Trigonometric bases for matrix weighted Lp-spaces

Abstract

We give a complete characterization of 2π-periodic matrix weights W for which the vector-valued trigonometric system forms a Schauder basis for the matrix weighted space Lp(T;W). Then trigonometric quasi-greedy bases for Lp(T;W) are considered. Quasi-greedy bases are systems for which the simple thresholding approximation algorithm converges in norm. It is proved that such a trigonometric basis can be quasi-greedy only for p=2, and whenever the system forms a quasi-greedy basis, the basis must actually be a Riesz basis.

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Contents lists available at ScienceDirect
Journal of Mathematical Analysis and
Applications
www.elsevier.com/locate/jmaa
Trigonometric bases for matrix weighted Lp-spaces
Morten Nielsen
Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg East, Denmark
a r t i c l ei n f o a b s t r a c t
Article history:
Received 18 September 2009
Available online xxxx
Submitted by L. Grafakos
Keywords:
Schauder basis
Trigonometric system
Matrix weight
Muchenhoupt condition
Quasi-greedy basis
Shift invariant system
We give a complete characterization of 2π-periodic matrix weights W for which the
vector-valued trigonometric system forms a Schauder basis for the matrix weighted space
Lp(T;W). Then trigonometric quasi-greedy bases for Lp(T;W) are considered. Quasi-
greedy bases are systems for which the simple thresholding approximation algorithm
converges in norm. It is proved that such a trigonometric basis can be quasi-greedy only
for p = 2, and whenever the system forms a quasi-greedy basis, the basis must actually be
a Riesz basis.
© 2010 Elsevier Inc. All rights reserved.
1. Introduction
A (periodic) matrix weight is a function W : T → CN×Ntaking values in the set of strictly positive definite Hermitian
forms. For technical reasons, we shall always assume that both W and W−1are integrable. The associated weighted space
Lp(T;W), 1 ? p < ∞, is the set of measurable (vector-)functions f : T → CNsatisfying
?f?p
T
where |·| denotes the Euclidean norm on CN. One can easily verify that Lp(T;W) is a Banach space for 1 < p < ∞, and a
Hilbert space for p = 2 with norm induced by the inner product
?
T
In this paper we study certain stability properties of a vector valued trigonometric system in Lp(T;W) in terms of properties
of the weight W. The trigonometric system is defined in a straightforward way. Let {ej}N
Then we simply define ej
T :=?ej
and an orthonormal basis for L2(T;Id), which can be deduced from the scalar case. It is also well known from the scalar
unimodular case that the trigonometric system cannot be unconditional in Lp(T) except in the Hilbert space case p = 2.
Lp(T;W):=
?
??W1/pf??pdx < ∞,
(1.1)
?f,g?L2(T;W):=
?Wf(t),g(t)?
?2(CN)dt.
(1.2)
j=1denote the standard basis for CN.
k(t) := e−2πiktej, t ∈ R, and let
??j = 1,2,...,N, k ∈ Z?.
k
(1.3)
For the trivial constant weight W := Id, it is easy to verify that T forms a (Schauder) basis for Lp(T;Id), 1 < p < ∞,
E-mail address: mnielsen@math.aau.dk.
0022-247X/$ – see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2010.06.015

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This follows from Orlicz’ Theorem, see [22]. However, when we consider T in a space with a matrix weight W with e.g.
unbounded spectrum on T, it is not so obvious what happens. In the scalar case, the seminal paper by Hunt, Muckenhoupt,
and Wheeden [9] demonstrates that the trigonometric system is stable in weighted Lp spaces precisely when the weight
satisfies a so-called Muckenhoupt Ap condition. An application of the results in [9] to Schauder bases problems related to
Gabor systems was recently studied by Heil and Powell [8] for p = 2.
The main contribution of the present paper is to give a complete characterization of matrix weights W for which T
forms a Schauder basis for Lp(T;W). The Schauder basis property turns out to be equivalent to W satisfying a so-called
Muckenhoupt Ap matrix condition. Furthermore, we prove that T can only be a quasi-greedy basis for Lp(T;W) when
p = 2. Quasi-greedy bases are systems for which the simple thresholding approximation algorithm converges in norm.
Moreover, we also show that whenever T is quasi-greedy in L2(T;W), then the basis must actually be a Riesz basis. In the
reduced scalar case, the quasi-greedy property for the univariate trigonometric system was studied by the present author
in [14].
To motivate the study of the vector valued system T , let us mention one important application where problems on
stability of T in Lp(T;W) arise naturally. For a finitely set of functions { fk}N
space S is given by
S := Span?
merely a Schauder basis, or a stronger condition such as unconditional or Riesz bases. The Schauder basis case was settled
by Nielsen and Šiki´ c [16] in the single generator case (i.e., N = 1). The finitely generated shift invariant case was settle by
the present author in [13], where it is proved that the system B forms a Schauder basis for S precisely when T forms a
basis for L2(T;G), with G, the N × N Gram matrix, given by
Gi,j:=
k∈Z
where the Fourier transform is given byˆf(ξ) :=?
The structure of the paper is as follows. In Section 2, we impose an ordering on T and define associated partial sum
operators. Then we study boundedness properties of these operators on Lp(T;W), and the main result of Section 2 shows
that T forms a Schauder basis for Lp(T;W) precisely when the weight W satisfies a Muckenhoupt Apmatrix condition. In
Section 3 we study conditions on the weight W that will make T a so-called quasi-greedy basis for Lp(T;W). Quasi-greedy
bases are Schauder bases for which approximations obtained by thresholding converge in norm. It is proved in Section 3
that T can be quasi-greedy in Lp(T;W) only when p = 2, and in the affirmative case, the system is actually a Riesz basis.
Section 4 concludes the paper with a selection of examples.
k=1in L2(R), the associated shift invariant
fk(·−l)??l ∈ Z, k = 1,2,...,N?⊂ L2(R).
A natural question to pose is whether B := Span{ fk(· − l) | l ∈ Z, k = 1,2,...,N} forms a basis for S. Here basis can mean
?
?fi(·−k)?fj(·−k),
Rf(t)e−2πitξdt. Moreover, T forms a basis for L2(T;G) precisely when
G is a Muckenhoupt A2matrix weight, a notion that will be discussed in details below.
2. Trigonometric Schauder bases for Lp(T T T;W)
In this section we characterize Schauder basis properties of T in Lp(T;W) in terms of properties of the matrix weight
W : T → CN×N. The main result is that T is a Schauder basis for Lp(T;W) precisely when W is a so-called Muckenhoupt
Ap matrix weight. We thus obtain Schauder bases for a large class of weighted spaces in the vector valued setting. In the
scalar case, the paper by Hunt, Muckenhoupt, and Wheeden [9] contains the first proof that the trigonometric system is
stable in weighted Lp spaces precisely when the weight satisfies a suitable Muckenhoupt condition. The connection to
Schauder bases was made precise by Heil and Powell [8] in the case of p = 2. The connection between Schauder bases for
shift invariant spaces and the trigonometric system was established by Nielsen and Šiki´ c [16].
Let us begin by making some observations about the system T in Lp(T;W). For p = 2, we notice that whenever
W,W−1∈ L1, then (W−1ej
?W−1ej
This fact insures that we can define partial sum operators associated with a fixed ordering of T which we will describe
now. First we choose the ordering ρ : N → Z of Z given by
{0,−1,1,−2,2,...}.
Then we induce an ordering η : N → {1,2,...,N}×Z as follows. For m ∈ N we write m − 1 = kN +r, 0 ? r < N, and define
η(m) := (r + 1,ρ(k + 1)). Moreover, we let e(η(m)) := eρ(k+1)
operators defined by
k,ej
k)k,jis a bi-orthogonal system in L2(T;W) in the sense that
?
T
k,ej?
k?
?
L2(T;W):=
?W W−1ej
k,ej?
k?
?
?2(CN)dt = δk,k?δj,j?.
r+1
. With the ordering in place, we can consider the partial sum
Sn(f) :=
n
?
s=1
?f,W−1e?η(s)??
L2(T;W)e?η(s)?.
(2.1)

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With the partial sum operators defined by (2.1), we can obviously study their boundedness properties in L2(T;W). More-
over, we claim that (2.1) can be made to make sense for f ∈ Lp(T;W), 1 < p < ∞, with one additional assumption. The
standing assumption W ∈ L1ensures that ej
that the dual space to Lp(T;W), induced by the inner product given by (1.2), is Lp?(T;W), with 1/p + 1/p?= 1, see [6].
We have,
k∈ Lp(T;W) for 1 < p < ∞, so we focus on the dual element W−1ej
k. We recall
??W−1ej
k
??
Lp?(T;W)=
??
T
??W1/p?−1(t)ej
k(t)??p?
dt
?1/p?
=
??
T
??W−1/p(t)ej
k(t)??p?
dt
?1/p?
.
Hence, if we make the assumptions that W,W−p?/p∈ L1, then (2.1) is well defined for f ∈ Lp(T;W). Notice that this
assumption is compatible with the definition of the Apclass below, see Definition 2.1.
It turns out that boundedness of (2.1) in Lp(T;W) is closely related to a study of certain singular integral operators on
Lp(T;W) related to the vector valued Hilbert transform. The vector-valued Hilbert transform was studied in the seminal
paper by Treil and Volberg [20], and this was later generalized to other types of singular integral operators by Goldberg [6].
Treil and Volberg showed that the correct condition to impose on W to obtain boundedness is the so-called Muckenhoupt
Apcondition.
The Muckenhoupt Ap-condition for matrix weights was introduced by Nazarov, Tre˘ ıl’ and Volberg in [12,21] to study
boundedness properties of the vector-valued Hilbert transform. Here we follow Roudenko [17] and give an equivalent and
more direct definition of matrix Ap weights. It is proved in [17] that the following definition is equivalent to the Ap
condition considered in [12,21]. We let I denote the family of all open intervals on R.
Definition 2.1. Let 1 < p < ∞, and let p?denote the conjugate exponent to p, i.e.,
matrix weight. We say that W belongs to the matrix Muckenhoupt class Approvided W,W−p?/pare integrable, and
1
p+
1
p? = 1. Let W : T → CN×Nbe a
A(p,W) := sup
I∈I
?
I
??
I
??W1/p(x)W−1/p(t)??p?dt
|I|
?p/p?
dx
|I|< ∞.
(2.2)
Remark 2.2. One can verify that W ∈ Apif and only if W−p?/p∈ Ap?, see [17].
We can now state the main result of this section characterizing boundedness of the partial sum operators {Sm}mgiven
by (2.1) in terms of an Apcondition on the matrix weight. A variation of Proposition 2.3 valid in the simpler scalar valued
case can be found in [16].
Proposition 2.3. Let 1 < p < ∞, and let W : T → CN×Nbe a matrix weight with W,W−min(1,p?/p)∈ L1, where p?is the dual
exponent to p, i.e.,1
if W ∈ Ap. Equivalently, the system T given by (1.3) is a Schauder basis for Lp(T;W) if and only if W ∈ Ap.
p+
1
p?= 1. Then the partial sum operators {Sm}mgiven by (2.1) are uniformly bounded on Lp(T;W) if and only
Before we can give the proof of Proposition 2.3, we need to state a few auxiliary results. The Hilbert transform H is
defined on L2(T) by
?
T
H( f )(x) := p.v.
f (t)cot?π(x−t)?
dt.
We lift H to a linear operator on L2(T;W) for any N × N matrix weight W on T by letting it act coordinate-wise, i.e.,
(Hf)i:= H( fi),
The fundamental result by Treil and Volberg [20], see also [19], is the following.
i = 1,...,N, f ∈ L2(T;W).
Theorem 2.4. (See [20].) Let W : T → CN×Nbe a matrix weight, and let 1 < p < ∞. Then the Hilbert transform extends to a bounded
operator on Lp(T;W) if and only if W ∈ Ap.
We recall that the univariate Dirichlet kernel DK is given by
DK(t) =sin2π(K + 1/2)t
sinπt
,
K ? 1,
(2.3)

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and for f ∈ Lp(T), we have
f ∗ DK:=
?
T
f (t)DK(·−t)dt =
K
?
k=−K
ˆf(k)e2πik·.
We lift f ∗ DK to an operator on Lp(T;W) by letting it act coordinate-wise.
We have the following immediate corollary to Theorem 2.4.
Corollary 2.5. Let 1 < p < ∞, and let W : T → CN×Nbe a matrix weight satisfying W,W−min(1,p?/p)∈ L1, with p?the dual
exponent to p, i.e.,1
p+1
p?= 1. Then the partial sum operators f → f∗ DKare uniformly bounded on Lp(T;W) if and only if W ∈ Ap.
Proof. Suppose W ∈ Ap. We let P+denote the Riesz projection onto H2for f ∈ L2(T;W). Recall that P+=1
where S0is the orthogonal projection onto constant vectors. It follows that P+is bounded on Lp(T;W), since H is bounded
on Lp(T;W) according to Theorem 2.4, and S0 is bounded on Lp(T;W) according to [6, Proposition 2.1]. Notice that
f → fe2πiM·is a unitary mapping on Lp(T;W), just as in the scalar case. Then we observe that
f∗ DK= e−2πiK·P+
and the uniform boundedness of f → f∗ DK follows.
Now, suppose f → f ∗ DK are uniformly bounded on Lp(T;W). Then we notice that for f a vector of trigonometric
polynomials,
P+(f) = e2πiK·??e−2πiK·f?∗ DK
bounded on Lp(T;W) and W ∈ Apby Theorem 2.4.
2(I + iH + S0),
?e2πiK·f?− e2πi(K+1)·P+
?e−2πi(K+1)·f?,
?,
for K large. It follows that P+extends to a bounded operator on Lp(T;W), and consequently, the Hilbert transform H is
2
With Corollary 2.5 at our disposal, we can now prove Proposition 2.3.
Proof of Proposition 2.3. First, suppose the partial sum operators {Sm}mgiven by (2.1) are uniformly bounded on Lp(T;W).
We notice that S(2L+1)N(f) = f∗ DL, so f → f∗ DK are uniformly bounded on Lp(T;W) and W ∈ Apby Corollary 2.5.
Now, suppose W ∈ Ap. Given m ∈ N, we choose K sufficiently large such that
Sm(f) = f∗ DK+ R(f),
where R(f) contains at most 2N − 1 terms of the type
?f,W−1ej
However, notice that1
???f,W−1ej
k
? ?f?Lp(T;W).
This, together with the uniform boundedness of f → f∗ DK from Corollary 2.5, shows that the partial sum operators {Sm}m
given by (2.1) are uniformly bounded on Lp(T;W).
Finally, we notice that uniform boundedness of {Sm}m on Lp(T;W) is equivalent to T being a Schauder basis for
Lp(T;W) provided that T is complete in Lp(T;W), see [11]. Suppose T is not complete. Then the Hahn–Banach theorem
provides a non-zero g ∈ Lq(T;W), 1/p + 1/q = 1, such that for all j,k,
?
T
However, one checks that g = W−1/qf for some f ∈ Lq(T;Id), so Wg = W1/pf ∈ Lp(T;Id). Hence, each entry of the vector
W1/pf is in Lp(T), and the completeness of the trigonometric system in Lp(T) implies that W1/pf = 0 a.e. This implies
g = 0, which is a contradiction, and consequently T is complete in Lp(T;W). This completes the proof.
k
?
L2(T;W)ek
j.
k
?
L2(T;W)ek
j
??
Lp(T;W)????f,W−1ej
k
?
L2(T;W)
????ek
j
Lq(R;W)·??ek
??
Lp(T;W)
? ?f?Lp(T;W)·??W−1ej
??
j
??
Lp(T;W)
?Wg(t),ek
j(t)?
?2(CN)dt = 0.
2
1The notation A ? B means that there exists a constant c (independent of any significant parameters) such that A ? cB. We use A ? B whenever A ? B
and B ? A.

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3. Beyond Schauder bases: Quasi-greedy bases
Proposition 2.3 tells us that T defined by (1.3) is a Schauder basis for Lp(T;W) if and only if W ∈ Ap. In this section,
we explore what happens when we impose slightly stronger conditions on T . The condition we have in mind is quasi-
greediness. A quasi-greedy basis is a quasi-normalized Schauder basis for which the thresholding approximation algorithm
converges, see Definition 3.1 below.
The main result of this paper stated as Theorem 3.8 below is that T can be quasi-greedy in Lp(T;W) only for p = 2,
and we completely characterize the weights W ∈ A2(T) for which T is quasi-greedy in L2(T;W). It turns out that T is
quasi-greedy in L2(T;W) precisely when the spectrum of W is bounded and bounded away from zero on T so, in particular,
a quasi-greedy system T in L2(T;W) is always a Riesz-basis.
We begin by introducing some notation and recalling some necessary results on quasi-greedy bases. First we give the
precise definition of a quasi-greedy system in a Banach space X. A bi-orthogonal system is a family (xn,x∗
such that x∗
is quasi-normalized, i.e., infn?xn?X> 0 and supn?x∗
Gm(x) :=
n∈A
where A is a set of cardinality m satisfying |x∗
we arbitrarily pick any such set. The definition of Gmleads directly to the definition of a quasi-greedy system, see [10].
n)n∈N⊂ X × X∗
n(xn) = δn,m. We fix a bi-orthogonal system (xn,x∗
n)n∈Nwith spann(xn) dense in X. We assume that the system
n?X∗ < ∞. For each x ∈ X and m ∈ N, we define
?
x∗
n(x)xn,
n(x)| ? |x∗
k(x)| whenever n ∈ A and k / ∈ A. Whenever A is not uniquely defined,
Definition 3.1. A quasi-normalized bi-orthogonal system (xn,x∗
greedy system provided
??Gm(x)??
n)n∈N⊂ X × X∗, with spann(xn) dense in X, is called a quasi-
X? ?x?X,
∀x ∈ X.
(3.1)
If the system is also a Schauder basis for X, we will use the term quasi-greedy basis.
Remark 3.2. It is straightforward to check that an unconditional basis is also quasi-greedy, but the converse statement is
false. There exist conditional quasi-greedy bases, see [10].
Remark 3.3. In was proved by Wojtaszczyk in [23] that a system is quasi-greedy if and only if for each x ∈ X, the sequence
Gm(x) converges to x in norm. In particular, if a Schauder basis is not quasi-greedy, then there exists x0∈ X such that
the Gm(x0) fails to converge to x0. Put another way, approximations obtained by decreasing rearrangements are not norm
convergent.
Let us explain our strategy to obtain information about the weight W. We are going to probe the weight with very
special vector functions induced by the Dirichlet kernel and its translates. The Dirichlet kernel has two important properties
that we will use extensively: It has unimodular coefficients relative to the trigonometric system, and its powers (properly
normalized) form approximations to the identity.
The first result we state is due to Wojtaszczyk [23], see also [5]. It shows that quasi-greedy bases are unconditional for
constant coefficients. Below in Lemma 3.7 we will use this fact to estimate the norm of vector functions induced by, e.g.,
the Dirichlet kernel.
Lemma 3.4. Suppose {bk}k∈Nis a quasi-greedy basis in a Banach space X. Then there exist constants 0 < c1? c2< ∞ such that for
every choice of signs εk= ±1 and any finite subset A ⊂ N, we have
????
For our purpose, Lemma 3.4 is not quite enough. When we consider translates of the Dirichlet kernel, we need to be
able to handle arbitrary unimodular complex coefficients and not only ±1 as covered by Lemma 3.4. For that purpose, we
need to use some facts about the Banach space geometry of Lp(T;W). For the definition of type and cotype of a Banach
space we refer to, e.g., [22, Chap. III]. We also recall that the scalar valued Lp(T) has type 2 for 2 ? p < ∞ and cotype 2 for
1 < p ? 2, see [22, Chap. III]. These properties are inherited by Lp(T;W) as the following lemma shows.
c1
?
k∈A
bk
????
X
?
????
?
k∈A
εkbk
????
X
? c2
????
?
k∈A
bk
????
X
.
(3.2)
Lemma 3.5. Let W : T → CN×Nbe an integrable matrix weight. The Banach space Lp(T;W) has type 2 for 2 ? p < ∞, and cotype 2
for 1 < p ? 2.
Proof. We focus on the type claim, so we assume that 2 ? p < ∞. Let r1,r2,... be the Rademacher functions on [0,1)
defined by rk(t) = sign(sin(2kπt)), and let {fk}∞
k=1be an arbitrary sequence of functions from Lp(T;W). We use [f]i to