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Low regularity classes and entropy numbers

Authors:
Andreas Seeger at University of Wisconsin–Madison
Andreas Seeger
Walter Trebels at Technical University of Darmstadt
Walter Trebels
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Abstract

We note a sharp embedding of the Besov space B^\infty_{0,q}(\bbT) into exponential classes and prove entropy estimates for the compact embedding of subclasses with logarithmic smoothness, considered by Kashin and Temlyakov. Comment: 11 pages
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arXiv:math/0703067v1 [math.CA] 2 Mar 2007
LOW REGULARITY CLASSES AND ENTROPY NUMBERS
ANDREAS SEEGER WALTER TREBELS
In memory of Eduard Belinsky (1947 – 2004)
Abstract. We note a sharp embedding of the Besov space B
∞
0,q
(T)
into exponential classes and prove entropy estimates for the compact
embedding of subclasses with logarithmic smoothness, considered by
Kashin and Temlyakov.
1. Introduction
We consider spaces of functions with low regularity and their embedding
properties with respect to the exponential classes exp(L
ν
). For simplicity
we work with functions on the torus T = R/Z (identified with 1-periodic
functions on R). We use the following characterization of the Luxemburg
norm in exp L
ν
(T), found for example in [15]. For ν > 0 set
(1) kfk
exp L
ν
(T)
= sup
1≤p<∞
p
−1/ν
kfk
L
p
(T)
;
this norm will be used in wh at follows.
We consider the Besov spaces B
∞
0,q
, defined via dyadic decompositions
as follows. Let Φ ≡ φ
0
be an even C
∞
function on R with the property
that Φ(s) = 1 for |s| ≤ 1 and Φ is supported in (−2, 2). For k ≥ 1 set
φ
k
(s) = Φ(2
−k
s) − Φ(2
−k+1
s) and, for k = 0, 1, 2, . . .
L
k
f(x) ≡ φ
k
(D)f(x) =
X
n
φ
k
(n)
c
f
n
e
2πinx
.
Then B
∞
0,q
is defined as the space of distributions for which
kfk
B
∞
0,q
=
∞
X
k=0
L
k
f
q
∞
1/q
is finite. It is well known that the class of functions defined in this way does
not depend on th e specific choice of Φ.
Date: March 1, 2007.
1991 Mathematics Subject Classification. 42B15.
A.S. was supported in part by an NSF grant.
1
2 ANDREAS SEEGER WALTER TREBELS
The space B
∞
0,q
consists of locally integrable functions if and only if q ≤ 2
(see [6 ], p. 112) and it follows easily from the d efi nition th at it embeds into
L
∞
if q ≤ 1. We shall sh ow for the interesting range 1 < q ≤ 2 a sharp
embedding result involving the exponential classes.
Theorem 1.1. Let 1 < q ≤ 2. Then the space B
∞
0,q
is continuously embedded
in exp L
q
′
, q
′
= q/(q − 1).
This can be read as a statement about the growth envelope of the space
B
∞
0,q
, defined by
(2) E
q
(t) = sup{f
∗
(t) : kfk
B
∞
0,q
≤ 1};
here f
∗
is the nonincreasing rearrangement of f. It is shown in Corollary
2.3 of [3] that kfk
exp L
q
′
≈ sup
t>0
f
∗
(t) log
−1/q
′
(e/t) so that Theorem 1.1
immediately implies an upper bound C|log t|
1/q
′
for E
q
(t) when t is small.
The corresponding lower boun d is proved in [6], Prop. 8.24 (there also the
nonoptimal upper bound C|log t| is derived). Thus we get
Corollary 1.2. For 1 ≤ q ≤ 2,
E
q
(t) ≈ |log t|
1/q
′
, |t| ≤ 1/2.
We shall now consider subclasses LG
γ
(T) of B
∞
0,2
which are compactly
embedded in Lebesgue an d exponential classes; these were introduced by
Kashin and Temlyakov [10]. For γ > 1/2 the class LG
γ
(T) is defined as the
class of L
1
(T) functions for w hich kL
k
fk
∞
= O((1 + k)
−γ
) and we set
kfk
LG
γ
(T)
= sup
k≥0
(1 + k)
γ
kL
k
fk
∞
.
Clearly, for γ > 1 the class LG
γ
(T) is embedded in L
∞
and if 1/2 < γ ≤ 1
then LG
γ
(T) is embedded in exp L
ν
(T) for ν < (1 − γ)
−1
, by Theorem 1.1.
We are interested in the compactness properties of this embedding and some
related quantitative statements.
We recall that given a Banach space X and a subspace Y ⊂ X one defines
the nth entropy number e
n
(Y ; X) as the infimum over all numbers ε > 0
for which there are 2
n−1
balls of r ad ius ε in X which cover the unit ball
{y ∈ Y : kyk
Y
≤ 1} embedded in X. It is easy to see that the embedding of
Y in X is a compact operator if and only if lim
n→∞
e
n
(Y ; X) = 0.
For γ > 1 the embedding of LG
γ
(T) into L
∞
is compact and Kashin and
Temlyakov [10] determined sh arp bounds for the entropy numbers for the
embedding into L
∞
and L
p
, p < ∞; they showed that for n ≥ 2 and γ > 1
(3) e
n
(LG
γ
, L
p
) ≈
(
(log n)
1/2−γ
, 1 ≤ p < ∞,
(log n)
1−γ
, p = ∞.
LOW REGULARITY CLASSES AND ENTROPY NUMBERS 3
We note that the restriction γ > 1 in [10] is only used to ensure the
imbedding into L
∞
; indeed it is implicitly in [10 ] that for p < ∞ the L
p
result (3) holds for all γ > 1/2. The hard part in the Kashin -Temlyakov
result are the lower boun ds. The L
p
lower bound is derived using Littlewood-
Paley theory from lower bounds for classes of trigonometric polynomials in
[9]. The L
∞
bounds requir e fine estimates for certain Riesz products (cf.
Theorem 2.3 in [10]).
It is desirable to explain the jump in the exponent that occurs in (3) when
p → ∞. To achieve this Belinsky and Trebels ([1], Theorem 5.3) studied
the entropy numbers e
n
(LG
γ
, exp L
ν
) for the natural embedding into the
exponential classes; they obtained the equivalence e
n
≈ (log n)
1/2−γ
for
ν ≤ 1. For ν ≥ 2 they obtained an almost sharp result, namely that e
n
is
essentially (log n)
1−γ−1/ν
, albeit with a loss of (log log n)
1/ν
for the upper
bound. A more substantial gap between lower and upper bounds remained
for 1 ≤ ν < 2. In [1] it was also noticed that this gap could be closed if
Pichorides conjecture [13] on the constant in the reverse Littlewood-Paley
inequality were proved; this however is still an open problem. Nevertheless
we shall use this insight to close the gap in [1].
Theorem 1.3. The embedding LG
γ
(T) → exp L
ν
(T) is compact if either
γ > 1/2, ν < 2, or ν ≥ 2, γ > 1 − ν
−1
, and there are the following upper
and lower bounds for the entropy numbers.
(i) For γ > 1/2, and ν < 2,
(4) e
n
(LG
γ
, exp L
ν
) ≈ (log n)
1/2−γ
.
(ii) For ν ≥ 2 and γ > 1 − ν
−1
,
(5) e
n
(LG
γ
, exp L
ν
) ≈ (log n)
1−γ−1/ν
.
The lower bounds are known; for ν ≤ 2 they follow immediately from
(3). It was pointed out in [1] that for ν > 2 the lower bounds f ollow from
the L
∞
lower bound in (3) and L
∞
→ exp(L
ν
) Nikolskii inequalities for
trigonometric polynomials.
We thus are left to establish the upper bounds for the entropy numbers.
The idea here is to embed the classes LG
γ
into slightly larger classes LG
γ
dyad
which contain discontinuous functions but satisfy the same entropy estimates
with respect to the exponential classes. I nstead of the Pichorides conjecture
we shall then use the well known bounds for a martingale analogue, due to
Chang, Wilson and Wolff [2]. This philosophy also applies to the proof of
Theorem 1.1; it has been used in other papers, among them [7], [8], [5 ] (see
also references contained in these papers).
4 ANDREAS SEEGER WALTER TREBELS
Notation. I f X, Y are normed linear spaces we use the notation Y ֒→ X
to indicate that Y ⊂ X and the embedding is continuous .
This paper. The proof of Theorem 1.1 is given in §2, and the proof of
Theorem 1.3 in §3.
2. Embedding into the expone ntial classes
We shall work with dyadic versions of the Besov spaces where the L ittle-
wood-Paley operators L
k
are replaced by martingale difference operators.
Let k be a n on negative integer. For a fu nction on [0, 1] we define the condi-
tional expectation operator
E
k
f(x) = 2
k
Z
m2
−k
(m−1)2
−k
f(t)dt, (m − 1)2
−k
≤ x < m2
−k
, m = 1, . . . , 2
k
,
and define
D
k
f(x) = E
k
f(x) − E
k−1
f(x), k ≥ 1,
D
0
f(x) = E
0
f(x);
clearly both E
k
f and D
k
f define 1-periodic functions and can be viewed as
functions on T. Note that the functions D
k
f are piecewise constant and
(typically) discontinuous at m2
−k
, m = 0, . . . , 2
k
− 1. We also observe that
f =
P
k≥0
D
k
f almost everywhere for f ∈ L
1
.
Definition 2.1. Let 1 ≤ q ≤ 2. The dyadic Besov-type spaces ℓ
q
(B
∞
dyad
)
consists of all f ∈ L
1
(T) for which the sequence {kD
k
fk
∞
}
∞
k=0
belongs to
ℓ
q
; the norm is given by
kfk
ℓ
q
(B
∞
dyad
)
=
∞
X
k=0
kD
k
fk
q
∞
1/q
.
Prop osition 2.2. Let 1 ≤ q ≤ 2. Then
B
∞
0,q
֒→ ℓ
q
(B
∞
dyad
) .
This is easily reduced to the f ollowing estimate on compositions of the
difference operators with the convolutions φ(D/λ) for large λ.
Lemma 2.3. Let λ ≥ 1 and k ≥ 0. Let ψ ∈ C
∞
be even, with support in
(−2, −1/2) ∪ (1/2, 2) and let L
λ
= ψ(λ
−1
D). Then
E
k
L
λ
L
∞
→L
∞
≤ C min{λ
−1
2
k
, 1}, k ≥ 0,(6)
D
k
L
λ
L
∞
→L
∞
≤ C min{λ
−1
2
k
, λ2
−k
}, k ≥ 1.(7)
LOW REGULARITY CLASSES AND ENTROPY NUMBERS 5
Proof. Use the notation ψ
−1
(s) = (2πis)
−1
ψ(s), ψ
1
(s) = s ψ(s) and observe
that ψ, ψ
−1
, ψ
1
are C
∞
–functions with compact support away from the ori-
gin so that by standard
c
L
1
-theory the sequences ℓ 7→ ψ(λ
−1
ℓ), ψ
−1
(λ
−1
ℓ),
ψ
1
(λ
−1
ℓ) define th e Fourier coefficients of L
1
(T) functions, with L
1
norms
uniformly in λ . Therefore,
(8) kψ(λ
−1
D)fk
∞
+ kψ
−1
(λ
−1
D)fk
∞
+ kψ
1
(λ
−1
D)fk
∞
≤ Ckfk
∞
.
In particular it is clear that kE
k
L
λ
k
L
∞
→L
∞
= O(1).
Now fix k so that 2
k
< λ and let x
m,k
= m2
−k
. Then for x ∈ [x
m,k
, x
m+1,k
),
E
k
L
λ
f(x) = 2
k
Z
x
m+1,k
x
m,k
X
ℓ∈Z
ψ(λ
−1
ℓ)
Z
1
0
e
−2πiℓy
f(y) dy e
2πiℓx
!
dx
= 2
k
X
ℓ∈Z
ψ(λ
−1
ℓ)
Z
1
0
e
2πiℓ(x
m+1,k
−y)
− e
2πiℓ(x
m,k
−y)
2πi ℓ
f(y) dy
= 2
k
λ
−1
ψ
−1
(D/λ)f(x
m+1,k
) − ψ
−1
(D/λ)f(x
m,k
)
and (6) follows by (8).
Inequality (7) for 2
k
< λ is an immediate consequence and it remains to
consider the case 2
k
≥ λ. Fix x, then E
k
L
λ
f(x) is the average of L
λ
f over
an interval of length 2
−k
containing x. Thus, by the mean value theorem
applied to E
k
L
λ
f(x) and E
k−1
L
λ
f(x), we can write for k ≥ 1
D
k
L
λ
f(x) = L
λ
f(x
′
) − L
λ
f(x
′′
) = (L
λ
f)
′
(˜x)(x
′
− x
′′
)
where x
′
, x
′′
, ˜x have distance at most 2
−k+1
from x. Now (L
λ
f)
′
= λψ
1
(D/λ)f
and thus
kD
k
L
λ
fk
∞
≤ 2
1−k
k(L
λ
f)
′
k
∞
≤ Cλ2
−k
kfk
∞
.
Proof of Proposition 2.2. Let Ψ
0
be a C
∞
function supported in (−4, 4)
which satisfies Ψ
0
(s) = 1 in (−2, 2) and let Ψ
n
= Ψ(2
−n
·) where Ψ is
supported in (−8, −1/8) ∪ (1/8, 8) so that Ψ(s) = 1 for |s| ∈ (1/2, 4). Then
Ψ
n
φ
n
= φ
n
for all n, so that Ψ
n
(D)L
n
= L
n
, and we can write
D
k
f
∞
=
D
k
∞
X
n=0
Ψ
n
(D)L
n
f
∞
≤
∞
X
n=0
D
k
Ψ
n
(D)
L
∞
→L
∞
L
n
f
∞
≤ C
∞
X
n=0
2
−|k−n|
kL
n
fk
∞
and therefore
kfk
ℓ
q
(B
∞
dyad
)
≤ C
∞
X
m=0
2
−m
kL
k+m
fk
∞
∞
k=−m
ℓ
q
≤ C
′
kfk
B
∞
0,q
.
6 ANDREAS SEEGER WALTER TREBELS
We now introduce the square-function and the maximal function
S(f) :=
X
k≥0
|D
k
f(x)|
2
1/2
, M
0
(f) := sup
k≥0
|E
k
f(x) − E
0
f(x)|,
resp., and recall the following deep “good λ inequality” due to Chang, Wilson
and Wolff (Corollary 3.1 in [2]): There are absolute constants c and C so
that for all λ > 0, 0 < ε < 1,
(9) meas

x : M
0
(f)(x) > 2λ, S(f) < ελ

≤ C exp(−
c
ε
2
)meas

x : sup
k≥0
|E
k
f(x)| > λ

.
It is standard that this implies the inequality
(10) kfk
p
≤ C
√
p kS(f)k
p
for all p ≥ 2, and some absolute constant C ≥ 1. Indeed, if we integrate out
the L
p
norms using the distribution fu nction, where we observe that
{x : M
0
(f)(x) > 2λ} ⊂ {x : M
0
(f) > 2λ, S(f ) < ελ} ∪ {x : S(f) ≥ ελ},
we obtain
sup
k
|E
k
f|
p
≤ kE
0
fk
p
+ 2C
1/p
e
−cε
−2
p
−1
ksup
k
|E
k
f|
p
+ 2ε
−1
S(f)
p
.
Now we cho ose ε = ap
−1/2
with a so small that 2Ce
−ca
−2
= 1/2. Since
D
0
= E
0
is incorporated in the definition of the square-function, |f(x)| ≤
sup
k
|E
k
g|(x) a.e., the asserted bound (10) follows.
The following interpolation result is a quick consequence of (10).
Lemma 2.4. There is a constant C so that for 1 ≤ s ≤ 2, s
′
= s/(s − 1),
2 ≤ p < ∞, and all sequences {f
k
} of L
p
(T) functions,
∞
X
k=0
D
k
f
k
L
p
(T)
≤ Cp
1/s
′
∞
X
k=0
kf
k
k
s
L
p
(T)
1/s
.
Proof. The statement is trivial for s = 1, because of the uniform L
p
bounds
for the operators D
k
. We thus only need to prove the statement for s = 2
since then the general case follows by complex interpolation. By a straight-
forward limiting argument we may assume that f
k
= 0 for all but finitely
many k.
We use that D
k
D
l
= 0 if k 6= l, and define g =
P
D
k
f
k
. Then by (10)
kgk
p
=
X
l
D
l
g
p
≤ C
√
p
X
l
|D
l
g|
2
1/2
p
,
LOW REGULARITY CLASSES AND ENTROPY NUMBERS 7
and since p ≥ 2 we can use Minkowski’s inequality to bound this by
C
√
p
X
l
kD
l
gk
2
p
1/2
= C
√
p
X
l
kD
l
f
l
k
2
p
1/2
≤ C
′
√
p
X
l
kf
l
k
2
p
1/2
.
Theorem 1.1 is an immediate consequence of Proposition 2.2 and the
following imbedding resu lt which is based on (10) (or rather the case s = 2
of Lemma 2.4).
Prop osition 2.5. Let 1 ≤ q ≤ 2. Then
ℓ
q
(B
∞
dyad
) ֒→ exp L
q
′
.
Proof. We m odify an argument from [1] (which was based there on the
Pichorides conjecture). Fix f ∈ ℓ
q
(B
∞
dyad
) and let n → k(n, f ) be a bijection
of N ∪ {0} so that the s equ en ce n → kD
k(n,f)
fk
∞
is nonincreasing (in other
words, we form the nonincreasing rearrangement of the sequence {kD
k
fk}).
For p ≥ 2 we need to estimate p
−1/q
′
kfk
p
. Thus fix p > 2 and let N ∈ N
so that p ≤ N < p + 1. We then split
f =
N
X
n=0
D
k(n,f)
f +
∞
X
n=N+1
D
k(n,f)
f := I
N
f + II
N
f.
By H¨older’s inequality
kI
N
fk
p
≤
N
X
n=0
kD
k(n,f)
fk
p
≤
N
X
n=0
kD
k(n,f)
fk
∞
≤ (N + 1)
1/q
′
N
X
n=0
kD
k(n,f)
fk
q
∞
1/q
≤ Cp
1/q
′
kfk
ℓ
q
(B
∞
dyad
)
.(11)
For the s econd term we get a bound in terms of the Lorentz-Besov type
space ℓ
q,2
(B
∞
dyad
) defined similarly as ℓ
q
(B
∞
dyad
), but with the s equ en ce space
ℓ
q
replaced by the Lorentz variant ℓ
q,2
. Since ℓ
q
⊂ ℓ
q,2
for q ≤ 2; this is a
better estimate. Note that
(12)
{D
k
f}
∞
k=0
ℓ
q,2
≈
∞
X
n=0
n
1/q
kD
k(n,f)
fk
∞
2
n
−1
1/2
.
8 ANDREAS SEEGER WALTER TREBELS
We now us e the case s = 2 of Lemma 2.4 to obtain
kII
N
fk
p
≤ Cp
1/2
∞
X
n=N+1
D
k(n,f)
f
2
p
1/2
≤ Cp
1/2
∞
X
n=N+1
D
k(n,f)
f
2
∞
1/2
≤ Cp
1/2
N
−1/2+1/q
′
∞
X
n=N+1
n
1−2/q
′
D
k(n,f)
f
2
∞
1/2
,
and, since 1 − 2/q
′
= 2/q − 1 and p ≈ N, we get f rom (12)
(13) p
−1/q
′
kII
N
fk
p
≤ Ckfk
ℓ
q,2
(B
∞
dyad
)
≤ C
′
kfk
ℓ
q
(B
∞
dyad
)
.
Estimates (11) and (13) yield
kfk
exp L
q
′
. kf k
ℓ
q
(B
∞
dyad
)
and thus the assertion.
3. Entropy number s for the Kashin-Temlyakov classes
We now give a proof of Th eorem 1.3. As discussed in the introduction
only the upper bounds have to be proved. It will be advantageous to define
larger “dyadic” analogues of the LG classes.
Definition 3.1. Let γ > 1/2 and let LG
γ
dyad
(T) denote the class of L
1
(T)
functions for which kD
k
fk
∞
= O(k
−γ
) as k → ∞. We set
kfk
LG
γ
dyad
= sup
k≥0
(k + 1)
γ
kD
k
fk
∞
.
We note that the classes LG
γ
(T) consist of continuous functions provided
that γ > 1. This is not the case for the dyadic analogue LG
γ
dyad
(T) as even
the building blocks D
k
f are piecewise constant and typically discontinuous
at m2
−k
, m = 0, . . . , 2
k
− 1. We prove the following embedding result.
Lemma 3.2. For γ > 1/2
LG
γ
(T) ֒→ LG
γ
dyad
(T) .
Proof. This follows easily from Lemma 2.3. Indeed let f ∈ LG
γ
(T), so that
kL
n
fk
∞
. kfk
LG
γ
(1 + n)
−γ
. As in §2 we can write L
n
= Ψ
n
(D)L
n
where
the operator D
k
Ψ
n
(D) has L
∞
→ L
∞
operator norm O(2
−|k−n|
). Thus
D
k
f
∞
=
D
k
∞
X
n=0
Ψ
n
(D)L
n
f
∞
≤ C
∞
X
n=0
2
−|k−n|
L
n
f
∞
≤ C
0
∞
X
n=0
2
−|k−n|
(1 + n)
−γ
kfk
LG
γ
≤ C
′
(1 + k)
−γ
kfk
LG
γ
.
This proves the assertion.
LOW REGULARITY CLASSES AND ENTROPY NUMBERS 9
We now state a crucial approximation result which will b e derived as a
quick consequence of Lemma 2.4.
Lemma 3.3. Let 1/2 < γ < 1 and 0 < ν < (1 − γ)
−1
or γ ≥ 1 and
0 < ν < ∞. There is a constant C = C(γ, ν) so that for M = 1, 2, . . .
sup
kfk
LG
γ
dyad
≤1
kf − E
M
fk
exp L
ν
≤ C
(
M
1/2−γ
, ν ≤ 2, γ > 1/2,
M
1−1/ν−γ
, ν ≥ 2, γ > 1 −ν
−1
.
Proof. Consider f ∈ LG
γ
dyad
, kfk
LG
γ
dyad
≤ 1, and write
f − E
M
f =
∞
X
k=M+1
D
k
D
k
f .
By Lemma 2.4 we have for 2 ≤ p < ∞, and sγ > 1
p
−1/ν
kf − E
M
fk
p
≤ Cp
1/s
′
−1/ν
∞
X
k=M+1
D
k
f
s
p
1/s
≤ Cp
1/s
′
−1/ν
∞
X
k=M+1
D
k
f
s
∞
1/s
≤ Cp
1/s
′
−1/ν
∞
X
k=M+1
(1 + k)
−sγ
1/s
≤ C(s, γ)p
1−1/ν−1/s
M
1/s−γ
.
If ν ≤ 2 th en we may apply this bound for s = 2, γ > 1/2 and get the bound
kf−E
M
fk
exp L
ν
= O(M
1/2−γ
). If ν > 2 we may apply it with s = ν/(ν−1) ∈
(1, 2), indeed we have sγ > 1 in view of our assumption γ > 1 − ν
−1
; the
result is the asserted bound kf − E
M
fk
exp L
ν
= O(M
1−1/ν−γ
).
We apply a result of Lorentz [11], cf. Theorem 3.1 in [12], p. 492. Here
one considers a Banach space X of functions, a sequ en ce G = {g
1
, g
2
, . . . } of
linearly independent fun ctions whose linear span is dense in X. Set X
0
= 0,
and let, for n ≥ 1, X
n
be the linear span of g
1
, . . . , g
n
. Let
D
n
(x) = inf{kx − yk : y ∈ X
n
}
and let d = (δ
0
, δ
1
, . . . ) be a nonincreasing sequence of positive numbers
with lim
n→∞
δ
n
= 0. Let
A(d) = {x ∈ X : D
n
(x) ≤ δ
n
, n = 0, 1, 2, . . . }
be the approximation set associated with d, G.
Next let N
ε
(A(d)) denotes the minimal number of balls of radius ε needed
to cover A(d). The following inequality for the natural logarithm of N
ε
(A(d))
is a special case of Lorentz’ result.
(14) log N
ε
(A(d)) ≤ 2n log
18δ
0
ε
, if ε ≥ δ
n
.
10 ANDREAS SEEGER WALTER TREBELS
We apply (14) to prove the d yadic analogue of the u pper bound in The-
orem 1.3.
Prop osition 3.4. The embedding LG
γ
dyad
(T) → exp L
ν
(T) is compact if
γ > 1/2, ν < 2 or ν ≥ 2, γ > 1 − ν
−1
and we have
e
n
(LG
γ
dyad
, exp L
ν
) ≤ C(log n)
1/2−γ
, γ > 1/2, ν ≤ 2,(15)
e
n
(LG
γ
dyad
, exp L
ν
) ≤ C(log n)
1−γ−1/ν
, γ > 1 − 1/ν, ν ≥ 2.(16)
Proof. We set X = exp L
ν
, and, for n = 2
M
+ j, j = 0, 1, . . . , 2
M
− 1, let
g
n
be th e characteristic function of the interval [j2
−M
, (j + 1)2
−M
). If X
n
,
D
n
(x) are defined as above then we note that Lemma 3.3 says that for f in
the un it ball of LG
γ
dyad
we have
D
n
(f) ≤ C
0
(log(n + 2))
−a
where a = γ − 1/2 if γ > 1/2 and ν ≤ 2, and a = γ + ν
−1
− 1 if ν ≥ 2 and
γ > 1 − 1/ν. We now note that (14) implies that
e
en
(LG
γ
dyad
, exp L
ν
) ≤ (log(n + 1))
−a
if en > Cn log log n. As log en ≈ log n the asserted inequalities follow.
Conclusion of the proof. By Lemma 3.2 we have
(17) e
n
(LG
γ
, exp L
ν
) ≤ Ce
n
(LG
γ
dyad
, exp L
ν
)
and the assertion of the Theorem 1.3 follows from Proposition 3.4.
Remark: We note that in the dyadic case, there are also similar lower
bounds matching (15), (16) for the entropy numbers e
n
(LG
γ
dyad
, exp L
ν
).
These follow from (17) and the known lower bounds for the entropy numbers
for LG
γ
.
References
[1] E. Belinsky, W. Trebels, Almost optimal estimates for entropy numbers of B
1/2
2,2
and
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[2] S.Y.A. Chang, M. Wilson and T. Wolff. Some weighted norm inequalities concerning
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LOW REGULARITY CLASSES AND ENTROPY NUMBERS 11
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Erg. Th. & Dyn. Sys. 18 (1998), 889–936.
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analysis, preprint, 2004.
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Mat. Inst. Steklov. 145 (1980), 111–116, 250. English transl.: Proc. Steklov Inst.
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Tsentra (AFTs), Moscow, 1999.
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(1966)
[12] G.G. Lorentz, M. v.Golitschek, Y . Makovoz, Constructive Approximation, Advanced
Problems. Grundlehren der mathematischen Wissenschaften, 304, Springer 1996.
[13] S. Pichorides, A note on the Littlewood-Paley square function inequality, Colloq.
Math. 60/61 (1990), 687–691.
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proximation numbers of integral operators (Russian), Mat. Zametk i 51 (1992), no. 5,
125–134, 159; English translation in Math. Notes 51 (1992), no. 5-6, 510–51
[15] H. Triebel, Approximation numbers and entropy numbers of embeddings of fractional
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A. Seeger, Department of Mathematics, University of Wisconsin, 480 Lin-
coln Drive, Madison, WI, 53706, USA
E-mail address: seeger@math.wisc.edu
W. Trebels, Fachbereich Mathematik, Technische Universit
¨
at Darmstadt,
Schloßgartenstraße 7, 64289 Darmstadt, Germany
E-mail address: trebels@mathematik.tu-darmstadt.de
... If E Ă Z d is a non-empty set satisfying the condition In turn, (1.3) will be a corollary of a higher-dimensional extension of a result due to Seeger and Trebels [12] concerning sharp bounds of sums involving "smooth" Littlewood-Paley projections on T d . To state this result, fix a Schwartz function η supported in p´2, 2q such that η| r´1,1s " 1 and consider φpξq " ηpξq´ηp2ξq. ...
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