

Corollary 3.21. We have 𝐹𝑁

𝔍=ℎ𝔍=−1/𝛼𝔍=𝑞𝑁

𝔍. Further, if 𝐹𝐷

𝔍=𝐹𝑁

𝔍, then

𝐹𝐷

𝔍=ℎ𝔍=𝑞𝐷

𝔍=𝑞𝑁

𝔍.

By Proposition 3.20, we always have

𝐹𝐷/𝑁

𝔍=𝑞𝐷/𝑁

𝔍

. It raises the question under

which conditions

𝐹𝐷/𝑁

𝔍

exists as a limit. For this purpose, we establish the following

regularity conditions for 𝔍.

Deﬁnition 3.22. We deﬁne two notions of regularity.

We call

𝔍

Neumann multifractal-regular (N-MF-regular) if

𝐹𝑁

𝔍=𝐹𝑁

𝔍

and

Dirichlet multifractal-regular (D-MF-regular) if 𝐹𝐷

𝔍=𝐹𝑁

𝔍.

We call

𝔍

Dirichlet/Neumann partition function regular (D/N-PF-regular) if

•𝑞𝐷/𝑁

𝔍>

0 and

𝜏𝐷/𝑁

𝔍(𝑞)=liminf𝑛→∞ 𝜏𝐷/𝑁

𝔍,𝑛 (𝑞)

for all

𝑞∈𝑞𝐷/𝑁

𝔍−𝜀,𝑞𝐷/𝑁

𝔍

for some 𝜀>0, or

•𝑞𝐷/𝑁

𝔍>

0 and

𝜏𝐷/𝑁

𝔍𝑞𝐷/𝑁

𝔍=liminf𝑛→∞ 𝜏𝐷/𝑁

𝔍,𝑛 𝑞𝐷/𝑁

𝔍

and

𝜏𝐷/𝑁

𝔍

is dif-

ferentiable at 𝑞𝐷/𝑁

𝔍.

Corollary 3.23. If 𝔍is Neumann N-MF-regular, then 𝐹𝑁

𝔍=𝐹𝑁

𝔍=ℎ𝔍=ℎ𝔍=𝑞𝑁

𝔍.

Proof. This follows from Proposition 3.1 and Proposition 3.20. □

Proposition 3.24. If 𝔍is Dirichlet/Neumann PF-regular, then

𝐹𝐷/𝑁

𝔍=𝑞𝐷/𝑁

𝔍=𝐹𝐷/𝑁

𝔍.

3.4. Coarse multifractal analysis

In particular, we have

𝔍is N-PF-regular =⇒𝔍is N-MF-regular.

Proof.

Due to Proposition 3.1, we can restrict our attention to the case

𝑞𝐷/𝑁

𝔍>

First, we assume there exists 𝜀>0 such that

𝜏𝐷/𝑁

𝔍(𝑞)=liminf

𝑛→∞ 𝜏𝐷/𝑁

𝔍,𝑛 (𝑞)

for all

𝑞∈𝑞𝐷/𝑁

𝔍−𝜀,𝑞𝐷/𝑁

𝔍

and set

[𝑎,𝑏]≔−𝜕𝜏 𝐷/𝑁

𝔍𝑞𝐷/𝑁

𝔍

. Then by the convexity

𝜏𝐷/𝑁

𝔍

we ﬁnd for every

𝜀∈0,𝑞𝐷/𝑁

𝔍

an element

𝑞∈𝑞𝐷/𝑁

𝔍−𝜀,𝑞𝐷/𝑁

𝔍

such that

𝜏𝐷/𝑁

𝔍

is diﬀerentiable at

𝑞

with

−𝜏𝐷/𝑁

𝔍′(𝑞)∈ [𝑏,𝑏 +𝜀]

since the points where

𝜏𝐷/𝑁

𝔍

is diﬀerentiable on

(

,∞)

lie dense in

(

,∞)

. This follows from the fact that

𝜏𝐷/𝑁

𝔍

a decreasing function and that the left-hand derivative of the convex function

𝜏𝐷/𝑁

𝔍

is left-hand continuous and non-decreasing (see Theorem A.5). Then by Proposition

3.18 we have

sup

𝛼≥dim(𝔍)

liminf

𝑛→∞

log+N𝐷/𝑁

𝛼,𝔍(𝑛)

𝛼log(2𝑛)≥sup

𝛼>−𝜏𝐷/𝑁

𝔍′(𝑞)

liminf

𝑛→∞

log N𝐷/𝑁

𝛼,𝔍(𝑛)

𝛼log(2𝑛)

≥−𝜏𝐷/𝑁

𝔍′(𝑞)𝑞+𝜏𝐷/𝑁

𝔍(𝑞)

−𝜏𝐷/𝑁

𝔍′(𝑞)≥

𝑏𝑞𝐷/𝑁

𝔍−𝜀

𝑏+𝜀.

Taking the limit

𝜀→

0 proves the claim in this situation. The case that

𝜏𝐷/𝑁

𝔍

exists

as a limit in 𝑞𝐷/𝑁

𝔍and is diﬀerentiable at 𝑞𝐷/𝑁

𝔍is covered by Proposition 3.18. □

Corollary 3.25. If 𝔍is Neumann PF-regular, then

𝐹𝑁

𝔍=𝑞𝑁

𝔍=ℎ𝔍=ℎ𝔍.

Proof. This follows immediately from Proposition 3.1 and Proposition 3.24. □

Chapter 4

Spectral dimension and spectral

asymptotic for Kre˘

ın–Feller

operators for the one-dimensional

case

Throughout this section, we consider a ﬁnite Borel measure

𝜈

(

)

. Further, we

assume

card(supp(𝜈)) =∞

to exclude trivial cases. This chapter is devoted to study

the spectral dimension and spectral asymptotic of Kre

ın–Feller operators for the

case

𝑑=

1 and

Ω=(

)

. Since the spectral dimension with respect to Dirichlet

or Neumann boundary conditions is the same (see Remark 4.5), we will restrict

our attention to study Kre

ın–Feller operators with respect to Dirichlet boundary

conditions. A big advantage of the case

𝑑=

1 is that by Proposition A.17 the Sobolev

space

𝐻1((𝑎,𝑏))

is compactly embedded into

C([𝑎, 𝑏])

with

𝑎<𝑏

. Thus, in this

chapter for

𝑓∈𝐻1=𝐻1(

)

we will always pick the continuous representative.

Recall that the set of dyadic intervals of (0,1]is given by

D={(2−𝑛𝑘, 2−𝑛(𝑘+1)]:𝑘=0, . . . ,2𝑛−1 with 𝑛∈N}.

Most of the main results of this chapter rely heavily on the theory developed in

Chapter 3 applied to

𝔍𝜈(𝑄)= 𝔍𝜈,1,1(𝑄)=Λ(𝑄)𝜈(𝑄)

𝑄∈ D

, which naturally arises

as optimal embedding constant of the embedding of 𝐻1

0(𝑄)into 𝐿2

𝜈(𝑄):

𝑄

𝑓2d𝜈≤𝔍𝜈(𝑄)𝑄(∇𝑓)2dΛwith 𝑓∈𝐻1

0(𝑄), 𝑄 ∈ D.

In the following, we collect further simpliﬁcations that arise in the case 𝑑=1:

1. 𝛽𝐷

𝜈(𝑞)=𝛽𝑁

𝜈(𝑞),𝑞≥0, by Proposition 2.41.

4.1. Lower bounds for the spectral dimension

2. 𝜏𝐷/𝑁

𝑛,𝔍𝜈(𝑞)=𝛽𝐷/𝑁

𝑛(𝑞)−𝑞,𝑞≥0, 𝑛∈N.

3. 𝜏𝑁

𝔍𝜈(𝑞)=𝛽𝐷/𝑁

𝜈(𝑞)−𝑞,𝑞≥0. In particular, we have 𝑞𝐷

𝔍𝜈=𝑞𝑁

𝔍𝜈.

Since, for all

𝛼>

𝑛∈N

, we have

N𝐷

𝔍𝜈,𝛼 (𝑛)≤ N𝑁

𝔍𝜈,𝛼 (𝑛)≤ N𝐷

𝔍𝜈,𝛼 (𝑛)+

2 and

it follows 𝐹𝑁

𝔍𝜈=𝐹𝐷

𝔍𝜈and 𝐹𝑁

𝔍𝜈=𝐹𝐷

𝔍𝜈.

This justiﬁes the following simpliﬁed notation:

𝛽𝜈≔𝛽𝐷/𝑁

𝜈, 𝑞𝔍𝜈≔𝑞𝐷/𝑁

𝔍𝜈, 𝐹 𝔍𝜈≔𝐹𝐷/𝑁

𝔍𝜈and 𝐹𝔍𝜈

≔𝐹𝐷/𝑁

𝔍𝜈.

The chapter is structured as follows. In Section 4.1, we establish lower bounds for

the lower and upper spectral dimension in terms of the lower and upper optimized

coarse multifractal dimension with respect to

𝔍𝜈

. For this reason, we introduce the

new notion of the lower and upper

𝑚

-reduced partition

𝜈

-entropy. In Section 4.2,

we present upper bounds of

𝑠𝐷

𝜈

and

𝑠𝐷

𝜈

in terms of

𝑞𝔍𝜈

and

dim𝑀(𝜈)/(

+dim∞(𝜈))

respectively. In Section 4.3, we present our main results – we prove that the upper

spectral dimension is given by

𝑞𝔍𝜈

which can be geometrically interpreted as the

unique intersection point of the

𝐿𝑞

-spectrum and the identity map, provided

𝑞𝔍𝜈>

Further, we impose regularity conditions to ensure the existence of the spectral

dimension and present general bounds in terms of fractal dimensions. In Section

4.4, we illustrate our general results, developed in Section 4.3, with a number of

examples. More precisely, in Section 4.4.1, we compute the spectral dimension of

weak Gibbs measures with or without overlap and obtain some reﬁnement estimates

under the assumption of the OSC. Moreover, in Section 4.4.2, we discuss an example

for which the spectral dimension does not exist. Finally, we conclude by computing

the spectral dimension for a class of purely atomic measures in Section 4.4.3.

4.1 Lower bounds for the spectral dimension

We start with the deﬁnition of an auxiliary target quantity.

Let

Π0

denote the set of ﬁnite disjoint collections of subintervals

𝐼

(

]

, and for

𝑚>1 and 𝑥>0, set

N𝐿

𝑚(𝑥)≔sup card (𝑃):𝑃∈Π0|min

𝐼∈𝑃𝜈𝐼1/𝑚Λ𝐼1/𝑚≥4

𝑥(𝑚−1).

Recall from Section 2.2.6 that

𝐼1/𝑚⊂𝐼

denotes the interval of length

Λ(𝐼)/𝑚

centered in

𝐼

. Then the lower and upper

𝑚

-reduced

𝜈

-partition entropy is given by

ℎ𝑚

𝜈≔liminf

𝑥→∞

log N𝐿

𝑚(𝑥)

log(𝑥)and ℎ𝑚

𝜈≔limsup

𝑥→∞

log N𝐿

𝑚(𝑥)

log(𝑥).

4.1. Lower bounds for the spectral dimension

Proposition 4.1. For all 𝑥>0, we have

N𝐿

𝑚(𝑥)≤𝑁𝐷

𝜈(𝑥).

Proof. Let 𝑃∈Π0such that

min

𝐼∈𝑃𝜈𝐼1/𝑚Λ𝐼1/𝑚≥4/(𝑥(𝑚−1))

and write

𝑃≔𝐼1, . . . ,𝐼 card(𝑃),𝐼𝑖=[𝑎𝑖,𝑏𝑖],and 𝐼𝑖1/𝑚=[𝑐𝑖,𝑑𝑖], 𝑖 =1, . . . , card(𝑃).

For each 𝑖=1, . . . ,card(𝑃), we deﬁne

𝑓𝑖(𝑦)≔𝑦−𝑎𝑖

𝑐𝑖−𝑎𝑖

[𝑎𝑖,𝑐𝑖)(𝑦)+

[𝑐𝑖,𝑑𝑖](𝑦)+ 𝑏𝑖−𝑦

𝑏𝑖−𝑑𝑖

(𝑑𝑖,𝑏𝑖](𝑦), 𝑦 ∈ [𝑎,𝑏],

which is an element of 𝐻1

0. Notice, by the deﬁnition of ⟨·⟩1/𝑚, we have

𝑐𝑖−𝑎𝑖=𝑏𝑖−𝑑𝑖=(𝑏𝑖−𝑎𝑖)(1−1/𝑚)/2.

Hence,

(0,1)∇𝑓2

𝑖(𝑦)dΛ(𝑦)

(0,1)𝑓2

𝑖(𝑦)d𝜈(𝑦)≤

𝑐𝑖−𝑎𝑖+1

𝑏𝑖−𝑑𝑖

𝜈𝐼1/𝑚

=4/(𝑚−1)

𝜈𝐼1/𝑚Λ𝐼1/𝑚

≤𝑥.

Since the intervals

(𝐼𝑖)𝑖

are disjoint, the

(𝑓𝑖)𝑖

are mutually orthogonal both in

𝐿2

𝜈

and in

𝐻1

, and we obtain that

span (𝑓𝑖:𝑖=1, . . ., card(𝑃))

is a

card(𝑃)

-dimensional

subspace of 𝐻1

0. Thus, we deduce from Lemma 2.19,

card(𝑃) ≤ 𝑁𝐷

𝜈(𝑥).

Taking the supremum over all 𝑃∈Π0with

min

𝐼∈𝑃𝜈𝐼1/𝑚Λ𝐼1/𝑚≥4/(𝑥(𝑚−1))

proves the claim. □

4.1. Lower bounds for the spectral dimension

Lemma 4.2. Let 𝛼>0. For every 𝑥>22+𝛼, we have

𝑁𝐷

𝜈(𝑥) ≥ N𝐿

3(𝑥)≥NN

𝔍𝜈,𝛼 𝑛𝛼

𝑥

3−3

with 𝑛𝛼

𝑥≔log 𝑥

2/(log(2)𝛼), and with 𝑥𝛼

𝑛≔2𝑛𝛼+1, for every 𝑛∈N, we have

𝑁𝐷

𝜈(𝑥𝛼

𝑛) ≥ NN

𝔍𝜈,𝛼 (𝑛)

3−3.

Proof.

For ﬁxed

𝑛∈N, 𝛼 >

0, let

(𝑐1,𝑑1], . . . , 𝑐N𝑁

𝔍𝜈,𝛼 (𝑛),𝑑 N𝑁

𝔍𝜈,𝛼 (𝑛)

denote the inter-

vals of

𝑀𝑁

𝔍𝜈,𝑛 (𝛼)

ordered in the natural way, i.e.

𝑐𝑖<𝑑𝑖≤𝑐𝑖+1<𝑑𝑖+1

. Further, we

deﬁne

𝐷𝑛,𝑖 ≔𝑐2+3𝑖−1

2𝑛,𝑐2+3𝑖∪(𝑐2+3𝑖,𝑑2+3𝑖]∪𝑑2+3𝑖,𝑑2+3𝑖+1

2𝑛

with

𝑖=

, . . . , N𝑁

𝔍𝜈,𝛼 (𝑛)/3−2−

1. Clearly, we have

𝐷𝑛,𝑖 ∩𝐷𝑛,𝑗 =∅

for all

𝑖≠𝑗

and

𝜈𝐷𝑛,𝑖 1/3Λ𝐷𝑛,𝑖 1/3≥𝜈((𝑐2+3𝑖,𝑑2+3𝑖])Λ((𝑐2+3𝑖,𝑑2+3𝑖]) ≥2−𝑛𝛼 .

Hence, for 𝑛𝛼

𝑥≔log 𝑥

2/(log(2)𝛼))and 𝑥>22+𝛼, we have

𝜈𝐷𝑛𝛼

𝑥,𝑖 1/3Λ𝐷𝑛𝛼

𝑥,𝑖 1/3≥2

𝑥.

In tandem with Proposition 4.1, we deduce

𝑁𝐷

𝜈(𝑥) ≥ N𝐿

3,𝜈 (𝑥)=sup card (𝑃):𝑃∈Π0: min

𝐼∈𝑃𝜈𝐼1/3Λ𝐼1/3≥2

𝑥

≥N𝑁

𝔍𝜈,𝛼 𝑛𝛼

𝑥

3−2≥N𝑁

𝔍𝜈,𝛼 𝑛𝛼

𝑥

3−3.□

Now, we can give a lower bound on

𝑠𝐷

𝜈

and

𝑠𝐷

𝜈

in terms of the lower and upper

optimize coarse multifractal dimension as well as the lower and upper

𝑚

-reduced

𝜈-partition entropy.

Proposition 4.3. As a general lower (upper) bound for the lower (upper) spectral

dimension for all 𝑚∈(1,3], we have

𝐹𝔍𝜈≤ℎ𝑚

𝜈≤𝑠𝐷

𝜈and 𝐹𝔍𝜈≤ℎ𝑚

𝜈≤𝑠𝐷

𝜈.

Proof.

First note that Proposition 4.1 gives

ℎ𝑚

𝜈≤𝑠𝐷

𝜈

and

ℎ𝑚

𝜈≤𝑠𝐷

𝜈

for all

𝑚>

1. Let

4.2. Upper bounds for the spectral dimension

𝑛𝛼

𝑥=log2𝑥

2/𝛼and 𝑥>22+𝛼, then by Lemma 4.2 for every 𝛼>0, we have

log 3N𝐿

3(𝑥)+3

log(𝑥)≥

log+N𝑁

𝔍𝜈,𝛼 𝑛𝛼

𝑥

log(𝑥).

Hence, for all 1 <𝑚≤3

ℎ𝑚

𝜈≥ℎ3

𝜈=liminf

𝑥→∞

log+3N𝐿

3(𝑥)+3

log(𝑥)≥lim inf

𝑥→∞

log+N𝑁

𝔍𝜈,𝛼 𝑛𝛼

𝑥

log(𝑥)

=liminf

𝑥→∞

log+N𝑁

𝔍𝜈,𝛼 𝑛𝛼

𝑥

𝛼log 2log2(𝑥

2)/𝛼+log (2)

≥liminf

𝑥→∞

log+N𝑁

𝔍𝜈,𝛼 𝑛𝛼

𝑥

𝛼log(2𝑛𝛼

𝑥)+log (2)(1+𝛼)

=liminf

𝑥→∞

log+N𝑁

𝔍𝜈,𝛼 𝑛𝛼

𝑥

𝛼log(2𝑛𝛼

𝑥)≥liminf

𝑛→∞

log+N𝑁

𝔍𝜈,𝛼 (𝑛)

𝛼log(2𝑛),

which implies ℎ𝑚

𝜈≥𝐹𝔍𝜈. We also have for 𝑥𝜂

𝑚≔2𝑚𝜂+1with 𝑚∈Nand 𝜂>0,

ℎ𝑚

𝜈≥ℎ3

𝜈≥limsup

𝑚→∞

log 3N𝐿

3𝑥𝜂

𝑚+3

log(𝑥𝜂

𝑚)≥limsup

𝑚→∞

log+N𝑁

𝔍𝜈,𝜂 𝑛𝜂

𝑥𝜂

𝑚

log(𝑥𝜂

𝑚)

≥limsup

𝑚→∞

log+N𝑁

𝔍𝜈,𝜂 𝑛𝜂

𝑥𝜂

𝑚

𝜂log 2𝑛𝜂

𝑥𝜂

𝑚=limsup

𝑚→∞

log+N𝑁

𝔍𝜈,𝜂 (𝑚)

𝜂log(2𝑚),

where we used 𝑛𝜂

𝑥𝜂

𝑚

=𝑚. Thus, ℎ𝑚

𝜈≥𝐹𝔍𝜈.□

4.2 Upper bounds for the spectral dimension

We begin with a slight generalization of

M𝔍𝜈(𝑥)=inf card (𝑃):𝑃∈Π𝜈|max

𝐼∈𝑃𝜈(𝐼)Λ(𝐼)<1/𝑥

by allowing left half-open intervals which do not necessarily consist of dyadic

intervals. This will be useful for the computation of the spectral dimension and the

determination of the spectral asymptotic of speciﬁc examples (see Section 4.4.1 and

Section 4.4.3).

4.2. Upper bounds for the spectral dimension

Let

denote the set of left half-open intervals contained in

(

]

. We call

𝑃⊂Γ

𝜈-partition of Qof ﬁnitely many left half-open intervals if

•card(𝑃)<∞,

•𝜈(𝐼∈𝑃𝐼)=𝜈(Q),

•𝐼1∩𝐼2=∅for all 𝐼1, 𝐼2∈𝑃with 𝐼1≠𝐼2,

•𝜈(𝐼)>0 for all 𝐼∈𝑃.

Let Πdenote the set of all 𝜈-partitions of left half-open intervals of Qand



M𝔍𝜈(𝑥)≔inf card (𝑃):𝑃∈Π|max

𝐼∈𝑃𝜈(𝐼)Λ(𝐼)<1/𝑥

for

𝑥>

/𝜈(Q)

. Before stating our main result of this section, we need some

preparations, where we follow [Nga11]. Let

((𝑎𝑖,𝑎𝑖+1])𝑖=0,...,𝑛

be a partition with

𝑎𝑖<𝑎𝑖+1of Qand 𝑛∈N. Consider the following closed subspace of 𝐻1

F≔𝑓∈𝐻1

0:𝑓(𝑎𝑖)=0, 𝑖 =1, . . ., 𝑛

and deﬁne the following equivalence relation on

𝐻1

, by

𝑥∼𝐻1

0/F 𝑦

if and only if

𝑥−𝑦∈ F. The associated quotient space is given by

𝐻1

0/F ≔[𝑢]𝐻1

0/F :𝑢∈𝐻1

0,

where

[𝑢]𝐻1

0/F

denotes the equivalence class of

𝑢∈𝐻1

with respect to

∼

. Further,

we deﬁne the addition and scalar multiplication on

𝐻1

0/F

in the standard way. For

𝑖=1, . . ., 𝑛, we deﬁne the following elements of 𝐻1

𝑓𝑖:𝑥↦→ 𝑥−𝑎𝑖−1

𝑎𝑖−𝑎𝑖−1

[𝑎𝑖−1,𝑎𝑖)(𝑥)+ 𝑎𝑖+1−𝑥

𝑎𝑖+1−𝑎𝑖

[𝑎𝑖,𝑎𝑖+1](𝑥).

Note that for all

𝑖, 𝑗 =

, . . . ,𝑛

we have

𝑓𝑖(𝑎𝑗)=𝛿𝑖 𝑗

, where

𝛿𝑖 𝑗

is the Kronecker delta,

i.e.

𝛿𝑖 𝑗 ≔1,for 𝑖=𝑗,

0,else.

Consequently, for any 𝑢∈𝐻1

0we have

𝑢−

𝑛



𝑖=1

𝑢(𝑎𝑖)𝑓𝑖∈ F.

4.2. Upper bounds for the spectral dimension

This implies

𝐻1

0/F =span [𝑓𝑖]𝐻1

0/F :𝑖=1, . . . ,𝑛 ,

dim 𝐻1

0/F =𝑛.

Now we are in the position to establish a link between

𝑁𝐷

𝜈

and



M𝔍𝜈

, allowing us to

use the results of Chapter 3.

Proposition 4.4. For all 𝑥>0, we have

𝑁𝐷

𝜈(𝑥) ≤ 2

M𝔍𝜈(𝑥)+1≤2M𝔍𝜈(𝑥)+1

and

𝑁𝐷

𝜈(𝑥) ≤ 

M𝔍𝜈(5𝑥) ≤ M𝔍𝜈(5𝑥)

Proof.

Here, we follow the proof of [Kig01, Theorem 4.1.7]. Let

((𝑎𝑖,𝑎𝑖+1])𝑖=0,...,𝑛

be a partition with 𝑎𝑖<𝑎𝑖+1of Qand deﬁne as above

F=𝑓∈𝐻1

0:𝑓(𝑎𝑖)=0, 𝑖 =1, . . ., 𝑛.

Then,

dim 𝐻1

0/F =𝑛

and for any subspace of

𝐿⊂𝐻1

with

dim(𝐿)=𝑖

, there exists

a linear, injective map Φ:𝐿/(𝐿∩F ) →𝐻1

0/𝐻1

0∩F=𝐻1

0/F with

Φ(ℓ+𝐿∩F) ≔ℓ+F, ℓ ∈𝐿.

Thus, the rank–nullity theorem yields

dim (𝐿∩F)≥dim(𝐿) −𝑁 ,

implying

𝑖−𝑁≤dim (𝐿∩F)≤𝑖.

Hence, if

L𝑖, 𝑗 ≔𝐿:𝐿is a subspace of 𝐻1

0,dim(𝐿)=𝑖, dim(𝐿∩ F) =𝑗,

then we obtain

𝐿:𝐿is a subspace of 𝐻1

0,dim(𝐿)=𝑖+𝑁=

𝑁



𝑘=0L𝑖+𝑁 ,𝑖+𝑘.

Now with



𝜆𝑖≔inf sup ⟨𝜓,𝜓 ⟩𝐻1

⟨𝜓,𝜓 ⟩𝜈

:𝜓∈𝐺\{0}:𝐺<𝑖F,⟨·,·⟩𝐻1

0,

4.2. Upper bounds for the spectral dimension

for every 𝐿∈ L𝑖+𝑁,𝑖 +𝑘with 𝑘∈ {0, . . ., 𝑁 }, we ﬁnd that

sup ⟨𝜓,𝜓 ⟩𝐻1

⟨𝜓,𝜓 ⟩𝜈

:𝜓∈𝐿\{0}≥sup ⟨𝜓,𝜓 ⟩𝐻1

⟨𝜓,𝜓 ⟩𝜈

:𝜓∈𝐿∩F \{0}

≥

𝜆𝑖+𝑘≥

𝜆𝑖.

We deduce from the min-max principle stated in Proposition 2.17

𝜆𝐷

𝑖+𝑁 ,𝜈 ≥

𝜆𝑖,

which implies

𝑁𝐷

𝜈(𝑥) ≤ card {𝑖∈N:

𝜆𝑖≤𝑥}+𝑁

. Furthermore, using Lemma 2.2,

for all 𝑢∈ F , we have

𝑢2d𝜈=

𝑛



𝑖=0,

𝜈((𝑎𝑖,𝑎𝑖+1])>0(𝑎𝑖,𝑎𝑖+1]

𝑢2d𝜈

≤max

𝑖=0,...,𝑛 𝜈((𝑎𝑖, 𝑎𝑖+1]) Λ((𝑎𝑖,𝑎𝑖+1])(0,1)(∇𝑢)2dΛ.

Now, assume max𝑖=0,...,𝑛 𝜈((𝑎𝑖, 𝑎𝑖+1])Λ((𝑎𝑖,𝑎𝑖+1]) <1/𝑥. Then,



𝜆1≥max

𝑖=0,...,𝑛 𝜈((𝑎𝑖, 𝑎𝑖+1]) Λ((𝑎𝑖,𝑎𝑖+1])−1

>𝑥.

This implies

𝑁𝐷

𝜈(𝑥) ≤ 𝑛.

Taking the inﬁmum over all 𝑃∈Πwith max𝐼∈𝑃𝜈(𝐼)Λ(𝐼)<1/𝑥yields

𝑁𝐷

𝜈(𝑥) ≤ 2

M𝔍𝜈(𝑥)+1.

The second inequality follows from the fact that

Π𝔍𝜈⊂Π

. Similarly, the second

claim follows from Corollary 2.3 by replacing Fwith

{𝑓∈𝐻1

0(0,1):𝐼

𝑓dΛ=0, 𝐼 ∈𝑃}

and the fact dim(𝐻1

0/F) =card(𝑃).□

Remark 4.5.Similarly, one can show that

dim 𝐻1/𝐻1

0=

2. In tandem with the

Poincar

e inequality

(PI)

it follows in a similar way as in Proposition 4.4 that

𝑠𝐷

𝜈=𝑠𝑁

𝜈

and 𝑠𝐷

𝜈=𝑠𝑁

𝜈.

4.2. Upper bounds for the spectral dimension

Corollary 4.6. We have

𝑠𝐷

𝜈≤ℎ𝔍𝜈=𝑞𝔍𝜈=inf {𝑞≥0 : 𝛽𝜈(𝑞)−𝑞≤0}≤dim𝑀(𝜈)

dim𝑀(𝜈)+1and 𝑠𝐷

𝜈≤ℎ𝔍𝜈.

Proof.

The ﬁrst and second inequality follows from Proposition 3.1 and Proposition

4.4 applied to 𝔍 = 𝔍𝜈. Moreover, notice that the convexity of 𝛽𝜈implies

𝛽𝜈(𝑞) ≤ dim𝑀(𝜈)(1−𝑞), 𝑞 ∈ [0,1].

This yields

inf {𝑞≥0 : 𝛽𝜈(𝑞)−𝑞≤0}≤dim𝑀(𝜈)

dim𝑀(𝜈)+1.□

Remark 4.7.The case

dim𝑀(𝜈)=

0 immediately gives

𝑠𝐷

𝜈=

0. If we use more

information on

𝛽𝜈

for the case

dim𝑀(𝜈)>

0, we ﬁnd a better upper bound; namely,

with 𝑞1≔inf {𝑠:𝛽𝜈(𝑠)≤0}, we have

𝑠𝐷

𝜈≤𝑞𝔍𝜈≤𝑞1dim𝑀(𝜈)

𝑞1+dim𝑀(𝜈).

The following proposition complements the connection of the Minkowski dimension

by establishing an upper bound of the lower spectral dimension in terms of the lower

Minkowski dimension dim𝑀(𝜈)of the support of 𝜈and the ∞-dimension of 𝜈.

Proposition 4.8. We always have

𝑠𝐷

𝜈≤ℎ𝔍𝜈≤dim𝑀(𝜈)

1+dim∞(𝜈).

Proof. This follows from Proposition 3.13 applied to 𝐽=𝜈and 𝑎=1. □

Proposition 4.9. Under the assumption that there exists a subsequence

(𝑛𝑘)𝑘

and a

constant 𝐾>0such that for all 𝑘∈N

max

𝐶∈D𝑛𝑘

𝜈(𝐶)𝑞𝑛𝑘≤𝐾

2𝛽𝑛𝑘(0)𝑛𝑘

𝐶∈D𝑛𝑘

𝜈(𝐶)𝑞𝑛𝑘

and

lim𝑘→∞𝑞𝑛𝑘=lim inf𝑛→∞𝑞𝑛

, where

𝑞𝑛≥

0is the unique solution to

𝛽𝑛(𝑞𝑛)=𝑞𝑛

we have

𝑠𝐷

𝜈≤ℎ𝔍𝜈≤liminf

𝑛→∞ 𝑞𝑛≤liminf

𝑛→∞

𝛽𝑛(0)

1+𝛽𝑛(0)=dim𝑀(𝜈)

1+dim𝑀(𝜈).

4.3. Main results

Proof.

Due to Proposition 4.8, we only have to consider the case

dim𝑀(𝜈)>

which implies

liminf𝑘→∞ 𝑞𝑛𝑘>

0. Now, Proposition 3.3 applied to

𝔍 = 𝔍𝜈

yields

the claim. □

4.3 Main results

In this section, we connect Proposition 4.3, Corollary 4.6, and the general results of

Chapter 3 to prove the main results of this chapter.

4.3.1

Upper spectral dimension and lower bounds for the lower spec-

tral dimension

In this , we compute the upper spectral dimension and obtain various lower and

upper bounds of the lower and upper spectral dimension.

Theorem 4.10. For all 1<𝑚≤3, we have

𝐹𝔍𝜈≤ℎ𝑚

𝜈≤𝑠𝐷

𝜈≤ℎ𝔍𝜈≤ℎ𝔍𝜈=ℎ𝑚

𝜈=𝑠𝐷

𝜈=𝑞𝔍𝜈=𝐹𝔍𝜈.(4.3.1)

In particular,

𝑠𝐷

𝜈=𝑞𝔍𝜈≤dim𝑀(𝜈)

dim𝑀(𝜈)+1≤1/2,

and the following necessary and suﬃcient conditions ensuring the existence of

spectral dimension:

𝑠𝐷

𝜈=𝑠𝐷

𝜈=⇒ℎ𝔍𝜈=ℎ𝔍𝜈=𝑠𝐷

𝜈and sup

𝑚>1

ℎ𝑚

𝜈=ℎ𝔍𝜈=⇒𝑠𝐷

𝜈=𝑠𝐷

𝜈=ℎ𝔍𝜈.

Proof. By Proposition 4.3 and Corollary 4.6, we have

𝐹𝔍𝜈≤ℎ𝑚

𝜈≤𝑠𝐷

𝜈≤ℎ𝔍𝜈≤𝑞𝔍𝜈

and

𝐹𝔍𝜈≤ℎ𝑚

𝜈≤𝑠𝐷

𝜈≤ℎ𝔍𝜈.

Moreover, Proposition 3.20 applied to

𝔍 = 𝔍𝜈

yields

𝐹𝔍𝜈=𝑞𝔍𝜈

, and thus, the

equalities in (4.3.1) are obtained. □

Remark 4.11.Theorem 4.10 shows that if the spectral dimension exists, then it

is given by purely measure-geometric data, which is encoded in the

𝜈

-partition

entropy

ℎ𝔍𝜈=ℎ𝔍𝜈

. We call

𝜈

regular if

sup𝑚>1ℎ𝑚

𝜈=ℎ𝔍𝜈

, in which case the spectral

dimension exists. If

𝔍𝜈

is Neumann MF-regular (i.e.

𝐹𝔍𝜈=𝐹𝔍𝜈

), then in the above

4.3. Main results

chain of inequalities

(4.3.1)

we have everywhere equality and especially

𝜈

is regular.

Moreover, if for some 𝑚>1 we have ℎ𝑚

𝔍𝜈≥1/2, then 𝑠𝐷

𝜈=1/2=ℎ𝔍𝜈.

4.3.2 Regularity results

Here, we investigate the question under which conditions the spectral dimension

exists. By combining the results of Section 3.4 and Theorem 4.10, we show that the

regularity conditions imposed in Deﬁnition 3.22 leads to the following suﬃcient

condition for existence of the spectral dimension.

Corollary 4.12. If

𝔍𝜈

is Neumann partition function regular, then the spectral

dimension exists and is given by 𝑠𝐷

𝜈=𝑞𝔍𝜈.

Proof.

This follows from Corollary 3.23 applied to

𝔍 = 𝔍𝜈

and Theorem 4.10.

□

Remark 4.13.Observe that due to

𝜏𝑁

𝔍𝜈,𝑛 (𝑞)=𝛽𝑛(𝑞)−𝑞

for all

𝑛∈N

𝑞≥

0, we infer

that

𝔍𝜈

is Dirichlet partition regular if and only if

𝛽𝜈

exists as a limit in

𝑞𝔍𝜈

and

𝛽𝜈

is diﬀerentiable at

𝑞𝔍𝜈

, or

𝛽𝜈(𝑞)=liminf𝑛→∞ 𝛽𝑛(𝑞)

for

𝑞∈𝑞𝔍𝜈−𝜀,𝑞𝔍𝜈

for some

𝜀>0.

Proposition 4.14. If for

𝑞∈ [

]

we have

𝛽𝜈(𝑞)=lim𝑛→∞ 𝛽𝑛(𝑞)

and

−𝜕𝛽𝜈(𝑞)=

[𝑎,𝑏], then

𝑎𝑞 +𝛽𝜈(𝑞)

1+𝑏≤𝑠𝐷

𝜈.

Proof. By Corollary 3.19 and Proposition 4.3 , we have

𝑎𝑞 +𝛽𝜈(𝑞)

1+𝑏≤sup

𝑡>𝑏

liminf

𝑛→∞

log N𝑁

𝔍𝜈,𝑡 (𝑛)

(1+𝑡)log(2𝑛)≤𝐹𝔍𝜈≤𝑠𝐷

𝜈,

where we used the fact that −𝜕𝜏𝔍𝜈(𝑞)=(𝑎+1,𝑏 +1].□

Remark 4.15.In the case that

𝛽𝜈(𝑞)=lim𝑛→∞ 𝛽𝑛𝑞𝔍𝜈

and

𝛽𝜈

is diﬀerentiable at

𝑞𝔍𝜈

, we infer

𝑞𝔍𝜈≤𝑠𝐷

𝜈

and hence obtain a direct proof of the regularity statement,

namely, 𝑞𝔍𝜈=𝑠𝐷

𝜈=𝑠𝐷

𝜈.

Also for measures without an absolutely continuous part we have the following

rigidity result in terms of reaching the maximum possible value 1

2 of the spectral

dimension.

Corollary 4.16. We have the following rigidity results:

1. If 𝑠𝐷

𝜈=1/2, then 𝛽𝜈(𝑞)=1−𝑞for all 𝑞∈ [0,1].

4.3. Main results

𝛽𝜈(𝑞)=lim𝑛→∞ 𝛽𝑛(𝑞)=

−𝑞

for some

𝑞∈(0,1)

, then

𝛽𝜈(𝑞)=

−𝑞

for all

𝑞∈ [0,1]and 𝑠𝐷

𝜈=1/2.

Proof.

𝑠𝐷

𝜈=

2 then it follows from Theorem 4.10 that 1

=𝑠𝜈=𝑞𝔍𝜈≤

The convexity of

𝛽𝜈

and the fact that

𝛽𝜈(1)=

0 and

𝛽𝜈(0)≤

1 forces

𝛽𝜈(𝑞)=

−𝑞

for all

𝑞∈ [

]

. The second statement is an immediate consequence of Proposition

4.14 by observing that, as in case (1), by convexity we have

𝛽𝜈(𝑞)=

−𝑞

for all

𝑞∈ [

]

. This implies the diﬀerentiability of

𝛽𝜈

in the particular point

𝑞∈(0,1)

where by our assumption

𝛽𝜈(𝑞)=lim𝑛→∞ 𝛽𝑛(𝑞)

. Now, applying Proposition 4.14

gives

2=𝑞+1−𝑞

1+1≤𝑠𝜈.

Since we always have 𝑠𝐷

𝜈≤1/2, our claim follows. □

4.3.3 General bounds in terms of fractal dimensions

As a consequence of Theorem 4.10 and Proposition 4.14 we improve the known

general upper bound of the spectral dimension of 1

2 as obtained in [BS66] in terms

of the upper Minkowski dimension. Furthermore, we obtain a general lower bound

of 𝑠𝐷

𝜈in terms of the left and right-hand derivative of 𝛽𝜈.

Corollary 4.17. For the lower and upper spectral dimension, we have the following

general lower and upper bounds depending on the topological support of

𝜈

, namely

dim𝑀(𝜈), and left and right derivative of 𝛽𝜈at 1:

−𝜕+𝛽𝜈(1)

1−𝜕−𝛽𝜈(1)≤𝑠𝐷

𝜈≤𝑠𝐷

𝜈≤dim𝑀(𝜈)

1+dim𝑀(𝜈)≤1

and

𝑠𝐷

𝜈=dim𝑀(𝜈)

1+dim𝑀(𝜈)⇐⇒ −𝜕−𝛽𝜈(1)=dim𝑀(𝜈).

Proof.

The ﬁrst inequalities follow from Corollary 4.6 and Proposition 4.14, using

the fact that

𝛽𝜈

always exists as a limit at 1. The last claim follows from the

fact that

𝛽𝜈

is linear on

[0,1]

if and only if

−𝜕−𝛽𝜈(

)=dim𝑀(𝜈)

and in this case

𝑞𝔍𝜈=dim𝑀(𝜈)/1+dim𝑀(𝜈).□

Remark 4.18.It is worth pointing out that these bounds have been ﬁrst observed

in the self-similar case under the open set condition in [SV95, p. 245]. In this

case the Minkowski dimension and the Hausdorﬀdimension of

supp(𝜈)

coincide as

well as

𝛽𝜈

is diﬀerentiable at 1 and

𝛽′

𝜈(

)

coincides with the Hausdorﬀdimension

𝜈

(see for instance [Heu07]). Furthermore, note that in the case that

𝜈

has an

100

4.4. Examples

atomic part, we always have

𝜕+𝛽𝜈(

0 (see Fact 2.32). Hence, the lower bound

−𝜕+𝛽𝜈(1)/(1−𝜕−𝛽𝜈(1))is only meaningful in the case of atomless measures.

Regarding Kac’s question if

𝛽𝜈

is diﬀerentiable at 1, then the spectral dimension is

determined by fractal-geometric quantities as follows

dim𝐻(𝜈)=𝛽′

𝜈(1) ≤ 𝑠𝐷

𝜈

1−𝑠𝐷

𝜈≤𝑠𝐷

𝜈

1−𝑠𝐷

𝜈≤dim𝑀(𝜈),

where dim𝐻(𝜈)≔inf{dim𝐻(𝐴):𝜈(𝐴∁)=0}.

4.4 Examples

4.4.1 C1-cIFS and weak Gibbs measures

In this section, we deﬁne weak-Gibbs measures with respect to not necessary linear

iterated function systems. We start by pointing out some simpliﬁcations of the notion

-cIFS in the one-dimension setting. Let

Φ≔{𝑇𝑖:[0,1]→[0,1]:𝑖=1, . . . ,𝑛 }

𝑛∈N

, be a

-cIFS as deﬁned in Deﬁnition 2.46. Observe that for every

𝜀>

0 we

can extend each

𝑇𝑖

to an injective contracting

-map

𝑇𝑖

(−𝜀,

+𝜀) → (−𝜀,

+𝜀)

via



𝑇𝑖(𝑥)≔









𝑇𝑖(0)+𝑇′

𝑖(0)𝑥, 𝑥 ∈ [−𝜀, 0),

𝑇𝑖(𝑥), 𝑥 ∈ [0,1],

𝑇𝑖(1)+𝑇′

𝑖(1)(𝑥−1), 𝑥 ∈ (1,1+𝜀].

Moreover, the notion of conformal maps becomes trivial in the one-dimensional

setting. This gives rise to the following equivalent conditions to be a C1-cIFS:

1. for all 𝑗∈𝐼we have 𝑇𝑗∈ C1([0,1])and

0<inf

𝑥∈[0,1]𝑇′

𝑗(𝑥)≤sup

𝑥∈[0,1]𝑇′

𝑗(𝑥)<1,

2. Φ

is non-trivial, i.e. there is more than one contraction and the

𝑇𝑖

’s do not

share a common ﬁxed point.

Deﬁnition 4.19. Let

Φ≔{𝑇𝑖:[0,1]→[0,1]:𝑖=1, . . . ,𝑛 }

𝑛∈N

, be a

-cIFS.

If additionally the

𝑇1, . . ., 𝑇𝑛

are

C1+𝛾

-maps with

𝛾∈(0,1)

, we call the system a

C1+𝛾

-conformal iterated function system (

C1+𝛾

-cIFS). Here

C1+𝛾

denotes the set of

diﬀerentiable maps with 𝛾-H¨

older continuous derivative.

101

4.4. Examples

In the remainder of this section we ﬁx a C1-IFS

Φ≔{𝑇𝑖:[0,1]→[0,1]:𝑖=1, . . . ,𝑛 }.

Recall that the unique non-empty compact invariant set

K ⊂ [0,1]

of a

- cIFS

is given by

K=

𝑖∈𝐼

𝑇𝑖(K)

with

𝐼={

, . . . ,𝑛 }

. Let

B𝐼N

denote the Borel

𝜎

-algebra of

𝐼N

. Note that

B𝐼N

is generated by the set of cylinder sets of arbitrary lengths. The set of

𝜎

-invariant

probability measures on

B𝐼N

is denoted by

M𝜎𝐼N

, where the measure

𝜇

called 𝜎-invariant if 𝜇=𝜇◦𝜎−1.

Deﬁnition 4.20. Let

C𝐼N

denote the space of continuous real valued functions on

𝐼N. For 𝑓∈ C 𝐼N,𝛼∈ (0,1), and 𝑛∈N0deﬁne

var𝑛(𝑓)≔sup |𝑓(𝜔)− 𝑓(𝑢)|:𝜔, 𝑢 ∈𝐼Nand 𝜔𝑖=𝑢𝑖for all 𝑖∈{1, . . ., 𝑛},

|𝑓|𝛼≔sup

𝑛≥0

var𝑛(𝑓)

𝛼𝑛and F𝛼≔𝑓∈ C 𝐼N:|𝑓|𝛼<∞.

Elements of

F𝛼

are called

𝛼

-H¨older continuous functions on

𝐼N

. Furthermore,

the Birkhoﬀsum of

𝑓

is deﬁned by

𝑆𝑛𝑓(𝑥)≔𝑛−1

𝑘=0𝑓◦𝜎𝑘(𝑥)

𝑥∈𝐼N

𝑛∈N

, and

𝑆0𝑓≔0.

Deﬁnition 4.21 (Geometric potential function).The geometric potential function

with respect to Φis given by

𝜑(𝜔1𝜔2···) ≔log 𝑇′

𝜔1(𝜋(𝜔2𝜔3···)).

Remark 4.22.We will make use of the following relation between

𝜑

and

𝑇′

𝜔

with

𝜔=𝜔1···𝜔𝑛∈𝐼𝑛, 𝑛 ∈N

. For any

𝑥∈ K

there exists

𝛼𝑥∈𝐼N

such that

𝜋(𝛼𝑥)=𝑥

Hence,

𝑇′

𝜔(𝑥)=e|𝜔|

𝑖=1log𝑇′

𝜔𝑖𝑇𝜎𝑖(𝜔)(𝜋(𝛼𝑥))=e|𝜔|

𝑖=1log𝑇′

𝜔𝑖(𝜎𝑖(𝜔𝜋 (𝛼𝑥)))=e𝑆𝑛𝜑(𝜔𝛼𝑥).

Note that

𝜑

is H

older continuous if the underlying IFS is a

C1+𝛾

-cIFS. Moreover, if

all the 𝑇𝑖are aﬃne, then 𝜑depends only on the ﬁrst coordinate.

Deﬁnition 4.23 (Perron-Frobenius operator).Let

𝜓∈ C(𝐼N)

(sometimes called

potential function). The Perron-Frobenius operator (with respect to 𝜓)

𝐿𝜓:C(𝐼N) → C(𝐼N)

102

4.4. Examples

is deﬁned by

𝐿𝜓𝑓(𝑥)≔

𝑦∈𝜎−1𝑥

e𝜓(𝑦)𝑓(𝑦), 𝑥 ∈𝐼N.

Deﬁnition 4.24 (Pressure function).For

𝑓∈ C 𝐼N

the pressure of

𝑓

is deﬁned by

𝑃(𝑓)≔lim

𝑛→∞

𝑛log 

𝜔∈𝐼𝑛

𝑒𝑆𝜔𝑓,(4.4.1)

with 𝑆𝜔𝑓:=sup𝑥∈[𝜔]𝑆|𝜔|𝑓(𝑥).

Remark 4.25.The existence of the limit in

(4.4.3)

follows from the subadditivity of

log(𝜔∈𝐼𝑛𝑒𝑆𝜔𝑓)and Fekete’s subadditive lemma.

Lemma 4.26. Let 𝜓∈ C(𝐼N)with 𝐿𝜓

≔

𝐼N. Then,

𝑃(𝜓)=0.

Proof. Using 𝐿𝜓

, for all 𝑛∈N,𝑥∈𝐼N, we deduce

=𝐿𝑛

𝜓

=

𝜔∈𝐼𝑛

𝑒𝑆𝑛𝜓(𝜔𝑥 ).

This leads to

−𝑛−1

𝑖=0var𝑖(𝜓)

𝑛+1

𝑛log 

𝜔∈𝐼𝑛

𝑒𝑆𝑛𝜓(𝜔𝑥 )≤1

𝑛log 

𝜔∈𝐼𝑛

𝑒𝑆𝜔𝜓

≤𝑛−1

𝑖=0var𝑖(𝜓)

𝑛+1

𝑛log 

𝜔∈𝐼𝑛

𝑒𝑆𝑛𝜓(𝜔𝑥 ).

Since

𝜓

is continuous, we have

lim𝑚→∞ var𝑚(𝜓)=

0 and thus

𝑛−1

𝑖=0var𝑖(𝜓)/𝑛

tend-

ing to zero for sending 𝑛to inﬁnity as a Ces`

aro limit. This gives 𝑃(𝜓)=0. □

Now, we are in the position to deﬁne weak Gibbs measures, which is subject of the

following proposition.

Proposition 4.27 ([Kes01, Proposition 1]).For any

𝜓∈ C(𝐼N)

with

𝐿𝜓

there

exists 𝜇∈ M𝜎𝐼Nsuch that

𝐿∗

𝜓𝜇=e𝑃(𝜓)𝜇=𝜇,

where

𝐿∗

𝜓

denotes the dual operator of

𝐿𝜓

acting on the set of Borel probability

measures supported on

𝐼N

. We call

𝜇

aweak

𝜓

-Gibbs measure and

𝜈≔𝜇◦𝜋−1

weak 𝜓-Gibbs measure with respect to the cIFS Φ.

103

4.4. Examples

Remark 4.28.The existence of the ﬁxed point in Proposition 4.27 follows from the

Schauder-Tychonov ﬁxed point theorem (see also [Kes01]) and the

𝜎

-invariance of

𝜇follows for 𝐸∈ B 𝐼N, by

𝜇(𝜎−1(𝐸)) =

𝜏∈𝐼

e𝑆𝑛𝜓(𝜏𝑦 )

𝜎−1(𝐸)(𝜏𝑦)d𝜇(𝑦)

=

𝜏∈𝐼

e𝑆𝑛𝜓(𝜏𝑦 )

𝐸(𝑦)d𝜇(𝑦)=𝜇(𝐸).

Remark 4.29.The following list of comments proves useful in our context.

1. By [Kes01, Lemma 3], for all 𝑢∈𝐼Nand 𝑛∈N, we have

e−𝑛−1

𝑖=0var𝑖(𝜓)≤𝜇([𝑢|𝑛])

e𝑆𝑛𝜓(𝑢)≤e𝑛−1

𝑖=0var𝑖(𝜓).(4.4.2)

In particular, the measure

𝜇

has no atoms, since

𝑛

𝑖=0var𝑖(𝜓)=𝑜(𝑛)

and

𝑆𝑛𝜓≤𝑛max𝜓.

The topological support

supp(𝜈)

𝜈

is equal to

. To see this, note that

is covered by the set

𝜔∈𝐼𝑛𝑇𝜔([0,1])

𝑛∈N

, and by

(4.4.2)

each

𝑇𝜔([0,1])

has positive 𝜈-measure 𝜈(𝑇𝜔([0,1])) ≥exp −𝑛−1

𝑖=0var𝑖(𝜓)𝜇([𝜔]).

𝜓

is additionally H

older continuous, then

𝜇

is the unique invariant ergodic

𝜓

-Gibbs measure and the bounds in the above inequality

(4.4.2)

can be chosen

to be positive constants.

For an arbitrary H

older continuous function

𝜓

𝐼N→R

(without assuming

𝐿𝜓

) there always exists a

𝜎

-invariant

𝜓

-Gibbs measure

𝜇

on the symbolic

space as a consequence of the general thermodynamic formalism and the

Perron-Frobenius theorem for H

older potentials (see e.g. [Bow08]). Let

ℎ

denote the only eigenfunction of the Perron-Frobenius operator for the

maximal eigenvalue

𝜆>

0, which is positive and in the same H

older class.

Then

𝜓1≔𝜓−log(𝜆) +log(ℎ)−log(ℎ◦𝜎)

deﬁnes another H

older continuous

function for which

𝐿𝜓1

and

𝜇

is the (unique)

𝜓1

-Gibbs measure, as

deﬁned here.

𝜓

depends only on the ﬁrst coordinate and is normalized such that we

have

𝑝𝑖≔exp(𝜓(𝑖 . . .))

𝑖∈𝐼

, deﬁnes a probability vector, then

𝜇

is in fact a

Bernoulli measure and the bounding constants in the above inequalities

(4.4.2)

can be chosen to be 1. Further,

𝜈

coincides with the self-conformal measure

as deﬁned in

(2.4.2)

. This can be seen as follows. For all

𝐸∈𝔅([

])

, we

104

4.4. Examples

have

𝜈(𝐸)=

𝑢∈𝐼

𝜇𝜋−1(𝐸)∩ [𝑢]

=

𝑢∈𝐼[𝑢]

𝐸◦𝜋d𝜇

=

𝑢∈𝐼𝐿|𝑢|

𝜓(

[𝑢](𝑥)

𝐸(𝜋(𝑥))) d𝜇(𝑥)

=

𝑢∈𝐼

𝜏∈𝐼|𝑢|

e𝑆|𝑢|𝜓(𝜏𝑥)

[𝑢](𝜏𝑥 )

𝐸(𝜋(𝜏𝑥 )) d𝜇(𝑥)

=

𝑢∈𝐼𝑝𝑢

𝐸(𝜋(𝑢𝑥)) d𝜇(𝑥)

=

𝑢∈𝐼

𝑝𝑢𝜈𝑇−1

𝑢(𝐸).

In particular, if additionally the

(𝑇𝑖)𝑖

are contracting similarities, then

𝜈

coincides

with the self-similar measure deﬁned in (2.4.2).

For

𝑚∈N

we will consider the accelerated shift-space

(𝐼𝑚)N

with natural shift map

𝜎

(𝐼𝑚)N→(𝐼𝑚)N

. Clearly,

(𝐼𝑚)N

can be identiﬁed with

𝐼N

allowing us to deﬁne

the accelerated ergodic sum for 𝑓∈ C 𝐼Nby



𝑆𝑛𝑓𝑚(𝑥)≔

𝑛−1



𝑖=0

𝑓𝑚(𝜎𝑖(𝑥)) with 𝑓𝑚(𝑥)≔

𝑚−1



𝑖=0

𝑓𝜎𝑖(𝑥).

For

𝜔∈(𝐼𝑚)∗

we let

|𝜔|𝑚

denote the word length of

𝜔

with respect to the alphabet

𝐼𝑚

. With this setup we have



𝑆𝑛𝑓𝑚=𝑆𝑚·𝑛𝑓

and



𝑆𝜔𝑓𝑚=sup𝑥∈[𝜔]

𝑆|𝜔|𝑚𝑓𝑚(𝑥)

for

𝜔∈(𝐼𝑚)∗.

Lemma 4.30. For every 𝑓∈ C 𝐼Nand 𝑚∈N, we have

𝑚𝑃 (𝑓)=𝑃𝜎(𝑓𝑚)≔lim

𝑛→∞

𝑛log

𝜔∈(𝐼𝑚)𝑛

exp 

𝑆𝜔𝑓𝑚.

Proof.

The assertion follows immediately from the following identity, for all

𝑛∈N

𝑛log

𝜔∈(𝐼𝑚)𝑛

exp 

𝑆𝜔

𝑓=1

𝑛log

𝜔∈(𝐼𝑚)𝑛

exp sup

𝑥∈[𝜔]

𝑆|𝜔|𝑚𝑓𝑚(𝑥)

=𝑚1

𝑚𝑛 log 

𝜔∈𝐼𝑚𝑛

exp sup

𝑥∈[𝜔]

𝑆|𝜔|𝑓(𝑥).□

105

4.4. Examples

In the following we show that the weak bounded distortion property (wBDP) holds

true for the IFS Φ=(𝑇1,. . . ,𝑇𝑛).

Lemma 4.31 (Weak Bounded Distortion Property).There exists a sequence of non-

negative numbers (𝑏𝑚)𝑚with 𝑏𝑚=𝑜(𝑚)such that for all 𝜔∈𝐼∗and 𝑥,𝑦 ∈ [0,1]

e−𝑏|𝜔|≤𝑇′

𝜔(𝑥)

𝑇′

𝜔(𝑦)≤e𝑏|𝜔|.

Proof.

Here, we follow the arguments in [KK12, Lemma 3.4]. For all

𝜔≔

𝜔1···𝜔𝑙∈𝐼∗,𝑥,𝑦 ∈ [0,1], we have

𝑇′

𝜔(𝑥)

𝑇′

𝜔(𝑦)≤exp 𝑙



𝑘=1log 𝑇′

𝜔𝑘𝑇𝜎𝑘𝜔(𝑥)−log 𝑇′

𝜔𝑘𝑇𝜎𝑘𝜔(𝑦)

≤exp 𝑙



𝑘=1

max

𝑥, 𝑦 ∈[0,1]max

𝑖=1,...,𝑛 log 𝑇′

𝑖𝑇𝜎𝑘𝜔(𝑥)−log 𝑇′

𝑖𝑇𝜎𝑘𝜔(𝑦)

ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ

≕𝐴𝑙−𝑘

.

Let 0

<𝑅<

1 be a common bound for the contraction ratios of the maps

𝑇1, . . ., 𝑇𝑛

Then we have

𝑇𝜎𝑘𝜔(𝑥) −𝑇𝜎𝑘𝜔(𝑦)≤𝑅𝑙−𝑘|𝑥−𝑦| ≤ 𝑅𝑙−𝑘.

Hence, we conclude

𝐴𝑙−𝑘≤max

𝑎,𝑏 ∈[0,1],

|𝑎−𝑏|≤𝑅𝑙−𝑘max

𝑖=1,...,𝑛 log 𝑇′

𝑖(𝑎)−log 𝑇′

𝑖(𝑏)≕𝐵𝑙−𝑘.

Using that each

𝑇′

1, . . . ,𝑇 ′

𝑛

is bounded away from zero and continuous, we ob-

tain

𝐵𝑘→

0 for

𝑘→ ∞

. With

𝑏𝑚≔𝑚−1

𝑘=0𝐵𝑘

we have

lim𝑚→∞𝑏𝑚/𝑚

equals

lim𝑘→∞𝐵𝑘=

0 as a Ces

aro limit and the second inequality holds. The ﬁrst in-

equality follows by interchanging the roles of 𝑥and 𝑦.□

4.4.1.1 Spectral dimension for weak Gibbs measures under the OSC

Let

𝜇

and

𝜈=𝜇◦𝜋−1

be weak

𝜓

-Gibbs measures with respect to a

-cIFS

deﬁned in Proposition 4.27. In this section we assume the open set condition (OSC)

with feasible open set

(0,1)

, i.e.

𝑇𝑖((0,1))∩𝑇𝑗((0,1)) =∅

and

𝑇𝑖((

)) ⊂ (

)

for

all

𝑖, 𝑗 ∈𝐼

with

𝑖≠𝑗

. An important quantity is

𝜉≔𝜑+𝜓

where

𝜑

is the geometric

potential with respect to Φas deﬁned in Deﬁnition 4.21. Since

𝑝:𝑡↦→𝑃(𝑡 𝜉 )(4.4.3)

106

4.4. Examples

is continuous, strictly monotonically increasing, convex, and

lim

𝑡→±∞𝑝(𝑡)=∓∞,

there exists a unique number

𝑧𝜈∈R

such that

𝑝(𝑧𝜈)=

0. This section is devoted to

identify the spectral dimension of

Δ𝐷

𝜈

with

𝑧𝜈

. We start with some basic observa-

tions.

Let

Kunique

be the set of points which have a unique preimage of the coding map

𝜋

The OSC implies that

K\Kunique

is a countable set. Notice that, due to

(4.4.2) 𝜇

has

no atoms, hence the OSC ensures that

𝜈

has also no atoms, implying

𝜈Kunique=

Lemma 4.32. For all 𝜔∈𝐼∗and 𝑓∈𝐻1, we have

𝐼𝜔(∇𝑓)2dΛ=[0,1](∇(𝑓◦𝑇𝜔))2|𝑇′

𝜔|−1dΛ

and 𝐼𝜔

𝑓2d𝜈=[0,1](𝑓◦𝑇𝜔)2e𝑆|𝜔|𝜓◦𝜋−1◦𝑇𝜔d𝜈,

with 𝐼𝜔≔𝑇𝜔([0,1]). In particular, we have

𝑒𝑠𝜔𝜓min

𝑥∈[0,1]|𝑇′

𝜔(𝑥)| ≤ 𝜈(𝑇𝜔([0,1]))Λ(𝑇𝜔([0,1])) ≤𝑒𝑆𝜔𝜓max

𝑥∈[0,1]|𝑇′

𝜔(𝑥)|.(4.4.4)

Proof. Clearly, by a change of variables

𝐼𝜔(∇𝑓)2dΛ=[0,1]((∇𝑓)◦𝑇𝜔)2|𝑇′

𝜔|dΛ

=[0,1](∇(𝑓◦𝑇𝜔))2|𝑇′

𝜔|−1dΛ.

Further using the deﬁnition of 𝜇and the OSC, we have

𝐼𝜔

𝑓2d𝜈=[𝜔]

𝑓2◦𝜋d𝜇

=𝐿|𝜔|

𝜓(

[𝜔](𝑥)𝑓2(𝜋(𝑥))) d𝜇(𝑥)

=

𝜏∈𝐼|𝜔|

e𝑆|𝜔|𝜓(𝜏𝑥 )

[𝜔](𝜏𝑥 )𝑓2(𝜋(𝜏𝑥 )) d𝜇(𝑥)

=e𝑆|𝜔|𝜓(𝜔𝑥 )𝑓2(𝜋(𝜔𝑥 )) d𝜇(𝑥)=[0,1](𝑓◦𝑇𝜔)2e𝑆|𝜔|𝜓◦𝜋−1◦𝑇𝜔d𝜈,

where we used the fact that 𝜋(𝜔𝑥 )=𝑇𝜔(𝜋(𝑥)).□

107

4.4. Examples

Lemma 4.33. There exists

𝐶>

0such that for

𝑚∈N

large enough, for all

𝑥>𝐶

𝑟𝑚,min

we have 𝑥𝑟𝑚,min

𝐶𝑢𝑚≤𝑁𝐷

𝜈(𝑥) ≤ 2𝑥𝑢𝑚

𝑅𝑢𝑚

𝑚,min +1

where, for

𝜔∈𝐼𝑚

, we set

𝑟𝜔≔exp(𝑠𝜔𝜑−𝑏𝑚+𝑠𝜔𝜓), 𝑅𝜔≔exp(𝑆𝜔𝜑+𝑏𝑚+𝑆𝜔𝜓)

𝑟𝑚,min ≔min𝑖∈𝐼𝑚𝑟𝑖

, and

𝑅𝑚,min ≔min𝑖∈𝐼𝑚𝑅𝑖

. Here

(𝑏𝑚)𝑚

is the sequence deﬁned

in Lemma 4.31, and 𝑢𝑚,𝑢𝑚∈R>0denotes the unique solutions of



𝜔∈𝐼𝑚

e𝑢𝑚(𝑆𝜔𝜑+𝑆𝜔𝜓+𝑏𝑚)=

𝜔∈𝐼𝑚

e𝑢𝑚(𝑠𝜔𝜑+𝑠𝜔𝜓−𝑏𝑚)=1.

Proof.

This proof follows the arguments used in [KL01, Lemma 2.7]. First, note

that for

𝑚∈N

suﬃciently large, for all

𝜔∈𝐼𝑚

we have

𝑆𝜔𝜑+𝑆𝜔𝜓+𝑏𝑚<

0 where

we used

𝑏𝑚=𝑜(𝑚)

and

𝑆𝜔𝜓+𝑆𝜔𝜑≤𝑚(max𝜓+max𝜑)

. Therefore, there exists

𝑢𝑚∈R>0such that 

𝜔∈𝐼𝑚

𝑅𝑢𝑚

𝜔=1.

For

𝜔≔𝜔1···𝜔𝑛∈(𝐼𝑚)𝑛

𝑛∈N

, deﬁne

𝑅𝜔≔|𝜔|

𝑖=1𝑅𝜔𝑖

and

𝑟𝜔≔|𝜔|

𝑖=1𝑟𝜔𝑖

. Let

𝑥>1 and deﬁne for 𝑚∈Nthe following partition of (𝐼𝑚)N

𝑃𝑚,𝑥 ≔𝜔∈(𝐼𝑚)∗:𝑅𝜔<1

𝑥≤𝑅𝜔−

with

𝑅𝜔−≔|𝜔|−1

𝑖=1𝑅𝜔𝑖

. Considering the Bernoulli measure on

(𝐼𝑚)N

given by the

probability vector

𝑅𝑢𝑚

𝜔𝜔∈𝐼𝑚

and using the fact that

𝑃𝑚,𝑥

deﬁnes a partition of

(𝐼𝑚)N, we obtain 

𝜔∈𝑃𝑚,𝑥

𝑅𝑢𝑚

𝜔=1,

which leads to

card (𝑃𝑚.𝑥 )≤𝑥𝑢𝑚/𝑅𝑚,min𝑢𝑚

. Combining Lemma 4.31,

(4.4.4)

, and

the chain rule for diﬀerentiation, for all 𝜔∈𝑃𝑚.𝑥 , we have

Λ(𝑇𝜔([0,1]))𝜈(𝑇𝜔([0,1])) ≤|𝜔|



𝑖=1

𝑒𝑆𝜔𝑖𝜓max

𝑥∈[0,1]|𝑇′

𝜔𝑖(𝑥)|

≤|𝜔|



𝑖=1

𝑒𝑆𝜔𝑖𝜓+𝑆𝜔𝑖𝜑+𝑏𝑚=𝑅𝜔<1/𝑥 .

108

4.4. Examples

We conclude from Proposition 4.4 and the OSC

𝑁𝜈(𝑥) ≤2 card 𝑃𝑚,𝑥 +1≤2𝑥𝑢𝑚

𝑅𝑚,min𝑢𝑚+1.

For the estimate from below we deﬁne for

𝑥>1

𝑟𝑚,min

the following partition of

(𝐼𝑚)N

𝛯𝑚,𝑥 ≔𝜔∈(𝐼𝑚)∗:𝑟𝜔<1

𝑥𝑟𝑚,min ≤𝑟𝜔−,

with

𝑟𝜔−≔|𝜔|−1

𝑖=1𝑟𝜔𝑖

. Again, there exists

𝑢𝑚∈R>0

such that

𝜔∈𝐼𝑚𝑟𝑢𝑚

𝜔=

1 and

we obtain 𝜔∈𝛯𝑚,𝑥 𝑟𝑢𝑚

𝜔=1. Hence, it follows

1=

𝜔∈𝛯𝑚,𝑥

𝑟𝑢𝑚

𝜔≤1

𝑥𝑟𝑚,min 𝑢𝑚

card 𝛯𝑚,𝑥 .(4.4.5)

Fix

𝑎∈ K \{

}

and choose

𝑢0∈ C∞

𝑐((

))

such that

𝑢0(𝑎)>

0. For

𝜔∈𝛯𝑚,𝑥

, we

deﬁne

𝑢𝜔(𝑥)≔𝑢0𝑇−1

𝜔(𝑥), 𝑥 ∈𝑇𝜔((0,1)),

0, 𝑥 ∈ [0,1]\𝑇𝜔((0,1)).

Clearly, we then have

𝑢𝜔∈ C∞

𝑐(𝑇𝜔((

))) ⊂ 𝐻1

. Using Lemma 4.31, Lemma 4.32,

and the chain rule for diﬀerentiation, we obtain

[0,1](∇𝑢𝜔)2dΛ

[0,1]𝑢2

𝜔d𝜇=[0,1](∇𝑢0)2|𝑇′

𝜔|−1dΛ

[0,1]𝑢2

0e𝑆|𝜔|𝜓◦𝜋−1◦𝑇𝜔d𝜈

≤[0,1](∇𝑢0)2dΛ

[0,1]𝑢2

0d𝜈

ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ

≕𝐶

|𝜔|



𝑖=1

min𝑥∈[0,1]|𝑇′

𝜔𝑖(𝑥)|𝑒−𝑠𝜔𝑖𝜓

≤𝐶|𝜔|



𝑖=1

𝑒−𝑠𝜔𝑖𝜑−𝑠𝜔𝑖𝜓+𝑏𝑚=𝐶

𝑟𝜔≤𝐶

(𝑟𝜔−)𝑟𝑚,min ≤𝐶·𝑥 .

Since the supports of

(𝑢𝜔)𝜔∈𝛯𝑚,𝑥

are disjoint, it follows that the

(𝑢𝜔)𝜔∈𝛯𝑚,𝑥

are

mutually orthogonal both in

𝐿2

𝜈

and in

𝐻1

. Consequently,

span(𝑢𝜔

𝜔∈𝐸𝑡)

is a

card 𝛯𝑚,𝑥 

-dimensional subspace of

𝐻1

. Therefore, for

𝑥≥𝐶/𝑟𝑚,min

, Lemma 2.19

and (4.4.5) give

𝑁𝐷

𝜈(𝐶·𝑥) ≥ card 𝛯𝑚,𝑥 ≥𝑥𝑟𝑚,min𝑢𝑚.□

In the case of self-similar measures, we obtain the following classical result of

109

4.4. Examples

[Fuj87].

Corollary 4.34. Assume

𝑇′

𝑖≡𝜎𝑖>

0and

𝜓(𝜔)=log(𝑝𝜔1)

, for

𝜔≔(𝜔1𝜔2...) ∈ 𝐼N

where

(𝑝𝑖)𝑖∈(0,1)𝑛

is a given probability vector. Then, for all

𝑥>𝐶(min𝑖∈𝐼𝑝𝑖𝜎𝑖)−1

we have

𝑥𝑢min𝑖∈𝐼𝑝𝑖𝜎𝑖

𝐶𝑢

≤𝑁𝐷

𝜈(𝑥) ≤ 𝑥𝑢

(min𝑖∈𝐼𝑝𝑖𝜎𝑖)𝑢,

where 𝑢is the unique solution of 𝑛

𝑖=1(𝜎𝑖𝑝𝑖)𝑢=1.

Lemma 4.35. For ﬁxed

𝑚∈N

large enough and

𝑢𝑚,𝑢𝑚∈R>0

denoting the unique

solutions of 

𝜔∈𝐼𝑚

e𝑢𝑚(𝑆𝜔𝜑+𝑏|𝜔|+𝑆𝜔𝜓)=

𝜔∈𝐼𝑚

e𝑢𝑚(𝑠𝜔𝜑−𝑏|𝜔|+𝑠𝜔𝜓)=1,

we have lim𝑚→∞𝑢𝑚=lim𝑚→∞𝑢𝑚=𝑧𝜈.

Proof. Deﬁne for 𝑚∈Nand 𝑡≥0

𝑃𝑚(𝑡)≔1

𝑚log 

𝜔∈𝐼𝑚

exp(𝑡(𝑠𝜔𝜑−𝑏𝑚+𝑠𝜔𝜓)),

𝑃𝑚(𝑡)≔1

𝑚log 

𝜔∈𝐼𝑚

exp(𝑡(𝑆𝜔𝜑+𝑏𝑚+𝑆𝜔𝜓)),

𝑃𝑚(𝑡)≔1

𝑚log 

𝜔∈𝐼𝑚

exp(𝑡𝑆𝜔𝜉).

We obtain

𝑃𝑚(𝑡) ≤ 𝑃𝑚(𝑡)

≤𝑃𝑚(𝑡)−𝑡𝑏𝑚

𝑚

𝑚log 

𝜔∈𝐼𝑚

exp(𝑡(𝑠𝜔𝜑+𝑠𝜔𝜓+𝑆𝜔𝜑−𝑠𝜔𝜑+𝑆𝜔𝜓−𝑠𝜔𝜓))−𝑡𝑏𝑚

𝑚

≤1

𝑚log 

𝜔∈𝐼𝑚

exp 𝑡(𝑠𝜔𝜑+𝑠𝜔𝜓)+𝑡𝑚−1



𝑗=0

var𝑗𝜓+

𝑚−1



𝑗=0

var𝑗𝜑−𝑡𝑏𝑚

𝑚

≤𝑃𝑚(𝑡)+ 𝑡

𝑚𝑚−1



𝑗=0

var𝑗𝜑+

𝑚−1



𝑗=0

var𝑗𝜓−𝑏𝑚.

Using the continuity of 𝜑, 𝜓 and lim𝑚→∞𝑏𝑚/𝑚=0, we deduce

lim

𝑚→∞ 𝑃𝑚(𝑡)=lim

𝑚→∞𝑃𝑚(𝑡)=𝑃(𝑡𝜉 ).

110

4.4. Examples

Furthermore, for all 𝑡≥0, we have

𝑃𝑚(𝑡) ≤ 𝑃𝑚(𝑡) ≤ 𝑡𝑏𝑚

𝑚+1

𝑚log 

𝜔∈𝐼𝑚

exp (𝑡𝑚 (max𝜓+max 𝜑))

=log(𝑛) +𝑡𝑏𝑚

𝑚+(max𝜓+max𝜑).

Observe that for

𝑚

large we have

𝑏𝑚/𝑚≤ −max𝜓/

2 and each of the maps

𝑡↦→

𝑃𝑚(𝑡), 𝑡 ↦→ 𝑃𝑚(𝑡)

and

𝑡↦→ 𝑃(𝑡)

is decreasing and has a unique zero lying in the

interval

[0,−log(𝑛)/(max𝜓/2+max𝜑)]

. Hence the statement follows from Lemma

3.5. □

We are now in the position to state our main result of this section which is an

immediate consequence of Lemma 4.33 and Lemma 4.35.

Theorem 4.36. The spectral dimension of

Δ𝐷

𝜈

exists and is equal to the unique zero

𝑧𝜈of the pressure function as deﬁned in (4.4.3). In particular 𝑧𝜈=𝑞𝔍𝜈.

Example 4.37. The natural choice of the potential

𝜓

is given by

𝑠𝜑

, where

𝑠≥

0 is

to be chosen such that

𝑃(𝑠𝜑)=

0. We then have

𝜉=(

+𝑠)𝜑

and

𝑃((𝑠/(𝑠+

))𝜉)=

Thus, in this case, the spectral dimension can be expressed by the simple formula

𝑠𝐷

𝜈=𝑠

𝑠+1.

4.4.1.2

Spectral asymptotics for Gibbs measures for

C1+𝛾

-cIFS under the

OSC

Let

𝜇

and

𝜈

be as deﬁned in Section 4.4.1. This section is devoted to improve

Theorem 4.36 to

𝑁𝐷

𝜈(𝑥) ≍ 𝑥𝑧𝜈

under the additional assumption that

𝜓

is H

older

continuous and the underlying IFS

{𝑇1, . . . ,𝑇𝑚}

C1+𝛾

, which implies that the asso-

ciated geometric potential

𝜑

is H

older continuous. In this situation, the following

reﬁned bounded distortion property holds (see [KK12, Lemma 3.4]).

Lemma 4.38 (Strong Bounded Distortion Property).Assume

𝑇1, . . ., 𝑇𝑛

are

C1+𝛾

-IFS

then we have the following strong bounded distortion property (sBDP). There exists

𝑎0>0such that for all 𝜔∈𝐼∗and 𝑥,𝑦 ∈ [0,1]we have

𝑎−1

0≤𝑇′

𝜔(𝑥)

𝑇′

𝜔(𝑦)≤𝑎0.

Using the sBDP, we can improve Lemma 4.32 in the following way.

111

4.4. Examples

Lemma 4.39. For all 𝑖∈N,𝜔∈𝐼∗, and 𝑥, 𝑦 ∈𝐼N, we have

e𝑆|𝜔|𝜑(𝜔𝑦 )−log(𝑎0)≤Λ(𝑇𝜔([0,1])) ≤e𝑆|𝜔|𝜑(𝜔 𝑦)+log (𝑎0),

and

e𝑆|𝜔|𝜓(𝜔𝑥 )−∞

𝑛=0var𝑛(𝜓)≤𝜈(𝑇𝜔([0,1])) ≤e𝑆|𝜔|𝜓(𝜔𝑥 )+∞

𝑛=0var𝑛(𝜓),

where 𝑎0is deﬁned as in Lemma 4.38. In particular, we have

e𝑆𝜔𝜉−𝑑0≤𝜈(𝑇𝜔([0,1]))Λ(𝑇𝜔([0,1])) ≤e𝑆𝜔𝜉+𝑑0,

with 𝑑0≔log(𝑎0) +∞

𝑛=0var𝑛(𝜓)and 𝜉≔𝜑+𝜓.

Proof. Note, that we have for all 𝜔∈𝐼∗and 𝑥, 𝑧 ∈𝐼N

𝑆|𝜔|𝜓(𝜔𝑥 )−𝑆|𝜔|𝜓(𝜔𝑧)≤∞



𝑘=0

var𝑘(𝜓)

and by Lemma 4.38, for all 𝑦,𝑣 ∈ [0,1], we obtain

log 𝑇′

𝜔(𝑦)−log 𝑇′

𝜔(𝑣)≤log (𝑎0).

Further, note that there exists a

𝑦∈K

such that

𝜋(𝑥)=𝑦

thus we obtain

log 𝑇′

𝜔(𝑦)=

𝑆|𝜔|𝜑(𝜔𝑥 ). Hence, the statement follows from Lemma 4.32. □

Lemma 4.40. For every 𝑡>𝑐>0, we have that

𝑡≔{𝜔∈𝐼∗:𝑆𝜔𝜉<log(𝑐/𝑡) ≤ 𝑆𝜔−𝜉}

is a partition of 𝐼N. In particular, for every 𝜔∈Γ

𝑡,𝑐 and 𝑥∈𝐼N, we have

log(𝑀e𝑑0𝑡/𝑐) ≥ −𝑆|𝜔|𝜉(𝜔𝑥 )

with

𝑀≔exp (max (−𝜉))

and

𝑑0≔log(𝑎0) + ∞

𝑘=0var𝑘(𝜓)

with

𝑎0

deﬁned as in

Lemma 4.38.

Proof.

First note, that two cylinder sets are either disjoint or one is contained in the

other. From 𝜔∈Γ

𝑡,𝑐 and all 𝜂∈𝐼∗, we have

𝑆𝜔𝜂 𝜉≤sup

𝑥∈𝐼N

𝑆|𝜔|𝜉(𝜔𝜂𝑥) ≤ sup

𝑥∈𝐼N

𝑆|𝜔|𝜉(𝜔𝑥 )=𝑆𝜔𝜉<log(𝑐/𝑡),

where we used

max𝜉<

0, which shows for

𝜂≠∅

that

𝜔𝜂 ∉Γ

𝑡,𝑐

. Moreover, since

min𝜉<

it follows that

𝑆𝜔𝜉

converge to

−∞

for

|𝜔| → ∞

. Consequently, the set

112

4.4. Examples

𝑡,𝑐

is ﬁnite. In particular, for every

𝜔∈𝐼N

, we have

𝑆𝜔|𝑛𝜉→ −∞

𝑛

tends to

inﬁnity. Therefore, there exists 𝑁∈Nsuch that

𝑆𝜔|𝑁𝜉<log(𝑐/𝑡) ≤ 𝑆𝜔|𝑁−1𝜉

and the ﬁrst statement follows. For the second claim ﬁx 𝜔∈Γ

𝑡,𝑐 , then

log(𝑡/𝑐) ≥ −(𝑆𝜔−𝜉)=−𝑆|𝜔−1|𝜉(𝜔𝑥 ) − 𝑆𝜔−𝜉−𝑆|𝜔−1|𝜉(𝜔𝑥)

≥ −𝑆|𝜔−1|𝜉(𝜔𝑥) −𝑑0

=−𝑆|𝜔−1|𝜉(𝜔𝑥) −𝜉𝜎|𝜔|−1(𝜔)𝑥+𝜉𝜎|𝜔|−1(𝜔)𝑥−𝑑0

≥ −𝑆|𝜔|𝜉(𝜔𝑥) −log(𝑀) −𝑑0,

and hence we obtain log 𝑀e𝑑0𝑡/𝑐≥ −𝑆|𝜔|𝜉(𝜔𝑥).□

Recall for 𝑚∈Nand 𝑥∈(𝐼𝑚)N

𝜉𝑚(𝑥)=

𝑚−1



𝑖=0

𝜉𝜎𝑖(𝑥).

Lemma 4.41. Set

𝑑0≔log (𝑎0)+∞

𝑘=0var𝑘𝜓

where

𝑎0

is deﬁned as in Lemma 4.38.

Then for

𝑡>𝑐>

0and

𝑚∈N

such that

−𝑚max𝜉−𝑑0>

0, and

𝑥∈(𝐼𝑚)N,

we have

that

Γ𝐿

𝑡,𝑚 ≔𝜔∈(𝐼𝑚)∗:−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 ) ≤ log (𝑡/𝑐)<min

𝑣∈𝐼𝑚−

𝑆|𝜔𝑣 |𝑚𝜉𝑚(𝜔 𝑣𝑥 )

deﬁnes a disjoint family, meaning

𝜔≠𝜔′

implies

[𝜔]∩[𝜔′]=∅

. With

𝑘𝑚≔

exp(−𝑚max𝜉)for every 𝜔∈(𝐼𝑚)∗

log 𝑡e𝑑0/(𝑘𝑚𝑐)<−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 ) ≤ log (𝑡/𝑐),

we have 𝜔∈Γ𝐿

𝑡,𝑚 .

Proof.

For every

𝜔∈Γ𝐿

𝑡,𝑚

and every

𝑣∈𝐼𝑚

we have

log(𝑡/𝑐)<−

𝑆|𝜔𝑣 |𝑚𝜉𝑚(𝜔 𝑣𝑥 )

implying

𝜔𝑣 ∉Γ𝐿

𝑡,𝑚

. Further, using

−𝑚max𝜉−𝑑0>

0 and the BDP, for every

113

4.4. Examples

𝜂∈(𝐼𝑚)∗\{∅}, we have

log(𝑐/𝑡)>

𝑆|𝜔𝑣 |𝑚𝜉𝑚(𝜔𝑣𝑥 )

≥

𝑆|𝜔𝑣 |𝑚𝜉𝑚(𝜔 𝑣𝜂𝑥 ) −𝑑0

=

𝑆|𝜔𝑣 |𝑚𝜉𝑚(𝜔𝑣𝜂𝑥 ) + |𝜂|𝑚−1



𝑖=0𝜉𝑚𝜎𝑖(𝜂)𝑥−𝜉𝑚𝜎𝑖(𝜂)𝑥−𝑑0

≥

𝑆|𝜔𝑣 𝜂 |𝑚𝜉𝑚(𝜔𝑣𝜂𝑥) −𝑚· |𝜂|𝑚max𝜉−𝑑0

≥

𝑆|𝜔𝑣 𝜂 |𝑚𝜉𝑚(𝜔𝑣𝜂𝑥) −𝑚·max 𝜉−𝑑0

>

𝑆|𝜔𝑣 𝜂 |𝑚𝜉𝑚(𝜔𝑣𝜂𝑥).

Thus, for every 𝜔∈Γ𝐿

𝑡,𝑚 and 𝜂′∈(𝐼𝑚)∗\{∅}it follows 𝜔𝜂 ′∉Γ𝐿

𝑡,𝑚 .

For the second assertion ﬁx 𝑥∈(𝐼𝑚)N,𝜔∈(𝐼𝑚)∗and assume

log 𝑡e𝑑0/(𝑘𝑚𝑐)<−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 ) ≤ log (𝑡/𝑐).

Using the BDP, for all 𝑣∈𝐼𝑚, 𝜔 ∈(𝐼𝑚)∗, we obtain



𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 ) −

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑣𝑥 )≤𝑑0,

and consequently,

log(𝑡/𝑐)<log(𝑘𝑚) −𝑑0−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 )

≤log(𝑘𝑚) +𝜉𝑚(𝑣𝑥 ) −𝜉𝑚(𝑣𝑥 ) −

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑣𝑥 )

≤ −

𝑆|𝜔𝑣 |𝑚𝜉𝑚(𝜔 𝑣𝑥 ).

Since −

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 ) ≤ log (𝑡/𝑐), we conclude 𝜔∈Γ𝐿

𝑡,𝑚 .□

Now we are in the position to prove the main theorem of this section.

Theorem 4.42. We have

𝑁𝐷

𝜈(𝑡) ≍𝑡𝑧𝜈,

where 𝑧𝜈is the unique zero of the pressure function as deﬁned in (4.4.3).

Proof.

Let

𝑡>𝑒−𝑑0

and

𝜔∈Γ

𝑡,𝑒 −𝑑0

. Then, by Lemma 4.39 and the deﬁnition of

𝜔∈Γ

𝑡,𝑒 −𝑑0, it follows

𝜈(𝑇𝜔([0,1]))Λ(𝑇𝜔([0,1])) ≤e𝑆𝜔𝜉+𝑑0<1/𝑡 .

114

4.4. Examples

Then from Lemma 4.40 and Proposition 4.4 we infer

𝑁𝐷

𝜈(𝑡) ≤ 2 card Γ

𝑡,𝑒 −𝑑0+1.

Hence, for the upper bound, we are left to show that

card Γ

𝑡,𝑐 ≪𝑡𝑧𝜈

. For this we

use [Kom18, Theorem 3.2] adapted to our situation, i.e.

𝑍(𝑥, 𝑡)≔

∞



𝑛=0

𝜎𝑛𝑦=𝑥

{−𝑆𝑛𝜉(𝑦)≤log (𝑡)}∼𝐺(𝑥 , log(𝑡))𝑡𝑧𝜈,

where

(𝑥, 𝑠)↦→𝐺(𝑥,𝑠)

, deﬁned on

𝐼N×R>0

, is bounded from above by inspecting the

corresponding function

𝐺

in [Kom18, Theorem 3.2], and

𝑠↦→𝐺(𝑥,𝑠)

is a constant

function in the aperiodic case and a periodic function in the periodic case. Therefore,

for ﬁxed

𝑦∈𝐼N

, with

𝑀≔exp (max−𝜉)

, we have by the second assertion of Lemma

4.40

card Γ𝑅

𝑡≤𝑍𝑦,𝑀e𝑑0𝑡/𝜆≪𝑡𝑧𝜈.

For the lower estimate we use an approximation argument involving the strong

bounded distortion property. Let

𝑎∈ K \ {

}

. Fix

𝑢0∈ C∞

𝑐((

))

such that

𝑢0(𝑎)>0 and deﬁne

𝑐0≔𝑒𝑑0[0,1](∇𝑢0)2dΛ

[0,1]𝑢2

0d𝜈,

where

𝑑0

is deﬁned as in Lemma 4.40. Applying Lemma 4.41 with

𝑥∈(𝐼𝑚)N

and

𝑚∈Nsuch that log(𝑘𝑚) −𝑑0>0 with 𝑘𝑚=exp(−max𝜉𝑚)yields

𝜔∈(𝐼𝑚)∗: log(𝑡e𝑑0/(𝑘𝑚𝑐0)) <−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 ) ≤ log (𝑡/𝑐0)⊂Γ𝐿

𝑡,𝑚

with

Γ𝐿

𝑡,𝑚 =𝜔∈ (𝐼𝑚)∗:−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 ) ≤ log (𝑡/𝑐0)<min

𝑣∈𝐼𝑚−

𝑆|𝜔𝑣 |𝑚𝜉𝑚(𝜔 𝑣𝑥 ).

For any 𝜔∈Γ𝐿

𝑡,𝑚 , we deﬁne

𝑢𝜔(𝑥)≔𝑢0◦𝑇−1

𝜔(𝑥), 𝑥 ∈𝑇𝜔((0,1)),

0, 𝑥 ∈ [0,1]\𝑇𝜔((0,1)),

115

4.4. Examples

which is an element of 𝐶∞

𝑐(𝑇𝜔(0,1)). Then, by Lemma 4.32, we have

𝐼𝜔(∇𝑢𝜔)2dΛ

𝐼𝜔𝑢2

𝜔d𝜈=[0,1](∇(𝑢𝜔◦𝑇𝜔))2𝑇′

𝜔−1dΛ

[0,1](𝑢𝜔◦𝑇𝜔)2e𝑆|𝜔|𝜓◦𝜋−1◦𝑇𝜔d𝜈

=[0,1](∇(𝑢0))2𝑇′

𝜔−1dΛ

[0,1]𝑢2

0e𝑆|𝜔|𝜓◦𝜋−1◦𝑇𝜔d𝜈

≤𝑐0𝑒−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 )≤𝑡 .

Since the supports of

(𝑢𝜔)𝜔∈Γ𝐿

𝑡,𝑚

are disjoint, it follows that the

(𝑢𝜔)𝜔∈Γ𝐿

𝑡,𝑚

are mu-

tually orthogonal both in

𝐿2

𝜈

and in

𝐻1

. Hence,

span 𝑢𝜔:𝜔∈Γ𝐿

𝑡,𝑚 

is a

card Γ𝐿

𝑡,𝑚 

dimensional subspace of 𝐻1

0. Hence, Lemma 2.19 yields

𝑁𝐷

𝜈(𝑡) ≥ card Γ𝐿

𝑡,𝑚 

≥card 𝜔∈(𝐼𝑚)∗: log(𝑡e𝑑0/(𝑘𝑚𝑐)) <−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 ) ≤ log (𝑡/𝑐).

We conclude

𝑁𝐷

𝜈(𝑡)

≥∞



𝑛=0

𝜔∈(𝐼𝑚)𝑛

{−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 )≤log(𝑡/𝑐)}−∞



𝑛=0

𝜔∈(𝐼𝑚)𝑛

{−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 )≤log(𝑡e𝑑0/(𝑘𝑚𝑐))}.

Moreover, by Lemma 4.30 we have 0

=𝑃(𝑧𝜈𝜉)=𝑃𝜎(𝑧𝜈𝜉𝑚)

as deﬁned in Lemma

4.30. Again, [Kom18, Theorem 3.2] applied to

𝜉𝑚

gives that there exists a function

(𝑥, 𝑠)↦→ 

𝐺(𝑥, 𝑠)

deﬁned on

(𝐼𝑚)N×R>0

, which is bounded away from zero by

inspecting the corresponding function 𝐺in [Kom18, Theorem 3.2], such that



𝑍(𝑥, 𝑡)≔

∞



𝑛=0

𝜔∈(𝐼𝑚)𝑛

{−

𝑆|𝜔|𝑚𝜉𝑚(𝜔𝑥 )≤log(𝑡)}∼

𝐺(𝑥, log(𝑡))𝑡𝑧𝜈.

In the aperiodic case

𝑠↦→ 

𝐺(𝑥, 𝑠)

is a constant function and hence in this case

we immediately get

𝑡𝑧𝜈≪𝑁𝜈,Λ(𝑡)

. In the periodic case,

𝑠↦→ 

𝐺(𝑥, 𝑠)

is periodic

with minimal period

𝑎>

0. To end this, we choose

𝑚

large enough such that

116

4.4. Examples

⌈𝑎/(log(𝑘𝑚)−𝑑0)⌉ =1 and exp(𝑑0)/𝑘𝑚<1. We ﬁnally then have

𝑁𝐷

𝜈(𝑡) ≥ 

𝑍(𝑥, 𝑡/𝑐)−

𝑍𝑥, e𝑑0/𝑘𝑚(𝑡/𝑐)

≥

𝑍(𝑥, 𝑡/𝑐)−

𝑍𝑥, e𝑑0/𝑘𝑚𝑎/(log(𝑘𝑚)−𝑑0)(𝑡/𝑐)

∼

𝐺(𝑥, log (𝑡/𝑐)) 𝑡

𝑐𝑧𝜈−

𝐺(𝑥, log (𝑡/𝑐)) 𝑡

𝑐𝑧𝜈e𝑑0/𝑘𝑚𝑎/(log(𝑘𝑚)−𝑑0)𝑧𝜈

=𝑡𝑧𝜈

𝐺(𝑥, log (𝑡/𝑐))

𝑐𝑧𝜈1−e𝑑0/𝑘𝑚𝑧𝜈𝑎/(log(𝑘𝑚)−𝑑0)

≫𝑡𝑧𝜈,

where we used log(𝑡/𝑐)−log e𝑑0/𝑘𝑚𝑎/(log (𝑘𝑚)−𝑑0)𝑡/𝑐=𝑎.□

4.4.1.3 Spectral dimension of weak Gibbs measures with overlap

This section relies on results from [PS00; Fen07; BF20] on the

𝐿𝑞

-spectrum together

with the regularity result stated in Section 4.3.2. Let

𝜈

and

𝜇

be deﬁned as in Section

4.4.1 such that

𝜈({0,1})=

0. Here we do not assume any separation conditions for

the C1-cIFS Φ.

Recall that

is non-trivial, i.e. there is more than one contraction and the

𝑇𝑖

’s do not

share a common ﬁxed point. Hence, by Proposition 2.48, it follows that self-similar

measures with or without OSC are atomless as long as

is non-trivial. However, it

is an open question under which condition the same applies to weak Gibbs measures

without OSC.

First, we will prove that the

𝐿𝑞

-spectrum of

𝜈

exists as a limit on

(

]

. Combining

this with Corollary 4.12 we conclude that the spectral dimension exists and is given

by 𝑞𝔍𝜈. To this end we need the following lemmas.

Lemma 4.43. We have for any 𝐺⊂𝐼∗with 𝑢∈𝐺[𝑢]=𝐼Nand 𝐸∈𝔅([0,1]) that

𝜈(𝐸)≥

𝑢∈𝐺

𝑐|𝑢|𝜇([𝑢])𝜈𝑇−1

𝑢(𝐸)

with 𝑐𝑛≔e−𝑛−1

𝑖=0var𝑖(𝜓)(and therefore log (𝑐𝑛)=𝑜(𝑛)).

117

4.4. Examples

Proof. For all 𝐸∈𝔅([0,1]) and 𝑢∈𝐼∗,we have

𝜇𝜋−1(𝐸)∩ [𝑢]=[𝑢]

𝐸◦𝜋d𝜇=𝐿|𝑢|

𝜓(

[𝑢](𝑥)

𝐸(𝜋(𝑥))) d𝜇(𝑥)

=

𝜏∈𝐼|𝑢|

e𝑆|𝑢|𝜓(𝜏𝑥)

[𝑢](𝜏𝑥 )

𝐸(𝜋(𝜏𝑥 )) d𝜇(𝑥)

=e𝑆|𝑢|𝜓(𝑢𝑥)

𝐸(𝜋(𝑢𝑥)) d𝜇(𝑥)

≥e−|𝑢|−1

𝑖=0var𝑖(𝜓)𝜈𝑇−1

𝑢(𝐸)𝜇([𝑢]) .

Setting 𝑐𝑛≔e−𝑛−1

𝑖=0var𝑖(𝜓)and summing over 𝑢∈𝐺, we obtain

𝜈(𝐸)=

𝑢∈𝐺

𝜇𝜋−1(𝐸)∩ [𝑢]≥

𝑢∈𝐺

𝑐|𝑢|𝜇([𝑢])𝜈𝑇−1

𝑢(𝐸).

Also, the continuity of the potential 𝜓implies log (𝑐𝑛)=𝑜(𝑛).□

For 𝑢∈𝐼∗let us deﬁne K𝑢≔𝑇𝑢(K). Then, for 𝑛≥2, the set

𝑊𝑛≔{𝑢∈𝐼∗: diam (K𝑢)≤2−𝑛<diam (K𝑢−)}

deﬁnes a partition of 𝐼N.

Lemma 4.44. For any 0

<𝑞<

1there exists a sequence

(𝑠𝑛)𝑛∈RN

such that

log(𝑠𝑛)=𝑜(𝑛)and for every 𝑛,𝑚 ∈Nand 

𝑄∈ D𝑁

𝑛, we have



𝐵∈D𝑁

𝑛

𝐵∼

𝑄

𝑄∈D𝑁

𝑚+𝑛,

𝑄⊂𝐵

𝜈(𝑄)𝑞≥𝑠𝑛𝜈(

𝑄)𝑞min

𝑢∈𝑊𝑛

𝑄∈D𝑁

𝑚+𝑛

𝜈𝑇−1

𝑢(𝑄)𝑞

where 𝐵∼

𝑄means that the closures of 𝐵and 

𝑄intersect.

Proof. As in [PS00] for 𝑛,𝑚 ∈N,𝑢∈𝑊𝑛, and 𝐴∈ D𝑁

𝑛, let us deﬁne

𝑤(𝑢,𝐴)≔

𝑄∈D𝑁

𝑛+𝑚:𝑄⊂𝐴

𝜈𝑇−1

𝑢(𝑄)𝑞

The interval

𝐴∈ D𝑁

𝑛

on which

𝑤(𝑢,𝐴)

attains its maximum will be called

𝑞

-heavy

for

𝑢∈𝑊𝑛

. We will denote the

𝑞

-heavy box by

𝐻(𝑢)

(if there are more than one

interval which maximizes

𝑤(𝑢, ·)

, we choose one of them arbitrarily). Note that

every

K𝑢

with

𝑢∈𝑊𝑛

intersects at most 3 intervals in

D𝑁

𝑛

. Hence, we obtain for all

118

4.4. Examples

𝑢∈𝑊𝑛

𝑄∈D𝑁

𝑛+𝑚

𝜈𝑇−1

𝑢(𝑄)𝑞=

𝐵∈D𝑁

𝑛

𝑄∈D𝑁

𝑛+𝑚:𝑄⊂𝐵

𝜈𝑇−1

𝑢(𝑄)𝑞

≤3

𝑄∈D𝑁

𝑛+𝑚:𝑄⊂𝐻(𝑢)

𝜈𝑇−1

𝑢(𝑄)𝑞

This leads to



𝑄∈D𝑁

𝑛+𝑚:

𝑄⊂𝐻(𝑢)

𝜈𝑇−1

𝑢(𝑄)𝑞≥

𝑄∈D𝑁

𝑛+𝑚

𝜈𝑇−1

𝑢(𝑄)𝑞

3≥min

𝑣∈𝑊𝑛

𝑄∈D𝑁

𝑛+𝑚

𝜈𝑇−1

𝑣(𝑄)𝑞

3.(4.4.6)

Further, for every 𝑄∈ D𝑁

𝑛+𝑚and 𝐵∈ D𝑁

𝑛, by Lemma 4.43, we have

𝜈(𝑄)≥

𝑢∈𝑊𝑛

𝑐|𝑢|𝜇([𝑢])𝜈𝑇−1

𝑢(𝑄)≥

𝑢∈𝑊𝑛:𝐵=𝐻(𝑢)

𝑐|𝑢|𝜇([𝑢])𝜈𝑇−1

𝑢(𝑄)

≥min

𝑢∈𝑊𝑛

𝑐|𝑢|

𝑢∈𝑊𝑛:𝐵=𝐻(𝑢)

𝜇([𝑢])𝜈𝑇−1

𝑢(𝑄).

Setting

𝑝−(𝐵)≔

𝑢∈𝑊𝑛:𝐵=𝐻(𝑢)

𝜇([𝑢]),

and if

𝑝−(𝐵)>

0, using the concavity of the function

𝑥↦→ 𝑥𝑞

for 0

<𝑞<

1, we

obtain

𝜈(𝑄)𝑞≥𝑝−(𝐵)𝑞min

𝑢∈𝑊𝑛

𝑐|𝑢|𝑞

𝑢∈𝑊𝑛:𝐵=𝐻(𝑢)

𝜇([𝑢])𝜈𝑇−1

𝑢(𝑄)

𝑝−(𝐵)

𝑞

≥𝑝−(𝐵)𝑞−1min

𝑢∈𝑊𝑛

𝑐|𝑢|𝑞

𝑢∈𝑊𝑛:𝐵=𝐻(𝑢)

𝜇([𝑢])𝜈𝑇−1

𝑢(𝑄)𝑞

Summing over 𝑄∈ D𝑁

𝑛+𝑚with 𝑄⊂𝐵, and using (4.4.6), we infer



𝑄⊂𝐵,

𝑄∈D𝑁

𝑛+𝑚

𝜈(𝑄)𝑞≥𝑝−(𝐵)𝑞−1min

𝑢∈𝑊𝑛

𝑐|𝑢|𝑞

𝑢∈𝑊𝑛:𝐵=𝐻(𝑢)

𝜇([𝑢]) 

𝑄⊂𝐵,

𝑄∈D𝑁

𝑛+𝑚

𝜈𝑇−1

𝑢(𝑄)𝑞

≥𝑝−(𝐵)𝑞

3min

𝑢∈𝑊𝑛

𝑐|𝑢|𝑞

min

𝑣∈𝑊𝑛

𝑄∈D𝑁

𝑛+𝑚

𝜈𝑇−1

𝑣(𝑄)𝑞

which is also valid in the case

𝑝−(𝐵)=

0. For



𝑄∈ D𝑁

𝑛

and

𝑢∈𝑊𝑛

with

K𝑢∩

𝑄≠∅

we have

K𝑢⊂𝐵∼

𝑄,𝐵 ∈D𝑁

𝑛𝐵

, as a consequence of

diam (K𝑢)≤

−𝑛

. In particular,

119

4.4. Examples

every

K𝑢

that intersects



𝑄

must have an interval

𝐵∈ D𝑁

𝑛

with

𝐵∼

𝑄

which is

𝑞-heavy for 𝑢. Hence, we obtain

𝜈

𝑄≤

𝑢∈𝑊𝑛:K𝑢∩

𝑄≠∅

𝜇([𝑢]) ≤

𝐵∼

𝑄:𝐵∈D𝑁

𝑛

𝑢∈𝑊𝑛:𝐵=𝐻(𝑢)

𝜇([𝑢]) =

𝐵∼

𝑄:𝐵∈D𝑁

𝑛

𝑝−(𝐵).

Using 0 <𝑞<1, we conclude

𝜈

𝑄𝑞≤

𝐵∼

𝑄:𝐵∈D𝑁

𝑛

𝑝−(𝐵)

𝑞

≤

𝐵∼

𝑄:𝐵∈D𝑁

𝑛

𝑝−(𝐵)𝑞.

Summing over all 𝐵∈ D𝑁

𝑛with 𝐵∼

𝑄gives



𝐵∼

𝑄:

𝐵∈D𝑁

𝑛



𝑄⊂𝐵:

𝑄∈D𝑁

𝑛+𝑚

𝜈(𝑄)𝑞≥

𝐵∼

𝑄:𝐵∈D𝑁

𝑛

𝑝−(𝐵)𝑞

3min

𝑢∈𝑊𝑛

𝑐|𝑢|𝑞

min

𝑣∈𝑊𝑛

𝑄∈D𝑁

𝑛+𝑚

𝜈𝑇−1

𝑣(𝑄)𝑞

≥

𝜈

𝑄𝑞

3min

𝑢∈𝑊𝑛

𝑐|𝑢|𝑞

min

𝑣∈𝑊𝑛

𝑄∈D𝑁

𝑛+𝑚

𝜈𝑇−1

𝑣(𝑄)𝑞

Note that for every 𝑢∈𝑊𝑛, by the deﬁnition of 𝑊𝑛, we have

|𝑢|<𝑛log(2) −log (𝛼max )

−log (𝛼max)

with

𝛼max ≔max𝑖=1,...,𝑛 max𝑥∈[0,1]𝑇′

𝑖(𝑥)

. Thus, setting

𝑠𝑛≔

−1min𝑢∈𝑊𝑛𝑐𝑞

|𝑢|

, we

ﬁnd that

lim𝑛→∞𝑛−1log(𝑠𝑛)=

0, where we used the elementary fact that for any

two sequences

(𝑥𝑛)𝑛∈RN

and

(𝑦𝑛)𝑛∈NN

with

𝑥𝑛=o(𝑛)

𝑦𝑛≪𝑛

, we have

𝑥𝑦𝑛=o(𝑛).□

Proposition 4.45. The 𝐿𝑞-spectrum 𝛽𝜈of 𝜈exists on (0,1]as limit.

Proof.

Let 0

<𝑞<

1. From [Fen07, Proposition 3.3] (which holds true for all

Borel probability measures with support

, see remark after Proposition 3.3 in

[Fen07]) it follows that there exists a sequence

𝑏𝑞,𝑛𝑛

of positive numbers with

log(𝑏𝑞,𝑛 )=𝑜(𝑛), such that for all 𝑚,𝑛 ∈Nand 𝑢∈𝑊𝑛

𝑏𝑞,𝑛 

𝑄∈D𝑁

𝑚

𝜈(𝑄)𝑞≤

𝐶∈D𝑁

𝑚+𝑛

𝜈𝑇−1

𝑢(𝑄)𝑞

120

4.4. Examples

In tandem with Lemma 4.44, for every 

𝑄∈ D𝑁

𝑛, we obtain



𝐵∈D𝑁

𝑛,𝐵∼

𝑄

𝑄∈D𝑁

𝑚+𝑛:𝑄⊂𝐵

𝜈(𝑄)𝑞≥𝑠𝑛𝜈(

𝑄)𝑞min

𝑢∈𝑊𝑛

𝑄∈D𝑁

𝑚+𝑛

𝜈𝑇−1

𝑢(𝑄)𝑞

≥𝑏𝑞,𝑛𝑠𝑛𝜈(

𝑄)𝑞

𝑄∈D𝑁

𝑚

𝜈(𝑄)𝑞.

Clearly,

log 𝑏𝑞,𝑛𝑠𝑛=𝑜(𝑛)

. Hence, we can apply [Fen07, Proposition 4.4], which

shows that 𝛽𝜈exists as a limit on (0,1].□

With this knowledge, we obtain the following theorem.

Theorem 4.46. The spectral dimension of Δ𝐷

𝜈exists and equals 𝑞𝔍𝜈.

Proof. The proof follows from Corollary 4.12 and Proposition 4.45. □

Corollary 4.47. Let

𝜈

be a self-conformal measure on

[

]

. Then the spectral

dimension of Δ𝐷

𝜈exists and equals 𝑞𝔍𝜈.

4.4.2 Homogeneous Cantor measures

Let us recall the construction of general homogeneous Cantor measures as in [Arz14;

Min20; BH97], allowing us to construct examples for which the spectral dimension

does not exist. Let

𝐽

be ﬁnite or countably inﬁnite subset of

. For every

𝑗∈𝐽

we deﬁne an iterated function system

S(𝑗)

. For

𝑖=

2 let

𝑆(𝑗)

𝑖

[𝑎,𝑏]→[𝑎,𝑏]

deﬁned by

𝑆(𝑗)

𝑖(𝑥)=𝑟(𝑗)

𝑖𝑥+𝑐(𝑗)

𝑖

with 𝑟(𝑗)

𝑖∈ (0,1)and 𝑐(𝑗)

𝑖∈Rare chosen such that

𝑎=𝑆(𝑗)

1(𝑎)<𝑆(𝑗)

1(𝑏) ≤ 𝑆(𝑗)

2(𝑎)<𝑆(𝑗)

2(𝑏)=𝑏.

This ensures the open set condition. We deﬁne

S(𝑗)=𝑆(𝑗)

1, 𝑆 (𝑗)

2

. Moreover let

us deﬁne an environment sequence

𝜉≔𝜉𝑖𝑖∈𝐽N

. Each

𝜉𝑖

represents an iterated

function system

S(𝜉𝑖)

. To give a suitable coding, we deﬁne the following word

space

𝑊𝑛≔{1,2}𝑛

of words with length

𝑛

. For

𝑛∈N

and

𝜔≔(𝜔1. . . 𝜔𝑛) ∈𝑊𝑛

we set

𝑆(𝜉)

𝜔≔𝑆(𝜉1)

𝜔1◦𝑆(𝜉2)

𝜔2◦···◦𝑆(𝜉𝑛)

𝜔𝑛.

Now, for any environment sequence

𝜉

, we construct a probability measure

𝜈(𝜉)

[𝑎,𝑏]with support 𝐾(𝜉)deﬁned by

𝐾(𝜉)≔

∞



𝑛=1

𝜔∈𝑊𝑛𝑆(𝜉1)

𝜔1◦𝑆(𝜉2)

𝜔2◦···◦𝑆(𝜉𝑛)

𝜔𝑛([𝑎,𝑏]).

121

4.4. Examples

For any

𝑗∈𝐽

, let

𝑝𝑗

1, 𝑝 𝑗

2∈ (

with

2

𝑖=1𝑝𝑗

𝑖=

1. Moreover, for

𝑛∈N

and

𝜔≔(𝜔1. . . 𝜔𝑛) ∈ 𝑊𝑛

, we deﬁne

𝑝(𝜉)

𝜔≔𝑛

𝑖=1𝑝(𝜉𝑖)

𝜔𝑖

. Then deﬁne the following

sequence of probability measures:

𝜈0≔1

𝑏−𝑎Λ|[𝑎,𝑏]

and for

𝑛∈N

and

𝐴∈𝔅([𝑎,𝑏])

𝜈𝑛(𝐴)≔

𝜔∈𝑊𝑛

𝑝(𝜉)

𝜔𝜈0𝑆(𝜉)

𝜔−1(𝐴).

Then we deﬁne 𝜈(𝜉)by

𝜈(𝜉)(𝐴)=lim

𝑛→∞𝜈𝑛(𝐴).

The theorem of Vitali-Hahn-Saks ensures that

𝜈(𝜉)

is a probability measure on

𝔅([𝑎,𝑏]) (see [Arz14, Lemma 3.1.2]) and for every 𝑛∈Nand 𝜔∈𝑊𝑛, we have

𝜈(𝜉)𝑆(𝜉)

𝜔([𝑎,𝑏])=𝑝(𝜉)

𝜔.

Example 4.48 (Homogeneous Cantor measure with non-converging 𝐿𝑞-spectrum).

Now, let us consider the following environment

𝜉𝑖≔1,∃ℓ∈N0: 22ℓ<𝑖≤22ℓ+1,

2,∃ℓ∈N0: 22ℓ+1<𝑖≤22ℓ+2,

𝑆(1)

1(𝑥)≔𝑥

4, 𝑆 (1)

2(𝑥)≔𝑥

4+3

𝑆(2)

1(𝑥)≔𝑥

, and

𝑆(2)

2(𝑥)≔𝑥

16 +15

for

𝑥∈ [

]

Furthermore, let

𝑝1,𝑝2∈ (

)

be with

𝑝1+𝑝2=

1. Then we deﬁne for every

𝑗∈{1,2}:𝑝𝑗

1≔𝑝1and 𝑝𝑗

2≔𝑝2. First, observe that for 𝑛∈Nand 𝜔∈𝑊𝑛,

Λ𝑆(𝜉)

𝜔([0,1])=2−(8/3)·22𝑙+1+10/3−4(𝑛−22𝑙+1),22𝑙+1<𝑛≤22𝑙+2,

2−10/3·22𝑙+2+10/3−2(𝑛−22𝑙+2),22𝑙+2<𝑛≤22𝑙+3.

Deﬁne for 𝑛∈N

𝑅(𝑛)≔−(8/3) · 22ℓ+1+10/3−4𝑛−22ℓ+1,22ℓ+1<𝑛≤22ℓ+2,

−(10/3) · 22ℓ+2+10/3−2𝑛−22ℓ+2,22ℓ+2<𝑛≤22ℓ+3.

122

4.4. Examples

Hence, for all 𝑛∈Nwith 22𝑙+1<𝑛≤22𝑙+2,𝑞≥0, and 𝜔∈𝑊𝑛, we obtain

−log Λ𝑆(𝜉)

𝜔([0,1])log 

2−𝑅(𝑛)−1



𝑙=0,

𝜈((𝑙2𝑅(𝑛),(𝑙+1)2𝑅(𝑛)])>0

𝜈(𝜉)(𝑙2𝑅(𝑛),(𝑙+1)2𝑅(𝑛)]𝑞

−log Λ𝑆(𝜉)

𝜔([0,1])log 

𝑙∈𝑊𝑛

𝜈(𝜉)𝑆(𝜉)

𝑙([𝑎,𝑏])𝑞

=log2𝑝𝑞

1+𝑝𝑞

2

−4·22𝑙+1

3𝑛+4−10/(3𝑛)

and, for 22𝑙+2<𝑛≤22𝑙+3,

−log Λ𝑆(𝜉)

𝜔([0,1])log 

𝑙∈𝑊𝑛

𝜈(𝜉)𝑆(𝜉)

𝑙([𝑎,𝑏])𝑞=log 𝑝𝑞

1+𝑝𝑞

2

−log Λ𝑆(𝜉)

𝜔([0,1])

=log2𝑝𝑞

1+𝑝𝑞

2

422𝑙+2

3𝑛+2−10/(3𝑛).

Therefore, by Fact 2.30,

𝛽𝜈(𝜉)(𝑞)=3

8log2𝑝𝑞

1+𝑝𝑞

2for 0≤𝑞≤1,

10 log2𝑝𝑞

1+𝑝𝑞

2for 𝑞>1

and

𝛽𝜈(𝜉)(𝑞)≔liminf

𝑛→∞ 𝛽𝑛(𝑞)=3

10 log2𝑝𝑞

1+𝑝𝑞

2for 0 ≤𝑞≤1

8log2𝑝𝑞

1+𝑝𝑞

2for 𝑞>1.

Hence, by Theorem 4.10, the upper spectral dimension of

Δ𝐷

𝜈(𝜉)

is given by the

unique solution of

𝑝

𝑞𝔍𝜈(𝜉)

1+𝑝

𝑞𝔍𝜈(𝜉)

2=28·𝑞𝔍𝜈(𝜉)/3.

Furthermore, the unique ﬁxed point of

𝛽𝜈(𝜉)

, denoted by

𝑞𝔍𝜈(𝜉)

, is the unique solution

of 2−10

3𝑝1𝑞𝔍𝜈(𝜉)+2−10

3𝑝2𝑞𝔍𝜈(𝜉)=1.

See Figure 4.4.1 for the two graphs.

123

4.4. Examples

𝑞

𝛽𝜈(𝜉)(𝑞)

dim𝑀𝜈(𝜉)

Figure 4.4.1 Illustration of 𝛽𝜈(𝜉)and liminf𝑛→∞ 𝛽𝜈(𝜉)

𝑛with 𝑝1=0.25.

For the special case 𝑝1=1/2, we obtain

𝛽𝜈(𝜉)(𝑞)=3

8(1−𝑞)for 0≤𝑞≤1,

10 (1−𝑞)for 𝑞>1.

and

𝛽𝜈(𝜉)(𝑞)≔liminf

𝑛→∞ 𝛽𝜈(𝜉)

𝑛(𝑞)=3

10 (1−𝑞)for 0 ≤𝑞≤1,

8(1−𝑞)for 𝑞>1.

In this special case, we have,

dim𝑀(𝜈)=

10 and

dim𝑀(𝜈)=

8. Now, applying

Proposition 4.9 in tandem with Theorem 4.10, we conclude

𝑠𝐷

𝜈(𝜉)≤dim𝑀(𝜈)

1+dim𝑀(𝜈)=3

13 <3

11 =dim𝑀(𝜈)

1+dim𝑀(𝜈)=𝑠𝐷

𝜈(𝜉).

Now, let us prove that

𝑠𝐷

𝜈(𝜉)=3

. Note that for

𝑚=

and all

𝜔∈𝑊𝑛

, there

exists 𝜂∈𝑊2such that

𝑆(𝜉)

𝜔𝜂 ([0,1]) ⊂𝑆(𝜉)

𝜔([0,1])1/𝑚.

Hence, we obtain

𝜈𝑆(𝜉)

𝜔([0,1])1/𝑚Λ𝑆(𝜉)

𝜔([0,1])1/𝑚≥2−𝑛+𝑅(𝑛)−2

𝑚.(4.4.7)

124

4.4. Examples

Recall that by Proposition 4.1, for 𝑥>0,

N𝐿

𝑚(𝑥)=sup card (𝑃):𝑃∈Π0: min

𝐶∈𝑃𝜈⟨𝐼⟩1/𝑚Λ⟨𝐼⟩1/𝑚≥4

𝑥(𝑚−1)≤𝑁𝜈(𝑥).

Now, with 𝑥𝑛=𝑚2𝑛−𝑅(𝑛)+4

𝑚−1, by (4.4.7), we have that N𝐿

𝑚(𝑥𝑛) ≥ 2𝑛, implying

log 𝑁𝐷

𝜈(𝑥𝑛)

log(𝑥𝑛)

≥1−𝑅(𝑛)

𝑛+4

𝑛+log (𝑚/(𝑚−1))

𝑛log(2)−1

=







1+(8/3) · 22𝑙+1−10/3+4(𝑛−22𝑙+1)+4

𝑛−log(𝑚/(𝑚−1))

𝑛log(2)−1

,22𝑙+1<𝑛≤22𝑙+2,

1+(10/3)·22𝑙+2−10/3+2(𝑛−22𝑙+2)+4

𝑛−log(𝑚/(𝑚−1))

𝑛log(2)−1

,22𝑙+2<𝑛≤22𝑙+3,

≥







1+8

3+(4−10/3)/𝑛−log(𝑚/(𝑚−1))

𝑛log(2)−1

,22𝑙+1<𝑛≤22𝑙+2,

1+10

3+(4−10/3)/𝑛−log(𝑚/(𝑚−1))

𝑛log(2)−1

,22𝑙+2<𝑛≤22𝑙+3.

This shows

liminf

𝑛→∞

log 𝑁𝐷

𝜈(𝑥𝑛)

log(𝑥𝑛)≥1

1+10/3.

Since 𝑥𝑛≤𝑥𝑛+1≤24𝑥𝑛, we infer

liminf

𝑥→∞

log 𝑁𝐷

𝜈(𝑥)

log(𝑥)≥lim inf

𝑛→∞

log 𝑁𝐷

𝜈(𝑥𝑛)

log(𝑥𝑛)≥1

1+10/3.

Indeed, for 𝑥>0, choose 𝑛∈Nsuch that 𝑥𝑛<𝑥≤𝑥𝑛+1and therefore

liminf

𝑥→∞

log 𝑁𝐷

𝜈(𝑥)

log(𝑥)≥lim inf

𝑛→∞

log 𝑁𝐷

𝜈(𝑥𝑛)

log(𝑥𝑛) +log(𝑥𝑛+1/𝑥𝑛)≥1

1+10/3.

4.4.3 Purely atomic case

In this section we give examples of singular measures

𝜂

(0,1)

of pure point

type such that the spectral dimension attains any value in

[0,1/2]

. To ﬁx notation,

throughout this section, we write

𝜂≔𝑘𝑝𝑘𝛿𝑥𝑘

with

(𝑝𝑘)𝑘∈(R>0)N

𝑘𝑝𝑘<∞

and (𝑥𝑘)𝑘∈(0,1)N.

The ﬁrst example shows that it is possible for the spectral dimension to be 0 even

though the Minkowski dimension is 1.

Example 4.49. In this example we consider purely atomic measures

𝜂

with

(𝑥𝑛)𝑛∈

(Q∩(0,1))N

such that

𝑥𝑛≠𝑥𝑚

for

𝑚≠𝑛

, and there exists

𝐶1>

0 such that

𝑝𝑛≤

125

4.4. Examples

𝐶1

−𝑛

for all

𝑛∈N

. We will show that the spectral dimension exists and equals

𝑠𝐷

𝜂=0. We deﬁne 𝐼𝑛

𝑘≔𝑥𝑘−𝑏𝑛

e𝑛,𝑥𝑘+𝑏𝑛

e𝑛∩[0,1]with

𝑏𝑛≔min 𝑥𝑙1−𝑥𝑙2:𝑙1≠𝑙2, 𝑙1,𝑙2≤𝑛

for all 𝑘=1, . . . ,𝑛. Then,

max

𝑘=1,...,𝑛

Λ𝐼𝑛

𝑘𝜈𝐼𝑛

𝑘≤∞



𝑙=1

𝑝𝑙max

𝑘=1,...,𝑛

Λ(𝐼𝑛

𝑘) ≤ 𝐶1

e−1

e𝑛.

Let

𝐴𝑛

𝑘

denote the disjoint half open intervals such that

(

]\𝑛

𝑘=1𝐼𝑛

𝑘=𝑚(𝑛)

𝑘=1𝐴𝑛

𝑘

with 𝑚(𝑛) ≤ 𝑛+1. Then we conclude

max

𝑘=1,...,𝑚 (𝑛)

Λ(𝐴𝑛

𝑘)𝜈(𝐴𝑛

𝑘) ≤ ∞



𝑙=𝑛+1

𝑝𝑙≤𝐶1

e−𝑛

e−1.

Now, Proposition 4.4 yields

𝑁𝐷

𝜈𝑒𝑛(𝑒−1)

2𝐶1≤2

M𝔍𝜈𝑒𝑛(𝑒−1)

2𝐶1≤2𝑛+1.

Hence, for all 𝑥≥𝑒(𝑒−1)/(2𝐶1), we have

𝑁𝐷

𝜈(𝑥)≤4𝐶1

𝑒−1log (𝑥)+1.

Thus, we obtain 𝑠𝐷

𝜈=0.

(𝑥𝑘)𝑘∈(0,1)N

is strictly decreasing, then in [FW17]

Δ𝐷

𝜂

is called Kre

ın–Feller

operator of Stieltjes type. We start with a general observation which is a consequence

of Proposition 4.4.

Lemma 4.50. Assume that

(𝑥𝑘)𝑘∈(0,1)N

is strictly decreasing such that for an

increasing function 𝑓:N>1→R+and all 𝑘∈N>1,

𝑥𝑘+𝑥𝑘−1

∞



𝑙=𝑘

𝑝𝑙≤1/𝑓(𝑘).

Then, for all

𝑥≥

0, we have

𝑁𝐷

𝜂(𝑥) ≤ min{

𝑓

ˇ−1(2𝑥)+

, 𝑓

ˇ−1(6𝑥)}

with

𝑓

ˇ−1(𝑥)≔

inf {𝑛∈N>1:𝑓(𝑛)≥𝑥}.

Proof. By our assumption, for all 𝑘∈N, we have

𝜈0,𝑥𝑘+𝑥𝑘−1

2Λ0,𝑥𝑘+𝑥𝑘−1

2=𝑥𝑘+𝑥𝑘−1

∞



𝑙=𝑘

𝑝𝑙.

126

4.4. Examples

Observe that for 𝑥>0

𝑥𝑘+𝑥𝑘−1

2∞

𝑙=𝑘𝑝𝑙≥𝑓(𝑘)≥2𝑥=⇒𝑘≥𝑓

ˇ−1(2𝑥).

For ﬁxed 𝑘∈Ndeﬁne the following 𝜈-partition

𝐼1≔𝑥1−min{𝑥1−𝑥2,1/𝑥}

4,𝑥1+min{1−𝑥1,1/𝑥})

4

and

𝐼𝑗≔𝑥𝑗−min 𝑥𝑗−𝑥𝑗+1,1/𝑥

4,𝑥𝑗+min 𝑥𝑗−𝑥𝑗−1,1/𝑥

4

for

𝑗=

, . . . ,𝑘 −

1 and

𝐼𝑘≔(

,(𝑥𝑘+𝑥𝑘−1)/

]

. Hence, for

𝑘=𝑓

ˇ−1(2𝑥)

, we see that

max

𝑖=1,...,𝑘 𝜈(𝐼𝑖)Λ(𝐼𝑖)<1/𝑥.

Now, Proposition 4.4 yields

𝑁𝐷

𝜂(𝑥) ≤ 2

M𝔍𝜈(𝑥)+1≤2𝑓

ˇ−1(2𝑥)+1.

Analogously, we obtain

𝑁𝐷

𝜂(𝑥) ≤ 

M𝔍𝜈(5𝑥) ≤ 𝑓

ˇ−1(6𝑥).□

Example 4.51 (Dirac comb with exponential decay).Observe that if

(𝑥𝑛)𝑛∈ (

𝑘≥𝑚𝑝𝑘𝑚

decays exponentially, then by Lemma 4.50, we have

𝑁𝐷

𝜂(𝑥) ≪

log(𝑥)

, hence the spectral dimension

𝑠𝐷

𝜂

equals 0. In particular, by Theorem 4.10,

we have 𝑞𝔍𝜈=0. Consequently, the (Neumann) 𝐿𝑞-spectrum is given by

𝛽𝜂(𝑞)=dim𝑀(𝜂), 𝑞 =0,

0, 𝑞 >0.

Example 4.52 (Dirac comb with at most power law decay).Assume that

(𝑥𝑘)𝑘∈

(0,1)Nis strictly decreasing and

𝑝𝑛≫𝑛−𝑢1𝑓1(𝑛),(𝑥𝑛−𝑥𝑛+1) ≫𝑛−𝑢2𝑓2(𝑛),

with 𝑢1,𝑢2≥1 and lim𝑛→∞ log(𝑓𝑖(𝑛))/log(𝑛)=0 for 𝑖=1,2. Then we have

𝑢1+𝑢2≤𝑠𝐷

𝜂

127

4.4. Examples

and, in particular if 𝑢1+𝑢2=2, we have 𝑠𝐷

𝜂=1/2. To see this, deﬁne

𝐼𝑘≔𝑥𝑘−min {𝑥𝑘−𝑥𝑘+1,𝑥𝑘−1−𝑥𝑘}

2,𝑥𝑘+min {𝑥𝑘−𝑥𝑘+1, 𝑥𝑘−1−𝑥𝑘}

2

for

𝑘=

, . . . ,𝑛

. We then have

𝜈(⟨𝐼𝑘⟩1/2)=𝜈(𝐼𝑘)

and for ﬁxed

𝜀>

0, we have for

𝑛

large enough

𝐶𝑛−(𝑢1+𝑢2)−𝜀≤𝐶𝑛−(𝑢1+𝑢2)𝑓1(𝑛)𝑓2(𝑛) ≤ min

𝑘=1,...,𝑛 𝜈⟨𝐼𝑘⟩1/2Λ⟨𝐼𝑘⟩1/2

with 𝐶>0 suitable, which implies for every 𝜀>0

N𝐿

2(𝑥)≫𝑥

𝑢1+𝑢2+𝜀.

This proves the claim. Since, in the case

𝑢1+𝑢2=

2, we always have

𝑠𝐷

𝜂≤

2. It

readily follows that 𝑠𝐷

𝜂=1/2.

Example 4.53 (Dirac comb with at most geometric decay and full dimension).

Assume that (𝑥𝑘)𝑘∈(0,1)Nis strictly decreasing and

𝑓1(𝑛)

𝑛≪𝑝𝑛,𝑓2(𝑛)

𝑛≪(𝑥𝑛−1−𝑥𝑛)

with

lim𝑛→∞ log(𝑓𝑖(𝑛))/log(𝑛)=

0 for

𝑖=

2. Then the spectral dimension exists

and equals

𝑠𝐷

𝜂=1/2

and, for 𝑞∈ [0,1],

𝛽𝜂(𝑞)=1−𝑞 , 𝑞 ∈[0,1],

0, 𝑞 >1.

Indeed, observe that Example 4.52 implies 1

≤𝑠𝐷

𝜂

and by and Corollary 4.17

Theorem 4.10 we also have

𝑠𝐷

𝜂≤𝑠𝐷

𝜂=𝑞𝔍𝜂≤

2, which shows

𝑠𝐷

𝜂=

2. The second

statement is then a direct consequence of the ﬁrst part of Corollary 4.16.

The following example shows that the spectral dimension attains every value in

(0,1/2).

Example 4.54 (Dirac comb with power law decay).If

lim

𝑛→∞−log(𝑝𝑛)/log(𝑛)=𝑢1>1 and 𝑥𝑘≔(𝑘+1)−𝑢2, 𝑘 ∈N, 𝑢2>0,

then the Neumann

𝐿𝑞

-spectrum exists as a limit on the positive half-line and we

have

𝛽𝜂(𝑞)=1

𝑢2+1−𝑞𝑢1

𝑢2+1for 𝑞∈[0,1/𝑢1],

0 for 𝑞>1/𝑢1.

128

4.4. Examples

Consequently,

𝜂

is Neumann

𝔍𝜂

-regular and the spectral dimension exists and equals

𝑠𝐷

𝜂=1

𝑢1+𝑢2+1.

In particular, for 𝑢1=𝑢2+1, we have

𝑠𝐷

𝜂=dim𝑀(𝜂)

This can be seen as follows. For every 𝜀>0, we have uniformly in 𝑛∈N

𝑛−(𝑢1+𝜀)≪𝑝𝑛≪𝑛−𝑢1+𝜀.

For suitable 𝐶>0,

𝑥𝑚−𝑥𝑚+1=(𝑚+1)𝑢2−𝑚𝑢2

𝑚𝑢2(𝑚+1)𝑢2=1

(𝑚+1)𝑢2𝑚𝑚+1

𝑚𝑢2−1

1/𝑚≥𝐶

𝑚𝑢2+1.

If 2

−𝑛<𝐶(𝑚+

)−(𝑢2+1)

, then

𝑚<(2𝑛𝐶)1

𝑢2+1

. Combining these observations, we

obtain



𝐶∈D𝑁

𝑛

𝜂(𝐶)𝑞≥

𝐶1/(𝑢2+1)2𝑛/(𝑢2+1)



𝑘=1

𝑝𝑞

𝑘

≫

𝐶1/(𝑢2+1)2𝑛/(𝑢2+1)



𝑘=1

𝑘−(𝑢1+𝜖)𝑞

≍2𝑛(−(𝑢1+𝜖)𝑞+1)/(𝑢2+1).

For 𝑞∈[0,1/(𝑢1+𝜀)), this gives

𝛽𝜂(𝑞)≥liminf

𝑛→∞ 𝛽𝜂

𝑛(𝑞)≥1

𝑢2+1−𝑞𝑢1+𝜀

𝑢2+1.

Letting 𝜀→0, showing for 𝑞∈ [0,1/𝑢1]

𝛽𝜂(𝑞)≥liminf

𝑛→∞ 𝛽𝜂

𝑛(𝑞)≥1

𝑢2+1−𝑞𝑢1

𝑢2+1,for 𝑞∈[0,1/𝑢1],

0,for 𝑞>1/𝑢1.

Moreover, for

𝑚≥(2𝑛𝐶)1

𝑢2+1

and

𝑘

−𝑛<𝑥𝑚

, we have

𝑘<

𝑛(2𝑛𝐶)−𝑢2

𝑢2+1

. From this

129

4.4. Examples

inequality, using the integral test for convergence, we obtain for 𝑞>1/(𝑢1−𝜀),



𝑄∈D𝑁

𝑛

𝜂(𝑄)𝑞=

2𝑛−1



𝑘=0

𝑚∈N:

𝑘2−𝑛<𝑥𝑚≤(𝑘+1)2−𝑛

𝑝𝑚

𝑞

≪

𝑚<𝐶1/𝑢22𝑛/𝑢2

𝑚𝑞(−𝑢1+𝜀)+

𝐶−𝑢2

𝑢2+12

𝑛

𝑢2+1



𝑘=0

𝑚∈N:

𝑘2−𝑛<𝑥𝑚≤(𝑘+1)2−𝑛

𝑝𝑚

𝑞

≪1+2𝑛1

𝑢2+1+𝑞−(𝑢1−𝜀−1)

𝑢2+(𝑢1−𝜀−1)−𝑢2

𝑢2(𝑢2+1)=1+2𝑛1

𝑢2+1−𝑞𝑢1−𝜀

𝑢2+1≪1.

Hence,

𝛽𝜂(𝑞)=

0 for

𝑞≥

/𝑢1

. Since

𝛽𝜂(0)=

/(𝑢2+

)

, by the convexity of

𝛽𝜂

, it

follows that for all 𝑞∈ [0,1/𝑢1],

𝑢2+1−𝑞𝑢1

𝑢2+1≤liminf

𝑛→∞ 𝛽𝜂

𝑛(𝑞)≤𝛽𝜂(𝑞)≤1

𝑢2+1−𝑞𝑢1

𝑢2+1.

Corollary 4.12 then gives 𝑠𝐷

𝜂=𝑞𝔍𝜂=1/(𝑢1+𝑢2+1).

The last example demonstrates how one can improve Example 4.51 if one knows

the exact exponential asymptotics of

𝑝𝑛

and

𝑥𝑛

, which in turn forces an logarithmic

asymptotic for the eigenvalue counting function. The following simple lemma is

provided for preparation.

Lemma 4.55. Let

𝜈≔𝑝𝛿𝑧

be with

𝑧∈(0,1)

𝑝>

0,0

<𝑎<𝑧

, and 0

<𝑏<

−𝑧

Then, 𝑓𝑎,𝑏,𝑧, 𝑓𝑎,𝑏,𝑧𝐻1

𝑓𝑎,𝑏,𝑧, 𝑓𝑎,𝑏,𝑧𝜈

=𝑎+𝑏

𝑝𝑎𝑏

with 𝑓𝑎,𝑏,𝑧 (𝑥)≔(𝑥−(𝑧−𝑎))

𝑎

[𝑧−𝑎,𝑧]+(𝑧+𝑏−𝑥)

𝑏

(𝑧,𝑧+𝑏].

Example 4.56 (Dirac comb with exponential decay – precise asymptotics).Let us

consider the case 𝑛

𝑘=1𝑝𝑘=1−e−𝛼𝑛 and 𝑥𝑘≔e−𝛾𝑘 for some 𝛼,𝛾 >0. Then

lim

𝑥→∞

𝑁𝐷

𝜂(𝑥)

log(𝑥)=1

𝛼+𝛾.

To see this, we consider the intervals

𝐴𝑛≔[𝑥𝑛−(𝑥𝑛−𝑥𝑛+1)/2,𝑥𝑛+(𝑥𝑛−1−𝑥𝑛)/2]

for 𝑛>1 and set

𝑓𝑛(𝑦)≔𝑓(𝑥𝑛−𝑥𝑛+1)/2,(𝑥𝑛−1−𝑥𝑛)/2,𝑥𝑛(𝑦)for 𝑦∈ [0,1].

130

4.4. Examples

Then, 𝑓𝑛∈𝐻1

0and by Lemma 4.55,

⟨𝑓𝑛, 𝑓𝑛⟩𝐻1

⟨𝑓𝑛, 𝑓𝑛⟩𝜂

=e(𝛼+𝛾)𝑛2(e𝛾−e−𝛾)

(1−e−𝛾) (e𝛼−1) (e𝛾−1).

Notice that for 𝑥>0, we have that

e(𝛼+𝛾)𝑛2(e𝛾−e−𝛾)

(1−e−𝛾) (e𝛼−1) (e𝛾−1)≤𝑥

implies

𝑛≤1

(𝛼+𝛾)log 𝑥(1−e−𝛾) (e𝛼−1) (e𝛾−1)

2(e𝛾−e−𝛾)

ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ

≕𝑛𝑥

Now, observe that

span (𝑓𝑖:𝑖=1, . . ., 𝑛𝑥)

is a

𝑛𝑥

-dimensional subspace of

𝐻1

and

the

(𝑓𝑖)𝑖

are mutually orthogonal both in

𝐿2

𝜈

and in

𝐻1

. Hence, by Lemma 2.19, we

obtain

𝑁𝐷

𝜂(𝑥)≥1

(𝛼+𝛾)log 𝑥(1−e−𝛾) (e𝛼−1) (e𝛾−1)

2(e𝛾−e−𝛾).

For the upper bound, we observe that for 𝑘>1,

𝑓(𝑘)≔1

𝑥𝑘+𝑥𝑘−1

2∞

ℓ=𝑘𝑝ℓ

e−𝛾𝑘+(𝑘−1)𝛼(1+e−𝛾)=2e(𝑘−1)𝛼+𝑘𝛾

1+e−𝛾.

Then,

𝑓(𝑘) ≥ 𝑥=⇒𝑘≥𝑚𝑥≔1

(𝛼+𝛾)log (1+e−𝛾)e𝛼𝑥

2.

Therefore, Lemma 4.50 applied to 𝑓yields for 𝑥large,

𝑁𝐷

𝜂(𝑥) ≤ 𝑓

ˇ−1(6𝑥)=𝑚6𝑥≤1

(𝛼+𝛾)log ((1+e−𝛾)e𝛼6𝑥)+1

and the claim follows.

131

Chapter 5

Spectral dimension for

Kre˘

ın–Feller operators in higher

dimensions

Throughout this chapter let

𝜈

denote a ﬁnite Borel measure on

with

𝜈(Q)>

𝑑>

1, and

dim∞(𝜈)>𝑑−

2. In Chapter 4, we studied the spectral dimension for

Kre

ın–Feller operators for the case

𝑑=

1. This chapter is dedicated to study the

spectral dimension of Kre

ın–Feller operators with respect to

𝜈

and

Ω=(

)𝑑

extending the ideas for the case

𝑑=

1 presented in Chapter 4. Similar to the one-

dimensional case, we use the results developed in Chapter 3 applied to the spectral

partition function

𝔍𝜈,2/𝑑−1,2/𝑡

with

𝑡>

2, which is crucial in our analysis of the

spectral dimension. The reason for the importance of

𝔍𝜈,2/𝑑−1,2/𝑡

becomes apparent

in Section 5.1.2 and Section 5.2.2.

This chapter is organized as follows. In Section 5.1, we establish a connection

between the scaling behavior on the embedding constants for the embedding

𝐶∞

𝑐(𝑄)

into

𝐿2

𝜈(𝑄)

𝑄∈ D

and the lower and upper spectral dimension (see Proposition

5.1). As an application of this general principle, we obtain upper bounds of the

lower and upper spectral dimension by using results of Adams (see [Maz11, p. 67]

and the references given there) and Maz’ya and Preobrazenskii [MP84]. Section

5.2 is devoted to obtain lower bounds of the lower and upper spectral dimension.

Similar to Section 5.1, we ﬁrst establish a relation to lower bounds on the embedding

constants for the embedding

𝐶∞

𝑐(𝑄)

into

𝐿2

𝜈(𝑄)

𝑄∈ D

and lower bounds of the

lower and upper spectral dimension (see Proposition 5.9). In Section 5.2.2, based

on this general observation, we obtain lower bounds of the lower and upper spectral

dimension by

𝐹𝐷/𝑁

𝔍𝜈

and

𝐹𝐷/𝑁

𝔍𝜈

, respectively. In Section 5.3, based on the results of

Section 5.1 and Section 5.2.2, we present our main results of this chapter. We ﬁrst

prove that the upper Neumann spectral dimension equals

𝑞𝑁

𝔍𝜈

. Further, we present

132

5.1. Upper bounds

conditions under which we can ensure that

𝑠𝐷

𝜈=𝑠𝑁

𝜈

. Moreover, in the case

𝑑=

and

𝜈(Q

˚)>

0, we show that

𝑠𝐷

𝜈=𝑠𝑁

𝜈=

1. In Section 5.3.2, we impose regularity

conditions on

𝔍𝜈

that guarantee the existence of the spectral dimension. We end

this chapter with a sequence of examples as an application of our general results

of Section 5.3. As a highlight, we conﬁrm the existence of the spectral dimen-

sion of self-conformal measures without any separation conditions (see Theorem

5.27). Finally, using Example 4.48, we construct an example for which the spectral

dimension does not exist for the case 𝑑=3.

5.1 Upper bounds

In this section, we obtain upper bounds for the spectral dimension.

5.1.1

Embedding constants and upper bounds for the spectral dimen-

sion

This section establishes a relation between embedding constants on sub-cubes and

the lower and upper spectral dimension.

Proposition 5.1. Suppose there exists a non-negative, uniformly vanishing, mono-

tone set function

𝔍

such that for all

𝑄∈ D

and all

𝑢∈ C∞

𝑏(𝑄)

with

𝑄𝑢

Λ=

we have

∥𝑢∥2

𝐿2

𝜈(𝑄)≤𝔍(𝑄)∥∇𝑢∥2

𝐿2

Λ(𝑄).

Then we have

𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤ℎ𝔍and 𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤ℎ𝔍.

Proof.

For a partition

𝛯∈Π𝔍

, let us deﬁne the following closed linear subspace

of 𝐻1

F𝛯≔𝑢∈𝐻1:𝑄

𝑢dΛ=0, 𝑄 ∈𝛯.

We deﬁne an equivalence relation

∼

𝐻1

induced by

F𝛯

as follows:

𝑢∼𝑣

if and

only if

𝑢−𝑣∈ F𝛯

. Note that we have

dim 𝐻1/F𝛯=card(𝛯)

. Further, by our

assumption, for all 𝑢∈𝐶∞

𝑏(Q)∩F𝛯we obtain

𝑢2d𝜈=

𝑄∈𝛯𝑄

𝑢2d𝜈≤

𝑄∈𝛯

𝔍(𝑄) ∥∇𝑢∥2

𝐿2

Λ(𝑄)

≤max

𝑄∈𝛯𝔍(𝑄)

𝑄∈𝛯∥∇𝑢∥2

𝐿2

Λ(𝑄)≤max

𝑄∈𝛯𝔍(𝑄) ∥∇𝑢∥2

𝐿2

Λ(Q).

133

5.1. Upper bounds

Next, we show that

𝐶∞

𝑏(Q)∩ F𝛯

lies dense in

F𝛯

with respect to

𝐻1

. Recall that

𝐶∞

𝑏(Q)

lies dense in

𝐻1

. Hence, for every

𝑢∈ F𝛯

, there exists a sequence

𝑢𝑛

𝐶∞

𝑏(Q)

such that

𝑢𝑛→𝑢

𝐻1

. The Cauchy-Schwarz inequality for all

𝑄∈𝛯

gives

𝑄

𝑢𝑛dΛ=𝑄

𝑢𝑛−𝑢dΛ≤(𝑢𝑛−𝑢)2dΛ→0.

It follows that

𝑄𝑢𝑛

Λ→

0. Furthermore, for every

𝑄∈𝛯

there exists

𝑢𝑄∈𝐶∞

𝑐(Q)

such that 𝑢𝑄|𝑄∁=0 and 𝑄𝑢𝑄dΛ=1. Then for

𝑢′

𝑛≔𝑢𝑛−

𝑄∈𝛯

𝑄𝜀𝑄,𝑛𝑢𝑄∈𝐶∞

𝑏(Q)∩F𝛯

with 𝜀𝑄,𝑛 ≔𝑄𝑢𝑛dΛwe have 𝑢′

𝑛→𝑢in 𝐻1. Thus, for 𝑢∈ F𝛯, we obtain

𝜄(𝑢)2d𝜈≤max

𝑄∈𝛯𝔍(𝑄)∥∇𝑢∥2

𝐿2

Λ(Q).

For 𝑖∈Ndeﬁne

𝜆𝑖

𝜈, F𝛯

≔inf sup 𝑅𝐻1(𝜓):𝜓∈𝐺★:𝐺<𝑖(F𝛯,⟨·,·⟩𝐻1)

with

𝑅𝐻1(𝜓)≔⟨𝜓,𝜓 ⟩𝐻1/⟨𝜄𝜓, 𝜄𝜓 ⟩𝜈

and

𝑁𝑁

𝜈(𝑦, F𝛯)≔card 𝑖∈N:𝜆𝑖

𝜈, F𝛯≤𝑦

with

𝑦>0. Thus, max𝑄∈𝛯𝔍(𝑄)<1/𝑥, implies

𝜆1

𝜈, F𝛯

>𝑥.

In view of the min-max principle as stated in Proposition 2.17, we deduce analo-

gously as in the proof of Proposition 4.4

𝑁𝑁

𝜈(𝑥) ≤ 𝑁𝑁

𝜈(𝑥, F𝛯)+card(𝛯)=card(𝛯),

implying 𝑁𝑁

𝜈(𝑥) ≤ M𝔍(𝑥), and hence 𝑠𝑁

𝜈≤ℎ𝔍and 𝑠𝑁

𝜈≤ℎ𝔍.□

Remark 5.2.The ideas underlying in Proposition 5.1 correspond to some extent to

those developed in [NS95; Sol94], [NS01, Chapter 5], that is, reducing the problem

of estimating the spectral dimension to an auxiliary counting problem. To illustrate

the parallel, we present an alternative proof of the upper estimate of the eigenvalue

counting function for self-similar measures under OSC (see [Sol94, Theorem 1]).

As in the setting in [Sol94], we let

𝜈

denote a self-similar measure under OSC with

contractive similitudes

𝑆1, . . . ,𝑆𝑚

and corresponding contraction ratios

ℎ𝑖∈ (

)

and probability weights

𝑝𝑖∈ (

)

with

𝑖=

, . . . ,𝑚

. We assume

𝜈(𝜕Q)=

0 and

dim∞(𝜈)>𝑑−

2, which is in this case equivalent to

max𝑖𝑝𝑖ℎ2−𝑑

𝑖<

1. For simplicity

134

5.1. Upper bounds

we assume that the feasible set is given by

, i.e.

𝑆𝑗(Q

˚) ⊂ Q

. Instead of

, we

will consider a symbolic partition by the cylinder sets



D≔{𝑆𝜔(Q

˚)

𝜔∈𝐼∗}

with

𝐼≔{1, . . ., 𝑚}

. Then

𝔍

will be replaced by



𝔍



D → R≥0

with



𝔍(𝑆𝜔(Q

˚)) ≔𝑝𝜔ℎ2−𝑑

𝜔

𝜔∈𝐼∗. Now, observe that for 0 <𝑡<min𝑖=1,...,𝑚 𝑝𝑖ℎ2−𝑑

𝑖, we have



𝑃𝑡≔𝜔∈𝐼∗:𝑝𝜔ℎ2−𝑑

𝜔<𝑡≤𝑝𝜔−ℎ2−𝑑

𝜔−

is a partition of

𝐼N

. Further, let

𝛿

denote the unique solution of

𝑚

𝑖=1𝑝𝑖ℎ(2−𝑑)

𝑖𝛿=

Thus, it follows that 

𝜔∈

𝑃𝑡𝑝𝜔ℎ(2−𝑑)

𝜔𝛿=1.

Furthermore, there exists

𝐾>

0 such that for all

𝑢∈𝐻1

with

𝑆𝜔(Q

˚)𝑢

Λ=

𝜔∈

𝑃𝑡

we have 𝜄(𝑢)2d𝜈≤𝐾max

𝜔∈

𝑃𝑡

𝔍𝑆𝜔Q

˚Q|∇𝑢|2dΛ<𝑡𝐾 Q|∇𝑢|2dΛ

(see [NS01, p. 502]). A similar computation as in the proof of Lemma 4.33 yields

the two-sided estimate

𝑡−𝛿≤card 

𝑃𝑡≤𝑡−𝛿

min𝑖=1,...,𝑚 𝑝𝑖ℎ2−𝑑

𝑖

The min-max principle gives

𝑁𝑁

𝜈(𝑡𝐾)−1≤card 

𝑃𝑡≤𝑡−𝛿

min𝑖=1,...,𝑚 𝑝𝑖ℎ2−𝑑

𝑖

hence the results of [NS01; NS95, Theorem 1] follow from this simple counting

argument without any use of renewal theory. The drawback of the ideas [NS95;

Sol94; NS01, Chapter 5] is that they rely heavily on the speciﬁc structure of the

self-similar measures, whereas our approach via dyadic cubes avoids the use of

speciﬁc properties of the underlying measure.

5.1.2

Upper bounds on the embedding constants and upper bounds

for the spectral dimension

In this section, up to multiplicative uniform constants, we make use of best em-

bedding constants for the embedding

𝐶∞

𝑐R𝑑

into

𝐿𝑡

𝜈R𝑑

𝑡>

2, to estimate the

135

5.1. Upper bounds

spectral dimension from above. More precisely, for 𝑑>2, the best constant 𝐶in

||𝑢||𝐿𝑡

𝜈|𝑄(R𝑑)≤𝐶||𝑢||𝐻1(R𝑑), 𝑢 ∈𝐶∞

𝑐R𝑑, 𝑄 ∈ D,(5.1.1)

is equivalent to

sup𝑥∈R𝑑,𝜌>0𝜌(2−𝑑)/2𝜈𝑄∩𝐵𝜌(𝑥)1/𝑡

in the sense that there exist

𝑐1,𝑐2>0 only depending on 𝑑and 𝑡such that

𝑐1𝐶≤sup

𝑥∈R𝑑,𝜌>0

𝜌(2−𝑑)/2𝜈𝑄∩𝐵𝜌(𝑥)1/𝑡≤𝑐2𝐶.

For 𝑑=2, the best constant 𝐶in (5.1.1) is equivalent to

sup

𝑥∈R𝑑,0<𝜌<1/2|log(𝜌)|1/2𝜈𝑄∩𝐵𝜌(𝑥)1/𝑡.

The result for the case

𝑑>

2 is a corollary of Adams’ Theorem on Riesz potentials

(see e.g. [Maz11, p. 67]) and the case

𝑑=

2 is due to Maz’ya and Preobrazenskii

and can be found in [Maz11, p. 83] or [MP84]. The following lemma establishes an

alternative representation of the best equivalent constant in terms of dyadic cubes.

Lemma 5.3. Let

𝑄∈ D

. Then, for

𝑎<

𝑏>

𝐶1≔2√𝑑𝑎

, and

𝐶2≔

3√𝑑𝑑𝑏 2−𝑎,

𝐶1𝔍𝜈,𝑎/𝑑,𝑏 (𝑄)≤sup

𝑥∈R𝑑,𝜌>0

𝜌𝑎𝜈𝑄∩𝐵𝜌(𝑥)𝑏≤𝐶2𝔍𝜈,𝑎/𝑑,𝑏 (𝑄).

For 𝑎=0,𝐶3≔𝑑−1, and 𝐶4≔3𝑏𝑑 ,

𝐶3𝔍𝜈,0,𝑏 (𝑄)≤sup

𝑥∈R𝑑,0<𝜌<1/2|log(𝜌)|𝜈(𝑄∩𝐵𝜌(𝑥))𝑏≤𝐶4𝔍𝜈,0,𝑏 (𝑄).

Proof. Let 𝑄∈ D𝑁

𝑛. Since 𝑎<0, we have

sup

𝑥∈R𝑑,𝜌>0

𝜌𝑎𝜈𝑄∩𝐵𝜌(𝑥)𝑏=sup

𝑥∈R𝑑,𝜌 ≤√𝑑2−𝑛+1

𝜌𝑎𝜈𝑄∩𝐵𝜌(𝑥)𝑏.

Thus, we assume without loss of generality, that 0

<𝜌≤√𝑑

−𝑛+1

. Then for

136

5.1. Upper bounds

𝑚≥𝑛−1 with √𝑑2−(𝑚+1)<𝜌≤√𝑑2−𝑚, and 𝑥∈R𝑑, we obtain

𝜌𝑎𝜈𝑄∩𝐵𝜌(𝑥)𝑏≤2−𝑎

𝑄′∈D𝑁

𝑚,

𝑄′∩𝑄∩𝐵𝜌(𝑥)≠∅

𝜈(𝑄∩𝑄′)

𝑏

2−𝑚𝑎

≤3√𝑑𝑑𝑏 2−𝑎max

𝑄′∈D𝑁

𝑚

𝜈(𝑄∩𝑄′)𝑏Λ(𝑄′)𝑎/𝑑

≤𝐶2sup

𝑄′∈D(𝑄)

𝜈(𝑄′)𝑏Λ(𝑄′)𝑎/𝑑=𝐶2𝔍𝜈,𝑎/𝑑,𝑏 (𝑄),

where we used the facts that

𝐵𝜌(𝑥)

can be covered by at most

3√𝑑𝑑

elements of

D𝑁

𝑚and if 𝑄′∩𝑄≠∅, then 𝑄′⊂𝑄for 𝑚≥𝑛, as well as

max

𝑄′∈D𝑁

𝑛−1

𝜈(𝑄∩𝑄′)𝑏Λ(𝑄′)𝑎/𝑑≤𝜈(𝑄)𝑏Λ(𝑄)𝑎/𝑑=max

𝑄′∈D𝑁

𝑛

𝜈(𝑄∩𝑄′)𝑏Λ(𝑄′)𝑎/𝑑.

Since 𝑥∈R𝑑and 𝜌>0 were arbitrary, the second inequality follows.

On the other hand, for

𝑄′∈ D𝑁

𝑚

with

𝑄′⊂𝑄

and

𝜌≔√𝑑

−𝑚+1

we ﬁnd

𝑥∈R𝑑

such that 𝑄′⊂𝐵𝜌(𝑥). Then

𝜈(𝑄′)𝑏Λ(𝑄′)𝑎/𝑑≤𝜈𝑄∩𝐵𝜌(𝑥)𝑏2−𝑚𝑎

≤√𝑑2−𝑎

𝜈𝑄∩𝐵𝜌(𝑥)𝑏𝜌𝑎

≤𝐶−1

1sup

𝑥∈R𝑑,𝜌>0

𝜌𝑎𝜈𝑄∩𝐵𝜌(𝑥)𝑏.

For the case 𝑎=0, for any 2−(𝑚+1)≤𝜌<2−𝑚,𝑚∈N, and 𝑥∈R𝑑, we have

|log(𝜌)|𝜈(𝑄∩𝐵𝜌(𝑥))𝑏≤|log (2)(𝑚+1)|𝜈(𝑄∩𝐵2−𝑚(𝑥))𝑏

≤log 2−𝑑𝑚

𝑄′∈D𝑁

𝑚,𝑄′∩𝑄∩𝐵2−𝑚(𝑥)≠∅

𝜈(𝑄∩𝑄′)

𝑏

≤3𝑑𝑏 max

𝑄′∈D𝑁

𝑚

𝜈(𝑄∩𝑄′)𝑏|log (Λ(𝑄′))|

≤3𝑑𝑏 max

𝑄′∈D(𝑄)𝜈(𝑄∩𝑄′)𝑏|log (Λ(𝑄′))|,

where we used that

max

𝑄′∈D𝑁

𝑚

𝜈(𝑄∩𝑄′)𝑏|log (Λ(𝑄′))| ≤𝜈(𝑄)𝑏log(2)𝑑𝑚 ≤max

𝑄′∈D𝑁

𝑛

𝜈(𝑄∩𝑄′)𝑏|log (Λ(𝑄′))|

137

5.1. Upper bounds

for

𝑚≤𝑛

. On the other hand, for

𝑄′∈ D𝑁

𝑚

with

𝑄′⊂𝑄

and

𝜌≔√𝑑

−𝑚+1

we ﬁnd

𝑥∈R𝑑such that 𝑄′⊂𝐵𝜌(𝑥). Then

𝜈(𝑄′)𝑏|log (Λ(𝑄′))| ≤𝜈𝑄∩𝐵𝜌(𝑥)𝑏𝑑𝑚 log(2)

≤𝑑𝜈 𝑄∩𝐵𝜌(𝑥)𝑏(𝑚+1)log(2) +log √𝑑

=𝑑𝜈 𝑄∩𝐵𝜌(𝑥)𝑏|log(𝜌)|

≤𝑑sup

𝑥∈R𝑑,𝜌>0|log(𝜌)|𝜈𝑄∩𝐵𝜌(𝑥)𝑏.□

Corollary 5.4. For

𝑡>

2there exists a constant

𝐶𝑡,𝑑 >

0such that for all

𝑢∈ C∞

𝑐R𝑑

and 𝑄∈ D, we have

∥𝑢∥𝐿2

𝜈|𝑄(R𝑑)≤𝐶𝑡,𝑑 𝔍1/2

𝜈,2/𝑑−1,2/𝑡(𝑄)∥𝑢∥𝐻1(R𝑑).

Proof.

Using [Maz85, Corollary, p. 54] or [Maz85, Theorem, p. 381–382] for

𝑑>

2 and [Maz85, Corollary 1, p. 382] for

𝑑=

2, for ﬁxed

𝑡>

2, we ﬁnd constants

𝑐1,𝑐2>0 independent of 𝑄∈ D and 𝜈such that for all 𝑢∈𝐶∞

𝑐R𝑑

∥𝑢∥𝐿2

𝜈|𝑄(R𝑑)≤∥𝑢∥𝐿𝑡

𝜈|𝑄(R𝑑)

≤𝑐1sup

𝑥∈R𝑑,0<𝑟<1/2|log(𝑟)|𝜈(𝑄∩𝐵𝑟(𝑥))2/𝑡1/2

∥𝑢∥𝐻1(R𝑑),

for

𝑑=

2 (note there is a typo, the constant

𝐶5

has to be replaced by

𝐶1/𝑝

, see also

for the correct version in [Maz11, p. 83]) and

∥𝑢∥𝐿2

𝜈|𝑄(R𝑑)≤∥𝑢∥𝐿𝑡

𝜈|𝑄(R𝑑)

≤𝑐2sup

𝑥∈R𝑑,𝜌>0

𝜌(2−𝑑)𝜈𝑄∩𝐵𝜌(𝑥)2/𝑡1/2

∥𝑢∥𝐻1(R𝑑),

for

𝑑>

2. Therefore, Lemma 5.3 (with

𝑎=

−𝑑

and

𝑏=

/𝑡

) proves the claim.

□

Lemma 5.5. Then for every

𝑡>

2there exists

𝑇𝑑,𝑡 >

0such that for all

𝑄∈ D

and

𝑢∈𝐶∞

𝑏𝑄with 𝑄𝑢dΛ=0, we have

∥𝑢∥𝐿2

𝜈|𝑄(𝑄)≤𝑇𝑑,𝑡 𝔍𝜈,2/𝑑−1,2/𝑡(𝑄)1/2∥∇𝑢∥𝐿2

Λ(𝑄).

138

5.2. Lower bounds

Proof. Combining Lemma 2.7 and Corollary 5.4, we have for all 𝑢∈𝐶∞

𝑏(𝑄)

∥𝑢∥𝐿2

𝜈|𝑄(𝑄)=𝔈𝑄(𝑢)𝐿2

𝜈|𝑄(R𝑑)

≤𝐶𝑡,𝑑 𝔍𝜈,2/𝑑−1,2/𝑡(𝑄)1/2𝔈𝑄(𝑢)𝐻1(R𝑑)

≤𝐶𝑡,𝑑 𝔈Q

𝐷Q

ˉˉˉˉˉˉˉˉˉˉ

≕𝑇𝑑,𝑡

𝔍𝜈,2/𝑑−1,2/𝑡(𝑄)1/2∥∇𝑢∥2

𝐿2

Λ(𝑄)+1

Λ(𝑄)𝑄

𝑢dΛ21/2

=𝑇𝑑,𝑡 𝔍𝜈,2/𝑑−1,2/𝑡(𝑄)1/2∥∇𝑢∥2

𝐿2

Λ(𝑄).□

Corollary 5.6. If dim∞(𝜈)>𝑑−2, then

𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤lim

𝑡↓2ℎ𝔍𝜈,𝑡 (2/𝑑−1)/2,1≤𝑞𝑁

𝔍𝜈and 𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤lim

𝑡↓2ℎ𝔍𝜈,𝑡 (2/𝑑−1)/2,1.

In particular, in the case 𝑑=2,we have 𝑠𝑁

𝜈≤1.

Proof.

Note that

dim∞(𝜈)>𝑑−

2 implies that for all

𝑡∈(2,2dim∞(𝜈)/(𝑑−2))

𝔍𝜈,2/𝑑−1,2/𝑡

is non-negative, monotone and uniformly vanishing on

. Combining

Proposition 3.6, Proposition 5.1, and Lemma 5.5, we obtain

𝑠𝑁

𝜈≤ℎ𝔍𝜈, (2/𝑑−1),2/𝑡=(𝑡/2)ℎ𝔍𝜈,𝑡(2/𝑑−1)/2,1≤ (𝑡/2)𝑞𝑁

𝔍𝜈,𝑡 (2/𝑑−1)/2,1

and

𝑠𝑁

𝜈≤ℎ𝔍𝜈,𝑡 (2/𝑑−1)/2,1

for all

𝑡∈(2,dim∞(𝜈)/(𝑑−2))

. The claim follows by letting

𝑡↘2 and Proposition 3.6. □

5.2 Lower bounds

5.2.1 Lower bound on the spectral dimension

Recall from Section 1.1.3, for 𝑛∈Nand 𝛼>0,

N𝐷/𝑁

𝛼,𝔍(𝑛)=card 𝑀𝐷/𝑁

𝛼,𝔍(𝑛)with 𝑀𝐷/𝑁

𝛼,𝔍(𝑛)=𝐶∈ D𝐷/𝑁

𝑛:𝔍(𝐶) ≥ 2−𝛼𝑛 .

As before, for

𝑠>

0, we let

⟨𝑄⟩𝑠

denote the cube centered and parallel with respect

𝑄

such that

Λ(𝑄)=𝑠−𝑑Λ⟨𝑄⟩𝑠

𝑠>

0. Recall that we always assume

dim∞(𝜈)>

𝑑−2. We start with the following simple geometric lemma.

Lemma 5.7. Let 𝑄, 𝑄 ′∈ D𝑁

𝑛,𝑛∈N, then

(1) 𝑄

˚5∩𝑄′

˚=∅implies 𝑄

˚3∩𝑄′

˚3=∅,

139

5.2. Lower bounds

(2) 𝑄

˚5∩𝑄′

˚=∅implies 𝑄′

˚5∩𝑄

˚=∅.

Proof. Let us write

𝑄

˚=

𝑑



𝑖=1(𝑎𝑖,𝑏𝑖)and 𝑄′

˚=

𝑑



𝑖=1(𝑐𝑖,𝑑𝑖)

and note that

⟨𝑄

˚⟩5=

𝑑



𝑖=1(3𝑎𝑖−2𝑏𝑖,3𝑏𝑖−2𝑎𝑖)and ⟨𝑄

˚⟩3=

𝑑



𝑖=1(2𝑎𝑖−𝑏𝑖,2𝑏𝑖−𝑎𝑖).

Now, if 𝑄

˚∩𝑄′

˚5=∅, then there exists 𝑗∈ {1, . . . ,𝑑 }such that

(𝑐𝑗,𝑑𝑗)∩(3𝑎𝑗−2𝑏𝑗,3𝑏𝑗−2𝑎𝑗)=∅.

We only consider the case

𝑑𝑗<

𝑎𝑗−

𝑏𝑗

, the case

𝑐𝑗>

𝑏𝑗−

𝑎𝑗

follows similarly.

Using 𝑑𝑗−𝑐𝑗=𝑏𝑗−𝑎𝑗=2−𝑛yields

2𝑑𝑗−𝑐𝑗=𝑑𝑗+𝑏𝑗−𝑎𝑗<3𝑎𝑗−2𝑏𝑗+𝑏𝑗−𝑎𝑗=2𝑎𝑗−𝑏𝑗,

implying

⟨𝑄

˚⟩3∩⟨𝑄′

˚⟩3=

𝑑



𝑖=1(2𝑎𝑖−𝑏𝑖,2𝑏𝑖−𝑎𝑖) ∩

𝑑



𝑖=1(2𝑐𝑖−𝑑𝑖,2𝑑𝑖−𝑐𝑖)=∅.

Further, we have

3𝑑𝑗−2𝑐𝑗=𝑑𝑗+2(𝑏𝑗−𝑎𝑗)<3𝑎𝑗−2𝑏𝑗=𝑎𝑗,

allowing us to infer that

⟨𝑄′

˚⟩5∩𝑄

˚=

𝑑



𝑖=1(3𝑐𝑖−2𝑑𝑖,3𝑑𝑖−2𝑐𝑖) ∩

𝑑



𝑖=1(𝑎𝑖,𝑏𝑖)=∅.□

We need also the following simple combinatorial lemma.

Lemma 5.8. For ﬁxed

𝛼>

0there exists a sequence

𝐸𝛼,𝑛 𝑛

with

𝐸𝛼,𝑛 ⊂𝑀𝐷/𝑁

𝛼,𝔍(𝑛)

𝑒𝛼,𝑛 ≔card 𝐸𝛼,𝑛 ≥N𝐷/𝑁

𝛼,𝔍(𝑛)/5𝑑

and for all cubes 𝑄,𝑄 ′∈𝐸𝛼 ,𝑛 with 𝑄≠𝑄′we have 𝑄

˚3∩𝑄′

˚3=∅.

140

5.2. Lower bounds

Proof.

We assume

𝑀𝐷/𝑁

𝛼,𝔍(𝑛)≠∅

. We construct inductively a subset

𝐸𝛼,𝑛

𝑀𝐷/𝑁

𝛼,𝔍(𝑛)

of cardinality

card 𝐸𝛼,𝑛 ≥N𝐷/𝑁

𝛼,𝔍(𝑛)/5𝑑

such that for all cubes

𝑄, 𝑄 ′∈

𝐸𝛼,𝑛

with

𝑄≠𝑄′

, we have

𝑄

˚3∩𝑄′

˚3=∅

. At the beginning of the induction we

set 𝐷(0)≔𝑀𝐷/𝑁

𝛼,𝔍(𝑛). Assume we have constructed

𝐷(0)⊃𝐷(1)⊃... ⊃𝐷(𝑗−1)

such that the following condition holds

𝑄𝑗

˚5∩𝑄

˚≠∅

for some

𝑄, 𝑄𝑗∈𝐷(𝑗−1)

with

𝑄≠𝑄𝑗. Then we set

𝐷(𝑗)≔𝐶∈𝐷(𝑗−1):𝐶

˚∩𝑄𝑗

˚5=∅∪𝑄𝑗.

By this construction, we have

card 𝐷(𝑗)<card 𝐷(𝑗−1)

, since

𝑄

˚∩𝑄𝑗

˚5≠∅

Further, by Lemma 5.7, for all

𝑄∈𝐷(𝑗)\{𝑄𝑗}

, we have

𝑄

˚𝑗∩𝑄

˚5=∅

, showing

𝑄𝑗∈𝐷(𝑘)

for all

𝑘≥𝑗

. If otherwise

𝑄

˚5∩𝑄

˚′=∅

for all

𝑄, 𝑄 ′∈𝐷(𝑗−1)

with

𝑄≠𝑄′

then we set

𝐸𝛼,𝑛 =𝐷(𝑗−1)

. In each inductive step, we remove at most 5

𝑑−

1 elements

𝐷(𝑗−1)

, while one element, namely

𝑄𝑗

, is kept. Moreover, by the construction of

𝐸𝛼,𝑛

, for each

𝑄′∈𝑀𝐷/𝑁

𝛼,𝔍(𝑛)

there exists

𝑄∈𝐸𝛼,𝑛

such that

𝑄

˚5∩𝑄′

˚≠∅

. This

implies

card 𝐸𝛼,𝑛 =

𝑄∈𝐸𝛼,𝑛

card {𝑄′∈𝑀𝐷/𝑁

𝛼,𝔍(𝑛):𝑄

˚5∩𝑄′

˚≠∅}

card {𝑄′∈𝑀𝐷/𝑁

𝛼,𝔍(𝑛):𝑄

˚5∩𝑄′

˚≠∅}

≥

𝑄∈𝐸𝛼,𝑛

5𝑑card {𝑄′∈𝑀𝐷/𝑁

𝛼,𝔍(𝑛):𝑄

˚5∩𝑄′

˚≠∅}

≥N𝐷/𝑁

𝛼,𝔍(𝑛)

5𝑑.

Finally, by Lemma 5.7, we obtain that if

𝑄

˚∩𝑄′

˚5=∅

for

𝑄, 𝑄 ′∈𝐸𝛼,𝑛

, then

𝑄

˚3∩𝑄′

˚3=∅.□

The lower estimate of the spectral dimension is based on the following abstract

observation, connecting the optimized coarse multifractal dimension and the spectral

dimension.

Proposition 5.9. Assume there exists a non-negative monotone set function

𝔍

with

dim∞(𝔍)>

0such that for every

𝑄∈ D

with

𝔍(𝑄)>

0there exists a non-

negative and non-zero function

𝜓𝑄∈ C∞

𝑐R𝑑

with support contained in

𝑄

˚3

such

that 𝜓𝑄2

𝐿2

𝜈≥𝔍(𝑄)∇𝜓𝑄2

𝐿2

Λ(R𝑑).

141

5.2. Lower bounds

Then for ﬁxed 𝛼>0and for 𝑥>0large, we have

N𝐷

𝛼,𝔍𝑛𝛼 ,𝑥 

5𝑑−1≤𝑁𝐷

𝜈(𝑥)and N𝑁

𝛼,𝔍𝑛𝛼 ,𝑥 

2·5𝑑−1≤𝑁𝑁

𝜈𝑥

𝐷Q,

with 𝑛𝛼,𝑥 ≔log2(𝑥)/𝛼. In particular, we have

𝐹𝑁

𝔍≤𝑠𝑁

𝜈and 𝐹𝑁

𝔍≤𝑠𝑁

𝜈, 𝐹𝐷

𝔍≤𝑠𝐷

𝜈and 𝐹𝐷

𝔍≤𝑠𝐷

𝜈.

Proof.

Fix

𝛼>

0 and let

𝐸𝛼,𝑛

𝑛∈N

, be given as in Lemma 5.8. Let us ﬁrst

consider the Dirichlet case. Since for each

𝑄∈ D𝐷

𝑛

, we have

𝜕Q∩𝑄=∅

, it

follows that

𝑄

˚3⊂Q

and therefore

𝜓𝑄∈𝐶∞

𝑐(Q

˚)

. Now, for

𝑥>

𝛼

we deﬁne

𝑛𝛼,𝑥 ≔log2(𝑥)/𝛼. Then, for each 𝑄∈𝐸𝑛𝛼,𝑥 , we have

Q∇𝜓𝑄2dΛ

𝜓2

𝑄d𝜈≤1

𝔍(𝑄)≤2𝛼𝑛𝛼 ,𝑥 ≤𝑥.

Hence, the

𝜓𝑄:𝑄∈𝐸𝑛𝛼,𝑥 ≕𝑓𝑖:𝑖=1, . . . ,𝑒𝑛𝛼 ,𝑥 

are mutually orthogonal both in

𝐿2

𝜈

and in

𝐻1

- Thus, we obtain that

span 𝑓𝑖:𝑖=1, . . ., 𝑒𝛼,𝑥 

is an

𝑒𝑛𝛼,𝑥

-dimensional

subspace of 𝐻1

0. Hence, we deduce from Lemma 2.19

N𝐷

𝛼,𝔍𝑛𝛼 ,𝑥 /5𝑑−1≤𝑒𝑛𝛼,𝑥 ≤𝑁𝐷

𝜈(𝑥).

In the Neumann case, we proceed similarly. For ﬁxed

𝛼>

0 again set

𝑛𝛼,𝑥 =

log2(𝑥)/𝛼

and write

𝐸𝑛𝛼,𝑥 =𝑄1, . . . ,𝑄 card(𝐸𝑛𝛼,𝑥 )

. For each

𝑖=

, . . . , 𝑒𝑛𝛼,𝑥 /2≕

𝑁𝛼,𝑥 we deﬁne

𝑓𝑖≔𝑎2𝑖−1𝜓𝑄2𝑖−1+𝑎2𝑖𝜓𝑄2𝑖|Q∈ C∞

𝑏Q,

where we choose

(𝑎2𝑖−1,𝑎2𝑖)∈R2\{(0,0)}

such that

Q𝑓𝑖

Λ=

0. Then, by Lemma

2.7, we have

⟨𝑓𝑖, 𝑓𝑖⟩𝐻1≤Q|∇𝑓𝑖|2dΛ

𝐷Q

Since we have

𝑄𝑗

˚3∩𝑄𝑘

˚3=∅

for

𝑗≠𝑘

, combined with the properties of medi-

ants, i.e.

𝑎+𝑏

𝑐+𝑑=𝑎

𝑐

𝑐+𝑑+𝑏

𝑑

𝑐+𝑑≤max 𝑎

𝑐,𝑏

𝑑for all 𝑎,𝑏, 𝑐,𝑑 >0,

142

5.2. Lower bounds

we obtain

⟨𝑓𝑖, 𝑓𝑖⟩𝐻1

𝑓2

𝑖d𝜈≤Q|∇𝑓𝑖|2dΛ

𝐷Q𝑓2

𝑖d𝜈≤|∇𝑓𝑖|2dΛ

𝐷Q𝑓2

𝑖d𝜈

≤1

𝐷Q

𝑎2

1∇𝜓𝑄2𝑖−12dΛ+𝑎2

2∇𝜓𝑄2𝑖2dΛ

𝑎2

1𝜓2

𝑄2𝑖−1d𝜈+𝑎2

2𝜓2

𝑄2𝑖d𝜈

≤1

𝐷Q

max ∇𝜓𝑄2𝑖2dΛ

𝜓2

𝑄2𝑖d𝜈,∇𝜓𝑄2𝑖−12dΛ

𝜓2

𝑄2𝑖−1d𝜈

≤1

𝐷Q

max 1

𝔍(𝑄2𝑖−1),1

𝔍(𝑄2𝑖)≤𝑥

𝐷Q

Hence, the

𝑓𝑖

are mutually orthogonal in

𝐻1

and also in

𝐿2

𝜈

, we obtain

span 𝑓1, . . . , 𝑓𝑁𝛼,𝑥 

is a

𝑁𝛼,𝑥

- dimensional subspace of

𝐻1

. Again, an application of Lemma 2.19 gives

N𝑁

𝛼,𝔍(𝑛𝛼 ,𝑥 )/2·5𝑑−1≤𝑁𝑁

𝜈𝑥/𝐷Q.

Consequently, analogous to the proof of Proposition 4.3, we conclude

𝑠𝐷

𝜈=liminf

𝑥→∞

log 𝑁𝐷

𝜈(𝑥)

log(𝑥)≥lim inf

𝑛→∞

log+N𝐷

𝛼,𝔍(𝑛)

𝛼log (2𝑛)=

𝐹𝐷

𝔍(𝛼)

𝛼.

Taking the supremum over all

𝛼>

0 gives

𝐹𝐷

𝔍≤𝑠𝐷

𝜈

. Furthermore, for

𝑥𝛼,𝑛 ≔

𝛼𝑛

with 𝑛∈N, we see that

𝑠𝐷

𝜈≥limsup

𝑛→∞

log 𝑁𝐷

𝜈(𝑥𝛼,𝑛 )

log(𝑥𝛼 ,𝑛)≥limsup

𝑛→∞

log+N𝐷

𝛼.𝔍(𝑛)

log(2𝑛)𝛼=𝐹𝐷

𝔍(𝛼)

𝛼,

implying 𝑠𝐷

𝜈≥𝐹𝐷

𝔍. In the Neumann case, using

N𝑁

𝛼,𝔍(𝑛𝛼 ,𝑥 )/2·5𝑑−1≤𝑁𝑁

𝜈𝑥/𝐷Q,

we obtain in the same ways as in the Dirichlet case that

𝐹𝑁

𝔍≤𝑠𝑁

𝜈

and

𝐹𝑁

𝔍≤𝑠𝑁

𝜈

□

5.2.2 Lower bound on the embedding constant

We need a slight modiﬁcation of 𝔍𝜈for the case 𝑑=2. We deﬁne

𝔍𝜈(𝑄)=sup

𝑄′∈D(𝑄)

𝜈(𝑄′)Λ(𝑄′)2/𝑑−1

143

5.2. Lower bounds

for

𝑄∈ D

. Hence, in the case

𝑑=

2, we have

𝔍𝜈(𝑄)=𝜈(𝑄)

. Clearly, we again have

dim∞(𝔍𝜈)>𝑑−

𝜏𝐷/𝑁

𝔍𝜈=𝜏𝐷/𝑁

𝔍𝜈

by Proposition 2.35 and for

𝑑>

𝐹𝐷/𝑁

𝔍𝜈

=𝐹𝐷/𝑁

𝔍𝜈

and 𝐹𝐷/𝑁

𝔍𝜈

=𝐹𝐷/𝑁

𝔍𝜈. The case 𝑑=2 is covered by the following lemma.

Lemma 5.10. In the case 𝑑=2, we have

𝐹𝐷/𝑁

𝔍𝜈

=𝐹𝐷/𝑁

𝔍𝜈and 𝐹𝐷/𝑁

𝔍𝜈

=𝐹𝐷/𝑁

𝔍𝜈.

Proof. We have always

𝐶∈ D𝐷/𝑁

𝑛: sup

𝑄′∈D(𝐶)

𝜈(𝑄′)|log (Λ(𝑄′)) | ≥ 2−𝛼 𝑛 ⊃𝐶∈ D𝐷/𝑁

𝑛:𝜈(𝐶) ≥ 2−𝛼𝑛 ,

and using dim∞(𝜈)>𝑑−2,we obtain for every 𝛿>1 and 𝑛∈Nlarge enough

𝜈(𝑄)|log(Λ(𝑄))| ≤ 𝜈(𝑄)1/𝛿, 𝑄 ∈ D𝐷/𝑁

𝑛.

Indeed, for

𝑑−

<𝑠<dim∞(𝜈)

, we have for all

𝑛

large and

𝑄∈ D𝐷/𝑁

𝑛

with

𝜈(𝑄)>

𝜈(𝑄) ≤ 2−𝑠𝑛 .

Further, for ﬁxed 0 <𝜀<1, we have for 𝑛large

𝑑𝑛 log(2) ≤ 2𝜀𝑠 𝑛 ≤𝜈(𝑄)−𝜀.

Hence, for all 𝑄∈ D𝐷/𝑁

𝑛, we obtain

𝜈(𝑄)|log(Λ(𝑄))| ≤ 𝜈(𝑄)1−𝜀.

This leads to

𝐶∈ D𝐷/𝑁

𝑛: sup

𝑄′∈D(𝐶)

𝜈(𝑄′)|log(Λ(𝑄′))| ≥ 2−𝛼 𝑛 ⊂𝐶∈ D𝐷/𝑁

𝑛:𝜈(𝐶) ≥ 2−𝛼𝛿 𝑛 .

Thus,

log+N𝐷/𝑁

𝛼,𝔍𝜈(𝑛)

𝛼𝑛 log(2)≤𝛿

log+N𝐷/𝑁

𝛼𝛿,𝔍𝜈(𝑛)

𝛼𝛿 𝑛 log(2).

Hence, the claim follows. □

Proposition 5.11. There exists a constant

𝐾>

0such that for every

𝑄∈ D

with

𝔍𝜈(𝑄)>

0there exists a function

𝜓𝑄∈ C∞

𝑐(R𝑑)

with support contained in

𝑄

˚3

144

5.3. Main results

and 𝜑𝑄𝐿2

𝜈

>0such that

𝜓𝑄2

𝐿2

𝜈≥𝐾𝔍𝜈(𝑄)∇𝜓𝑄2

𝐿2

Λ(R𝑑).

Proof.

Since

dim∞(𝔍𝜈)>

0, it follows that for each

𝑄∈ D

there exists

𝐶𝑄∈ D(𝑄)

such that

𝔍𝜈(𝑄)=𝜈(𝐶𝑄)Λ(𝐶𝑄)2/𝑑−1

. Now, choose

𝜓𝑄≔𝜑⟨𝐶𝑄⟩3,3

as in Lemma

2.21. Then 𝜓𝑄·

𝐶𝑄=

𝐶𝑄, supp(𝜓𝑄) ⊂ 𝐶𝑄

˚3⊂𝑄

˚3, and

∇𝜓𝑄2dΛ

𝜓𝑄2d𝜈≤𝐶2−231−2/𝑑

Λ𝐶𝑄31/31−2/𝑑

𝜈𝐶𝑄31/3

=𝐶2−231−2/𝑑Λ𝐶𝑄1−2/𝑑

𝜈𝐶𝑄=𝐶2−231−2/𝑑𝔍𝜈(𝑄)−1.□

Proposition 5.12. Assume

𝜈(Q

˚)>

0. Then for ﬁxed

𝛼>

0and for

𝑥>

0large, we

have

N𝐷

𝛼,𝔍𝜈(𝑛𝛼 ,𝑥 )

5𝑑−1≤𝑁𝐷

𝜈(𝑥𝐾)

with 𝑛𝛼,𝑥 ≔log2(𝑥)/𝛼. In particular, 𝐹𝐷

𝔍𝜈≤𝑠𝐷

𝜈and 𝐹𝐷

𝔍𝜈≤𝑠𝐷

𝜈.

Proof. This follows from Proposition 5.9, Lemma 5.10, and Proposition 5.11. □

In the same way we obtain the following proposition for the Neumann case.

Proposition 5.13. For ﬁxed 𝛼>0, we have for 𝑥>0large

N𝑁

𝛼,𝔍𝜈(𝑛𝛼 ,𝑥 )

2·5𝑑−1≤𝑁𝑁

𝜈𝑥𝐾 /𝐷Q

with 𝑛𝛼,𝑥 ≔log2(𝑥)/𝛼. In particular, 𝐹𝑁

𝔍𝜈≤𝑠𝑁

𝜈and 𝐹𝑁

𝔍𝜈≤𝑠𝑁

𝜈.

5.3 Main results

In this section, we combine the results of Section 5.1 and Section 5.2.2 to prove the

main results of this chapter.

145

5.3. Main results

5.3.1

Upper spectral dimension, and lower and upper bounds for the

lower spectral dimension

To break up the main result (Theorem 5.15) of this section, we start with the

following proposition.

Proposition 5.14. We have

𝑞𝑁

𝔍𝜈=𝐹𝑁

𝔍𝜈=ℎ𝔍𝜈=𝑠𝑁

𝜈and 𝑠𝑁

𝜈≤lim

𝑡↓2ℎ𝔍𝜈,𝑡 (2/𝑑−1)/2,1.

Proof. From Proposition 3.20 and Proposition 5.13 applied to 𝔍 = 𝔍𝜈, we obtain

𝑞𝑁

𝔍𝜈=𝐹𝑁

𝔍𝜈≤𝑠𝑁

𝜈.

Moreover, Corollary 3.21 and Corollary 5.6 yield

𝑠𝑁

𝜈≤𝑠𝑁

𝜈≤lim

𝑡↓2ℎ𝔍𝜈,𝑡 (1/𝑑−1)/2,1≤𝑞𝑁

𝔍𝜈and 𝐹𝑁

𝔍𝜈=ℎ𝔍𝜈=𝑞𝑁

𝔍𝜈

which proves the claimed equalities. □

Theorem 5.15. Let

𝜈

be a Borel probability measure on

such that

dim∞(𝜈)>𝑑−

1. Under Neumann boundary conditions we have

𝐹𝑁

𝔍𝜈≤𝑠𝑁

𝜈≤lim

𝑡↓2ℎ𝔍𝜈,𝑡 (2/𝑑−1)/2,1≤ℎ𝔍𝜈=𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈=𝐹𝑁

𝔍𝜈.(5.3.1)

2. Under Dirichlet boundary conditions and 𝜈(Q

˚)>0we have

𝐹𝐷

𝔍𝜈≤𝑠𝐷

𝜈and 𝐹𝐷

𝔍𝜈=𝑞𝐷

𝔍𝜈≤𝑠𝐷

𝜈≤𝑞𝑁

𝔍𝜈.

In particular, if

𝑑=

2, then

𝑠𝑁

𝜈=

1, and under the assumption

𝜈(Q

˚)>

0, we

also have 𝑠𝐷

𝜈=1.

𝜏𝑁

𝔍𝜈𝑞𝐷

𝔍𝜈=

0, or equivalently

𝐹𝑁

𝔍𝜈=𝐹𝐷

𝔍𝜈

, then the upper Dirichlet and

Neumann spectral dimensions have the common value

𝑠𝐷

𝜈=𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈

. This

assumption is particularly fulﬁlled if

dim𝑀(supp (𝜈)∩𝜕Q)

dim𝑁\𝐷

∞(𝜈)−𝑑+2

<𝑞𝑁

𝔍𝜈.(5.3.2)

Proof.

The ﬁrst and second claim follow from Proposition 5.12, Proposition 5.13,

and Proposition 5.14. Furthermore, by Proposition 2.35, we always have in the case

146

5.3. Main results

𝑑=

2 that

𝑞𝑁

𝔍𝜈=

1. For the third claim note that

𝜈(Q

˚)>

0 implies that there exists

an open cube

𝑄

such that

𝑄⊂Q

𝜈(𝑄)>

0, and

dim∞(𝜈|𝑄)>𝑑−

0. Hence, we

obtain

1=𝑞𝑁

𝔍𝜈|𝑄≤𝐹𝑁

𝔍𝜈|𝑄=𝐹𝐷

𝔍𝜈|𝑄≤𝐹𝐷

𝔍𝜈=𝑞𝐷

𝔍𝜈≤𝑠𝐷

𝜈≤𝑠𝑁

𝜈=1.

To prove the last claim, we note that

(5.3.2)

fulﬁlls the assumption of Lemma 2.38,

which implies 𝑞𝐷

𝔍𝜈=𝑞𝑁

𝔍𝜈.□

Remark 5.16.By Corollary 5.22 we have

𝑞𝑁

𝔍𝜈≥dim𝑀(𝜈)/(dim𝑀(𝜈)−𝑑+

)

. Hence,

we can replaces

𝑞𝑁

𝔍𝜈

dim𝑀(𝜈)/(dim𝑀(𝜈)−𝑑+

)

on the right hand side in

(5.3.2)

making this condition independent of

𝑞𝑁

𝔍𝜈

. Moreover,

(5.3.2)

can easily be veriﬁed

for particular measures 𝜈such that

1. dim𝑀(supp (𝜈)∩𝜕Q)<dim𝑀(𝜈)dim∞(𝜈)−𝑑+2

dim𝑀(𝜈)−𝑑+2,in particular, for 𝜈with

dim∞(𝜈)>𝑑−1 and dim𝑀(supp (𝜈)∩𝜕Q)≤dim𝑀(𝜈)/2.

2. dim𝑀(supp (𝜈)∩𝜕Q)=0, particularly for supp(𝜈)⊂Q

˚,

3. 𝜈

is given by the

𝑑

-dimensional Lebesgue measure

Λ|Q

restricted to

(then

the left-hand side in (5.3.2) is equal to (𝑑−1)/2).

Let us also remark that in Section 5.4.4 we present an example for which 𝑠𝑁

𝜈<𝑠𝑁

𝜈.

Remark 5.17.The above theorem and the notion of regularity give rise to the

following list of observations for measures 𝜈with dim∞(𝜈)>𝑑−2:

If the Neumann spectral dimension with respect to

𝜈

exists, then it is given

by purely measure-geometric data encoded in the

𝜈

-partition entropy, namely

we have

ℎ𝔍𝜈=lim𝑡↓2ℎ𝔍𝜈,𝑡(2/𝑑−1)/2,1

and this value coincides with the spectral

dimension.

N-MF-regularity implies equality everywhere in the chain of inequalities

(5.3.1)

and in particular the Neumann spectral dimension exists. If

𝔍𝜈

D-MF-regular, then we have equality everywhere in all chains of inequalities

above and in particular both Neumann and Dirichlet spectral dimensions

exist.

5.3.2 Regularity results

The following theorem shows that the spectral partition function is a valuable

auxiliary concept to determine the spectral behavior for a given measure 𝜈.

147

5.3. Main results

Theorem 5.18. Under the assumption

dim∞(𝜈)>𝑑−

2we have the following

regularity result:

𝔍𝜈

is N-PF-regular, then it is N-MF-regular and the Neumann spectral

dimension 𝑠𝑁

𝜈exists.

𝔍𝜈

is D-PF-regular and

𝜏𝑁

𝔍𝜈𝑞𝐷

𝔍𝜈=

0, then both the Dirichlet and Neumann

spectral dimension exist and coincide, i.e. 𝑠𝐷

𝜈=𝑠𝑁

𝜈.

Proof.

Under the assumption that

𝔍𝜈

is D/N-PF-regular, we obtain from Proposition

3.24 applied to 𝔍 = 𝔍𝜈, Lemma 2.18, and Theorem 5.15

𝑞𝑁

𝔍𝜈=𝐹𝑁

𝔍𝜈=𝑠𝑁

𝜈≥𝑠𝐷/𝑁

𝜈≥𝐹𝐷/𝑁

𝔍𝜈=𝑞𝐷/𝑁

𝔍𝜈and 𝑠𝐷

𝜈≤𝑠𝑁

𝜈,

proving the claims. □

We will see in Section 5.4.4 that the result is optimal in the sense that there is an

example derived from a similar example for

𝑑=

1 in

𝜈

which is not

𝔍𝜈

N-PF-regular

and for which

𝑠𝑁

𝜈>𝑠𝑁

𝜈

. It should be noted that PF-regularity is easy accessible if

the spectral partition function is essentially given by the 𝐿𝑞-spectrum of 𝜈.

In the following proposition we present lower bounds of the lower spectral dimen-

sion in terms of the subdiﬀerential of 𝜏𝐷/𝑁

𝔍𝜈at 𝑞.

Proposition 5.19. If dim∞(𝜈)>𝑑−2and if for 𝑞∈0,𝑞𝐷/𝑁we have

𝜏𝐷/𝑁

𝔍𝜈(𝑞)=lim

𝑛→∞𝜏𝐷/𝑁

𝔍𝜈,𝑛 (𝑞)and −𝜕𝜏𝐷/𝑁

𝔍𝜈(𝑞)=[𝑎,𝑏],

then

𝑎𝑞 +𝜏𝐷/𝑁

𝔍𝜈(𝑞)

𝑏≤𝑠𝐷/𝑁

𝜈.

Proof.

This follows from Proposition 3.18, Proposition 5.12, and Proposition 5.13.

□

Remark 5.20.In the case that

𝜏𝑁

𝔍𝜈(𝑞𝑁

𝔍𝜈)=lim𝑛→∞𝜏𝑁

𝔍𝜈,𝑛 (𝑞𝑁

𝔍𝜈)

and

𝜏𝑁

𝔍𝜈

is diﬀerentiable

𝑞𝑁

𝔍𝜈

, we infer

𝑞𝑁

𝔍𝜈≤𝑠𝑁

𝜈

and hence we obtain a direct proof of the regularity

statement, namely, 𝑞𝑁

𝔍𝜈=𝑠𝑁

𝜈=𝑠𝑁

𝜈.

Also, if

𝜏𝐷/𝑁

𝔍𝜈(

)=lim𝑛→∞𝜏𝐷/𝑁

𝔍𝜈,𝑛 (1)=𝑑−

2, we have the lower bound in terms of

left-sided, respectively right-sided, derivative of 𝛽𝑁

𝜈given by

−𝜕+𝜏𝐷/𝑁

𝔍𝜈(1)−𝑑+2

−𝜕−𝜏𝐷/𝑁

𝔍𝜈(1)≤𝑠𝐷/𝑁

𝜈.

148

5.3. Main results

Corollary 5.21. For 𝑑=2we have:

1. It holds

−𝜕+𝛽𝑁

𝜈(1)

−𝜕−𝛽𝑁

𝜈(1)≤𝑠𝑁

𝜈≤𝑠𝑁

𝜈=1.

2. If 𝛽𝐷/𝑁

𝜈is diﬀerentiable at 1, then 𝑠𝐷/𝑁

𝜈=1.

𝜈(Q

˚)>

0and

𝛽𝑁

𝜈

is diﬀerentiable at 1, then also

𝛽𝐷

𝜈

is diﬀerentiable at 1.

In particular, 𝑠𝐷

𝜈=𝑠𝑁

𝜈=1.

5.3.3 General bounds in terms of fractal dimensions

We obtain general bounds for

𝑠𝑁

𝜈

in terms of the upper Minkowski dimension

dim𝑀(𝜈)

and the possibly smaller lower

∞

-dimension

dim∞(𝜈)

𝜈

(see also

Figure 5.3.1).

Corollary 5.22. Assume

dim∞(𝜈)>𝑑−

2. Then for the Neumann upper spectral

dimension we have

𝑑

2≤dim𝑀(𝜈)

dim𝑀(𝜈)−𝑑+2≤𝑠𝑁

𝜈≤dim∞(𝜈)

dim∞(𝜈)−𝑑+2.

Proof.

Theorem 5.15 gives

𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈

, hence the claim follows from the estimates of

𝑞𝑁

𝔍𝜈obtained in Fact 2.37. □

Remark 5.23.Note that by choosing measures with

dim𝑀(𝜈)

close to

𝑑−

2 we can

easily ﬁnd examples where 𝑠𝑁

𝜈becomes arbitrarily large.

It is also worth mentioning that the analogous situation in the dimension

𝑑=

1 is

quite diﬀerent (cf. Section 4.3.3), namely the lower bound becomes an upper bound,

𝑠𝐷/𝑁

𝜈≤dim𝑀(𝜈)

dim𝑀(𝜈)+1≤1

This inequalities in Corollary 5.22 naturally links to the famous question by M. Kac

[Kac66], “Can one hear the shape of a drum?” This question has been modiﬁed

by various authors e.g. in [Ber79; Ber80; BC86; Lap91], and closer to our context

by Triebel in [Tri97]. In the plane, the spectral dimension does not encode any

information about the fractal-geometric nature of the underlying measure as we

always have

𝑠𝐷/𝑁

𝜈=

1 for any bounded Borel measure with

𝜈(Q

˚)>

0. This has been

observed in [Tri97] for the special case of

𝛼

-Ahlfors–David regular measures. For all

other dimensions, our results show that the upper spectral dimension

𝑠𝑁

𝜈

is uniquely

149

5.3. Main results

1 2 𝑞𝑁

𝔍𝜈

dim𝑀(𝜈)

𝑞

𝜏𝑁

𝔍𝜈(𝑞)

Figure 5.3.1 Partition function

𝜏𝑁

𝔍𝜈

in dimension

𝑑=

3 for the self-similar measure

𝜈

supported on the Sierpi´nski tetraeder with all four contraction ratios equal 1

2 and

with probability vector

(0.36,0.36,0.2,0.08)

. Natural bounds for

𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈

in this set-

ting are the zeros of the dashed line

𝑥↦→ −𝑥𝜏𝑁

𝔍𝜈(0)−1+𝜏𝑁

𝔍𝜈(0)

and the dotted line

𝑥↦→ (1−𝑥)dim∞(𝜈)+𝑥

as given in Corollary 5.22. In this case

𝜏𝑁

𝔍𝜈(0)=dim𝑀(𝜈)=

2 and

dim∞(𝜈)=−log (0.36)/log (2)≈1.47

150

5.4. Examples

determined by the spectral partition function

𝜏𝑁

𝔍𝜈

, which in turn reﬂects many

important fractal-geometric properties of

𝜈

. For the case

𝑑>

2, this common ground

provides interesting bounds on the upper Minkowski dimension of the support of

𝜈

and the lower ∞-dimension of 𝜈in terms of the upper spectral dimension given by

dim𝑀(𝜈)≥𝑠𝑁

𝜈(𝑑−2)

𝑠𝑁

𝜈−1≥dim∞(𝜈).

So the answer to Kac’s question is “partially yes”. If additionally the

𝐿𝑞

-spectrum

𝛽𝑁

𝜈

is an aﬃne function, we obtain

𝛽𝑁

𝜈(𝑞)=dim𝑀(𝜈)+dim𝑀(𝜈)(

−𝑞)

and with

Corollary 5.22

dim∞(𝜈)=dim𝑀(𝜈)=𝑠𝑁

𝜈(𝑑−2)

𝑠𝑁

𝜈−1.

In this case, Kac’s question regarding dimensional quantities must be answered in

the aﬃrmative.

5.4 Examples

Finally, we give some leading examples where the spectral partition function is

essentially given by the

𝐿𝑞

-spectrum of

𝜈

(see Section 2.4.3.1 and Section 2.4.3.4)

and in this case we are able to provide the following complete picture.

5.4.1 Absolutely continuous measures

As a ﬁrst application of our results, we present the case of absolutely continuous

measures.

Proposition 5.24. Let

𝜈

be absolutely continuous with respect to

with

𝑟

-integrable

density for some

𝑟≥𝑑/

2. Then the Dirichlet and Neumann spectral partition

function exists as a limit with

𝜏𝑁

𝔍𝜈(𝑞)=𝜏𝐷

𝔍𝜈(𝑞)=𝑑−2𝑞, for 𝑞∈[0,𝑟 ],

𝜈

is D/N-PF-regular, and the Dirichlet and Neumann spectral dimension exist and

equal 𝑠𝐷

𝜈=𝑠𝑁

𝜈=𝑑/2.

Proof. We immediately obtain from Proposition 2.44 and Theorem 5.18 that

𝑠𝐷

𝜈=𝑠𝑁

𝜈=𝑑/2.□

Also for absolutely continuous measures we have the following rigidity result in

terms of reaching the minimal possible value 𝑑/2 of the spectral dimension.

151

5.4. Examples

Proposition 5.25. Let

𝜈

be an absolutely continuous measure. If

dim∞(𝜈)>𝑑−

then the following rigidity result holds:

1. If 𝑠𝑁

𝜈=𝑑/2, then 𝜏𝑁

𝔍𝜈(𝑞)=𝑑−2𝑞for all 𝑞∈[0,𝑑/2].

𝜏𝑁

𝔍𝜈(𝑞)=𝑑−

𝑞

for some

𝑞>𝑑/

2, then

𝜏𝑁

𝔍𝜈(𝑞)=𝑑−

𝑞

for all

𝑞∈ [

,𝑑/

]

and 𝑠𝑁

𝜈=𝑑/2.

Proof.

Suppose

𝑠𝑁

𝜈=𝑑/

2. Then by Theorem 5.15, we have

𝑞𝑁

𝔍𝜈=𝑑/

2. Moreover,

by Lemma 2.43, for all 0 ≤𝑞≤1, we have

𝑑−2𝑞=𝛽𝑁

𝜈(𝑞)+(𝑑−2)𝑞≤𝜏𝑁

𝔍𝜈(𝑞),

and the convexity of 𝜏𝑁

𝔍𝜈yields for all 𝑞∈ [0,𝑑/2]

𝜏𝑁

𝔍𝜈(𝑞) ≤𝑑−2𝑞.

Furthermore, the convexity of 𝛽𝑁

𝜈for all 𝑞∈ [1,𝑑 /2]yields

𝑑−2𝑞≤𝛽𝑁

𝜈(𝑞)+(𝑑−2)𝑞≤𝜏𝑁

𝔍𝜈(𝑞).

This proves the ﬁrst claim. For the second claim assume

𝜏𝑁

𝔍𝜈(𝑞)=𝑑−

𝑞

for some

𝑞>𝑑/2. Again, for all 𝑞′∈ [0,𝑞],we deduce

𝑑−2𝑞′≤𝛽𝑁

𝜈(𝑞′)+(𝑑−2)𝑞′≤𝜏𝑁

𝔍𝜈(𝑞′)≤𝑑−2𝑞′.

In particular, 𝜏𝑁

𝔍𝜈(𝑑/2)=0, which implies 𝑠𝑁

𝜈=𝑑/2. □

5.4.2 Ahlfors–David regular measure

As a second application, we consider a class of measures with linear spectral

partition functions, namely we treat

𝛼

-Ahlfors–David regular measures

𝜈

for

𝛼>

0, i.e. there exist constants

𝐾>

0 such that for every

𝑥∈supp (𝜈)

and

𝑟∈(0,diam(supp(𝜈))]we have

𝐾−1𝑟𝛼≤𝜈(𝐵𝑟(𝑥)) ≤𝐾𝑟 𝛼.

Recall that

𝐵𝑟(𝑥)

denotes the open ball with center

𝑥

and radius

𝑟>

0. Note that for

𝛼-Ahlfors–David regular measures 𝜈we have 𝛼=dim𝑀(𝜈)=dim∞(𝜈).

Proposition 5.26. Assume that

𝜈

𝛼

-Ahlfors–David regular with

𝛼∈(𝑑−2,𝑑]

such that

𝜈(Q

˚)>

0. Then both Neumann and Dirichlet spectral dimensions exist

and are given by 𝑠𝐷

𝜈=𝑠𝑁

𝜈=𝛼/(𝛼−𝑑+2).

152

5.4. Examples

Proof.

We immediately obtain from Theorem 5.18 and the properties of the partition

function provided in Section 2.4.3.3 that 𝑠𝐷

𝜈=𝑠𝑁

𝜈=𝛼/(𝛼−𝑑+2).□

This proposition recovers some of the major achievements on isotropic

𝛼

-sets

(in our terms this means that the

𝛼

-dimensional Hausdorﬀmeasure restricted to

𝛼

-Ahlfors–David regular) as investigated by Triebel in his book [Tri97]. This

follows in our framework from the fact that the partition function is linear and exists

as a limit (see Section 2.4.3.3).

5.4.3 Self-conformal measures

As a third application, we treat self-conformal measures with possible overlaps,

following up on a question explicitly posed in [NX21, Sec. 5].

Theorem 5.27. Let

𝜈

be a self-conformal measure as deﬁned in Section 2.4.3.4

with

𝜈(𝜕Q)=

0and

dim∞(𝜈)>𝑑−

2. Then the spectral partition function exists as

a limit and is given by

𝜏𝐷/𝑁

𝔍𝜈(𝑞)=𝛽𝑁

𝜈(𝑞)+(𝑑−2)𝑞.

Further,

𝜈

is D/N-PF-regular and the Dirichlet and Neumann spectral dimension

exist and equal

𝑠𝐷

𝜈=𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈

. In particular, in the case

𝑑=

2, we always have

𝑠𝐷

𝜈=𝑠𝑁

𝜈=1.

Proof.

Let

𝜈

be a self-conformal measure with

dim∞(𝜈)>𝑑−

2. Then it follows

from Corollary 2.53 that 𝔍𝜈is D/N-PF-regular and

𝜏𝐷

𝔍𝜈𝑞𝐷

𝔍𝜈=𝜏𝑁

𝔍𝜈𝑞𝐷

𝔍𝜈=0.

Now, Theorem 5.15 and Theorem 5.18 give 𝑠𝐷

𝜈=𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈.□

Remark 5.28.In general, it is diﬃcult to verify the condition dim∞(𝜈)>𝑑−2, but

in the case

𝑑=

2 a suﬃcient condition is that the measure

𝜈

is invariant with respect

to an IFS given by a system of bi-Lipschitz contractions such that the attractor is not

a singleton (see [HLN06, Lemma 5.1]). This carries over to self-similar measures,

provided that the contractive similitudes do not share the same ﬁxed point, so that

dim∞(𝜈)>0 and the spectral dimension is given by 𝑠𝐷

𝜈=𝑠𝑁

𝜈=1.

5.4.4 Non-existence of the spectral dimension

Here, we present an example for which lower and upper spectral dimension diﬀer.

153

5.4. Examples

Example 5.29. Let us consider the homogeneous Cantor measure

𝜇

(0,1)

form

Example 4.48 with non-converging 𝐿𝑞-spectrum, for which we have

𝑠𝐷/𝑁

𝜇=3/13 <3/11 =𝑠𝐷/𝑁

𝜇, 𝛽𝑁

𝜇(𝑞)=3

8(1−𝑞), 𝑞 ∈[0,1],

10 (1−𝑞), 𝑞 >1,

and

𝛽𝜇(𝑞)≔liminf

𝑛→∞ 𝛽𝑁

𝜇,𝑛 (𝑞)=3

10 (1−𝑞), 𝑞 ∈[0,1],

8(1−𝑞), 𝑞 >1.

Take the one-dimensional Lebesgue-measure

Λ1

restricted to

[0,1]

and deﬁne the

product measure on

𝜈≔𝜇⊗Λ1⊗Λ1

. Due to the product structure, we have

for the 𝐿𝑞-spectrum of 𝜈

𝛽𝑁

𝜈(𝑞)=𝛽𝑁

𝜇(𝑞)+𝛽𝑁

Λ2(𝑞)=𝛽𝑁

𝜇(𝑞)+2(1−𝑞), 𝑞 ≥0,

and hence

dim∞(𝜈)=

1. Let

𝜋1

denote the projection onto the ﬁrst

coordinate. Then for 𝑡∈ [2,4), we have

𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡,𝑛 (𝑞)=1

log(2𝑛)log 

𝑄∈D𝑁

𝑛

sup

𝑄′∈D(𝑄)𝜈(𝑄′)2/𝑡Λ(𝑄′)−1/3𝑞

log(2𝑛)

log(2𝑛)log 

𝑄∈𝜋1(D𝑁

𝑛)

𝜇(𝑄)𝑞2/𝑡2𝑞𝑛(−4/𝑡+1)22𝑛

=𝛽𝑁

𝜇,𝑛 (𝑞2/𝑡)−𝑞(4/𝑡−1) +2

and the spectral partition function 𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡is given by

𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡(𝑞)=𝛽𝑁

𝜇(𝑞)−𝑞(𝑡−1)+2

and therefore

𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡(𝑞)≠liminf

𝑛→∞ 𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡,𝑛 (𝑞)=𝛽𝑁

𝜇(𝑞)−𝑞(𝑡−1)+2

for

𝑞∈R≥0\{1}

. This gives for the upper spectral dimension

𝑠𝑁

𝜈=𝑞𝑁

𝔍𝜈=

13.

Furthermore, by Example 4.48, with

𝑛𝑘≔

2𝑘+1

−

3 and 2

<𝑡<

4, we have

𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡,𝑛𝑘(𝑞)=𝛽𝑁

𝜇,𝑛𝑘(𝑞2/𝑡)−𝑞(4/𝑡−1)+2=(1−𝑞2/𝑡)

8/3−10/(3𝑛𝑘)−𝑞(4/𝑡−1)+2.

154

5.4. Examples

Therefore, it follows

𝑞𝑡,𝑛𝑘≔inf{𝑞≥0 : 𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡,𝑛𝑘(𝑞)<0}

=1

8/3−10/(3𝑛𝑘)+2 2/𝑡

8/3−10/(3𝑛𝑘)+4

𝑡−1−1

Moreover, by the construction of 𝜇, we have

max

𝑄∈D𝑁

𝑛𝑘

𝔍𝜈, (2/3−1),2/𝑡(𝑄)𝑞𝑛𝑘=max

𝑄∈D𝑁

𝑛𝑘𝜇(𝜋1(𝑄))𝑞2/𝑡2𝑞𝑛𝑘(−4/𝑡+1)22𝑛𝑘𝑞𝑛𝑘

=𝑄∈D𝑁

𝑛𝑘𝜇(𝜋1(𝑄))𝑞2/𝑡2𝑞𝑛𝑘(−4/𝑡+1)22𝑛𝑘𝑞𝑛𝑘

2𝑛𝑘𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡,𝑛𝑘(0)

=𝑄∈D𝑁

𝑛𝑘

𝔍𝜈, (2/3−1),2/𝑡(𝑄)𝑞𝑛𝑘

2𝑛𝑘𝜏𝑁

𝔍𝜈, (2/3−1),2/𝑡,𝑛𝑘(0).

Hence, the assumptions of Proposition 3.3 are fulﬁlled. Thus, combining Proposition

3.3 and Corollary 5.6 yield

𝑠𝑁

𝜈≤lim

𝑡↓2ℎ𝔍𝜈,(2/3−1),2/𝑡≤lim

𝑡↓2lim

𝑘→∞𝑞𝑡 ,𝑛𝑘≤lim

𝑡↓2

838

𝑡8−1−1

=19

11 .

Furthermore, by Theorem 5.15 and the result of Section 2.4.3.2, we ﬁnd that

𝑠𝐷

𝜈=𝑠𝑁

𝜈

To summarize, we obtain from the consideration above that

𝑠𝐷

𝜈≤𝑠𝑁

𝜈≤19/11 <𝑠𝐷

𝜈=𝑠𝑁

𝜈=23

13 .

It should be remarked that this example can be easily modiﬁed to construct an

example for non-existing spectral dimension for any 𝑑≥3.

155

Chapter 6

Quantization Dimension

In this chapter, we study the lower and upper quantization dimension with respect

to a compactly supported Borel probability measure

𝜈

R𝑑

. Let us ﬁrst recall the

deﬁnition of the lower and upper quantization dimension as given in Section 1.1.2:

𝐷𝑟(𝜈)=liminf

𝑛→∞

log(𝑛)

−log(𝔢𝑛,𝑟 (𝜈)), 𝐷𝑟(𝜈)=lim sup

𝑛→∞

log(𝑛)

−log(𝔢𝑛,𝑟 (𝜈)),

with 𝑟>0 and

𝔢𝑛,𝑟 (𝜈)=inf

𝛼∈A𝑛min

𝑦∈𝛼∥𝑥−𝑦∥𝑟d𝜈(𝑥)1/𝑟

=inf

𝑓∈F𝑛|𝑥−𝑓(𝑥)|𝑟d𝜈(𝑥)1/𝑟

where A𝑛is the set of subsets of R𝑑with cardinality less then or equal to 𝑛and

F𝑛={𝑓:R→𝐴, 𝐴 ∈ A𝑛}.

Note that without loss of generality, we can (and for ease of exposition, we will)

assume that the support of

𝜈

is contained in

(

)𝑑

. To see this, ﬁx

𝑎≠

,𝑏 ∈R𝑑

and let Φ𝑎,𝑏 (𝑥)≔𝑎𝑥 +𝑏,𝑥∈R𝑑, such that Φ𝑎,𝑏 (supp(𝜈)) ⊂ (0,1)𝑑. Then,

𝔢𝑛,𝑟 𝜈◦Φ−1

𝑎,𝑏 =inf

𝑓∈F𝑛𝑎𝑥 +𝑏−𝑓Φ𝑎,𝑏 (𝑥)𝑟d𝜈(𝑥)1/𝑟

=|𝑎|inf

𝑓∈F𝑛𝑥+Φ−1/𝑎,𝑏/𝑎𝑓Φ𝑎,𝑏 (𝑥)𝑟d𝜈(𝑥)1/𝑟

=|𝑎|inf

𝑓∈F𝑛|𝑥−𝑓(𝑥)|𝑟d𝜈(𝑥)1/𝑟

=|𝑎|𝔢𝑛,𝑟 (𝜈),

where we used that

𝑓↦→ Φ−1/𝑎,𝑏/𝑎◦𝑓◦Φ𝑎,𝑏

deﬁnes a surjection on

F𝑛

. Again, as

for the computation of the spectral dimension, the main strategy is to reduce the

156

6.1. Upper bounds for the quantization dimension

problem of the determination of the quantization dimension to the combinatorial

problems considered in Chapter 3 applied to 𝔍𝜈,𝑎 :𝑄↦→𝜈(𝑄)Λ(𝑄)𝑟/𝑑with 𝑄∈ D.

This chapter is structured as follows. Section 6.1 is devoted to provide upper bounds

of the lower and upper quantization dimension; we obtain bounds in terms of the

lower and upper

𝔍𝜈,𝑟/𝑑

-partition entropy. In Section 6.2, we obtain lower bounds

of the lower and upper quantization dimension in terms of the lower and upper

optimized coarse multifractal dimension with respect to

𝔍𝜈,𝑟/𝑑

. The main results

of this chapter are presented in Section 6.3. Thereby, we combine the results of

Section 6.1 and Section 6.2 to compute the upper quantization dimension and impose

regularity conditions that guarantee the existence of the quantization dimension.

Finally, we conﬁrm the existence of the quantization dimension for self-conformal

measures where no separation conditions are assumed.

6.1 Upper bounds for the quantization dimension

In this section, building on the results of Section 3.3, we establish upper bounds

of the quantization dimension in terms of the

𝐿𝑞

-spectrum of

𝜈

. For this purpose,

we recall the notation from Section 3.3:

𝔍𝜈,𝑎 (𝑄)=𝜈(𝑄)Λ(𝑄)𝑎

𝑎>

0 and

𝑄∈ D

Notice that

𝜏𝑁

𝔍𝜈,𝑎 (𝑞)=𝛽𝑁

𝜈(𝑞)−𝑎𝑑𝑞, 𝑞 ≥0.

Further, we deﬁne

𝑞𝑟≔𝑞𝔍𝜈,𝑟 /𝑑=inf 𝑞>0 : 𝛽𝑁

𝜈(𝑞)<𝑟𝑞 .

The following proposition establishes an upper bound of the quantization dimension

in terms of the

𝐿𝑞

-spectrum with respect to

𝜈

and the lower

𝔍𝜈,𝑟/𝑑

-partition entropy.

Proposition 6.1. For all 𝑛∈N, we have

𝑒𝑛,𝑟 (𝜈)𝑟≤√𝑑𝑛𝛾𝔍𝜈,𝑟 /𝑑,𝑛.

In particular, we have

𝐷𝑟(𝜈)≤𝑟ℎ𝔍𝜈,𝑟 /𝑑

1−ℎ𝔍𝜈,𝑟 /𝑑

=𝑟𝑞𝑟

1−𝑞𝑟≤dim𝑀(𝜈),

and

𝐷𝑟(𝜈)≤

𝑟ℎ𝔍𝜈,𝑟 /𝑑

1−ℎ𝔍𝜈,𝑟 /𝑑

157

6.1. Upper bounds for the quantization dimension

Proof.

We only consider the case

𝑞𝑟>

0. The case

𝑞𝑟=

0 follows analogously.

Note that we always have

𝑞𝑟<

1. Let

𝑃∈Π𝜈

with

card(𝑃) ≤ 𝑛

. Let us write

𝑃=𝑄1, . . ., 𝑄card(𝑃)

and let

𝑚𝑖

denote the middle point of the dyadic cube

𝑄𝑖

for

𝑖≤card(𝑃)and set 𝛼𝑛≔(𝑚1, . . .,𝑚 card(𝑃)). Then we have

𝔢𝑛,𝑟 (𝜈) ≤ 𝑑(𝑥, 𝛼𝑛)𝑟d𝜈(𝑥)1/𝑟

=card(𝑃)



𝑖=1𝑄𝑖

𝑑(𝑥, 𝛼𝑛)𝑟d𝜈(𝑥)1/𝑟

≤card(𝑃)



𝑖=1𝑄𝑖

𝑑(𝑥, {𝑚𝑖})𝑟d𝜈(𝑥)1/𝑟

≤√𝑑card(𝑃)



𝑖=1

𝜈(𝑄𝑖)Λ(𝑄𝑖)𝑟/𝑑1/𝑟

≤√𝑑𝑛1/𝑟max

𝑄∈𝑃𝜈(𝑄)Λ(𝑄)𝑟/𝑑1/𝑟

Now, taking the inﬁmum over all 𝑃∈Π𝜈with card(𝑃) ≤ 𝑛yields

𝔢𝑟

𝑛,𝑟 (𝜈) ≤ √𝑑𝑟𝑛𝛾𝔍𝜈,𝑟/𝑑,𝑛 .

Note that by Proposition 3.11, for every 𝜀∈ (0,1/𝑞𝑟−1), we have for 𝑛large

𝑛𝛾𝔍𝜈,𝑟 /𝑑,𝑛 ≤𝑛1−1/𝑞𝑟+𝜀,

and, if ℎ𝔍𝜈,𝑟 /𝑑

>0, there exists a subsequence (𝑛𝑘)𝑘such that

𝑛𝑘𝛾𝔍𝜈,𝑟 /𝑑,𝑛𝑘≤𝑛

1−1/ℎ𝔍𝜈,𝑟/𝑑+𝜀

𝑘.

The case ℎ𝔍𝜈,𝑟 /𝑑=0 follows again similar. This implies

limsup

𝑛→∞

−log(𝑛)

log 𝔢𝑛,𝑟 (𝜈)≤𝑟𝑞𝑟

1−𝑞𝑟

and 𝐷𝑟(𝜈)≤liminf

𝑘→∞ −log(𝑛𝑘)

log 𝔢𝑛𝑘,𝑟 (𝜈)≤

𝑟ℎ𝔍𝜈,𝑟 /𝑑

1−ℎ𝔍𝜈,𝑟 /𝑑

where we used that lim𝑛→∞𝔢𝑛,𝑟 (𝜈)=0. Moreover, Proposition 3.11 implies

−1/𝑞𝑟≤ −dim𝑀(𝜈)+𝑟/dim𝑀(𝜈),

which proves the last inequality. □

158

6.2. Lower bounds for the quantization dimension

Corollary 6.2. If 𝜈is singular, then

lim

𝑛→∞𝑛1/𝑑𝔢𝑛,𝑟 (𝜈)=0.

Proof. Since 𝜈is singular, by Corollary 3.10 and Proposition 6.1, we have

𝔢𝑛,𝑟 (𝜈) ≤ 𝑛𝛾𝔍𝜈 ,𝑟 /𝑑,𝑛1/𝑟=𝑜𝑛−1/𝑑.□

By P¨

otzelberger [P¨

ot01], for all 𝑟>0, we have

𝐷𝑟(𝜈)≤dim𝑀(𝜈).(6.1.1)

The following corollary gives rise to a slight improvement to the estimate in

(6.1.1)

Corollary 6.3. For 𝑟>0such that dim𝑀(𝜈)/(𝑟+dim∞(𝜈))<1, we have

𝐷𝑟(𝜈)≤𝑟dim𝑀(𝜈)

𝑟+dim∞(𝜈)−dim𝑀(𝜈).

Proof. By Proposition 3.13, we have

ℎ𝔍𝜈,𝑟 /𝑑≤dim𝑀(𝜈)

𝑟+dim∞(𝜈).

Now, the claim follows from Proposition 6.1 and the fact that

𝑥↦→ 𝑥/(

−𝑥)

increasing on (0,1).□

6.2 Lower bounds for the quantization dimension

Recall, for 𝑛∈Nand 𝛼>0,

N𝑁

𝛼,𝔍(𝑛)=card 𝑀𝑁

𝛼,𝔍(𝑛)with 𝑀𝑁

𝛼,𝔍(𝑛)=𝐶∈ D𝑁

𝑛:𝔍(𝐶) ≥ 2−𝛼𝑛 ,

as well as

𝐹𝑁

𝔍(𝛼)=limsup

𝑛→∞

log+N𝑁

𝛼,𝔍(𝑛)

log(2𝑛)and 𝐹𝑁

𝔍(𝛼)=liminf

𝑛→∞

log+N𝑁

𝛼,𝔍(𝑛)

log(2𝑛),

and

𝐹𝑁

𝔍=sup

𝛼>0

𝐹𝑁

𝔍(𝛼)

𝛼and 𝐹𝑁

𝔍=sup

𝛼>0

𝐹𝑁

𝔍(𝛼)

𝛼.

Recall, for

𝑠>

0 we let

⟨𝑄⟩𝑠

denote the cube centered and parallel with respect to

𝑄

such that Λ(𝑄)=𝑠−𝑑Λ⟨𝑄⟩𝑠.

159

6.2. Lower bounds for the quantization dimension

Proposition 6.4. We have,

𝐷𝑟(𝜈)≥𝑟𝑞𝑟

1−𝑞𝑟

and 𝐷𝑟(𝜈)≥

𝑟 𝐹 𝑁

𝔍𝜈,𝑟 /𝑑

1−𝐹𝑁

𝔍𝜈,𝑟 /𝑑

Proof. Fix 𝛼>0 such that 𝐹𝔍𝜈,𝑟/𝑑(𝛼)>0. Further, let (𝑛𝑘)𝑘be such that

𝐹𝑁

𝔍𝜈,𝑟 /𝑑(𝛼)=lim

𝑘→∞

log+N𝑁

𝛼,𝔍𝜈 ,𝑟/𝑑(𝑛𝑘)

log(2𝑛𝑘)

and let

𝑐𝛼,𝑛𝑘≔card(𝐸𝛼,𝑛𝑘)

be given as in Lemma 5.8 for

𝔍𝜈,𝑟/𝑑

. Notice that by

our assumption

𝐹𝑁

𝔍𝜈,𝑟 /𝑑(𝛼)>

0, we infer that

lim𝑘→∞𝑐𝛼,𝑛𝑘=∞

. Let

𝐴⊂R𝑑

be of

cardinality at most 𝑐𝛼,𝑛𝑘/2 and

𝐸′

𝛼,𝑛𝑘

≔𝑄∈𝐸𝛼,𝑛𝑘: min

𝑎∈𝐴𝑑(𝑎,𝑄)≥2−𝑛𝑘.

Since, for all 𝑄1,𝑄2∈𝐸𝛼,𝑛𝑘we have 𝑄

˚13∩𝑄

˚23=∅. Hence, it follows

𝑑(𝑄1,𝑄2) ≥ 2−𝑛𝑘.

Thus, if

𝑑(𝑎,𝑄)<

−𝑛𝑘

for some

𝑎∈𝐴

and

𝑄∈𝐸𝛼,𝑛𝑘

, then

𝑑(𝑎,𝑄 ′)≥

−𝑛𝑘

for all

𝑄′∈𝐸𝛼,𝑛𝑘\{𝑄}and therefore,

card 𝑄∈𝐸𝛼,𝑛𝑘: min

𝑎∈𝐴𝑑(𝑎,𝑄)<2−𝑛𝑘≤card (𝐴).

Hence, card 𝐸′

𝛼,𝑛𝑘≥𝑐𝛼,𝑛𝑘−card(𝐴) ≥ 𝑐𝛼,𝑛𝑘/2 and

𝑑(𝑥, 𝐴)𝑟d𝜈(𝑥) ≥ 

𝑄∈𝐸′

𝛼,𝑛𝑘𝑄

𝑑(𝑥, 𝐴)𝑟d𝜈(𝑥)

≥

𝑄∈𝐸′

𝛼,𝑛𝑘

𝜈(𝑄)2−𝑛𝑘

≥card(𝐸′

𝛼,𝑛𝑘)2−𝛼𝑛𝑘≥𝑐𝛼 ,𝑛𝑘2−𝛼𝑛𝑘−1.

160

6.2. Lower bounds for the quantization dimension

Hence, 𝔢𝑟

⌊𝑐𝛼,𝑛𝑘/2⌋,𝑟 (𝜈) ≥𝑐𝛼,𝑛𝑘2−𝛼𝑛𝑘−1and we obtain for the ﬁrst claim

limsup