# A Remark on Wavelet Bases in WeightedLp Spaces

Authors: Agnieszka Wojciechowska
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The paper deals with unconditional wavelet bases in weightedLp spaces. Inhomogeneous wavelets of Daubechies type are considered. Necessary and sufficient conditions for weights are found for which the wavelet system is an unconditional basis in weighted Lp spaces in dependence on p.


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## Body

## 1. Introduction

Wavelet systems in weightedLp spaces were investigated by several authors. Lemarié-Rieusset considered one-dimensional case, compare [1]. He proved that the homogeneous wavelet system of Daubechies type is an unconditional basis in Lp(ℝn,dμ),  1<p<∞, if and only if dμ=wdx, where w is a weight belonging to the Muckenhoupt class 𝒜p. He also found a sufficient and necessary condition for inhomogeneous systems to be unconditional bases. Other one-dimensional systems were investigated by Kazarian [2] and García-Cuerva and Kazarian [3]. Multidimensional homogeneous wavelet systems were considered by Aimar et al., compare [4]. They proved that a homogeneous wavelet system satisfying certain regularity conditions is an unconditional basis in Lp(dμ) if and only if dμ=wdx with w∈𝒜p.The aim of the paper is to prove the counterpart of Lemarié-Rieusset result for multidimensional inhomogeneous wavelet systems. To formulate the necessary and sufficient condition we use the class of local Muckenhoupt weights𝒜∞loc introduced by Rychkov in 2001, compare [5]. In his work Rychkov developed the theory of weighted Besov and Triebel-Lizorkin spaces with local Muckenhoupt weight. Wavelets characterizations of the function spaces with so-called admissible weights was given by Haroske and Triebel in [6]. Later Haroske and Skrzypczak proved the characterization for the spaces with Muckenhoupt weights, compare [7]. Recently Izuki and Sawano have proved the result for the function spaces with weights from class 𝒜∞loc, compare [8], and another approach can be found in [9].The main Theorem of the paper asserts that the inhomogeneous wavelet system of Daubechies type is an unconditional basis inLp(dμ) if and only if dμ=wdx with w∈𝒜ploc.


## 2. Classes of Weights

Letw be a nonnegative and locally integrable function on ℝn. These functions are called weights, and, for measurable set E, w(E) denotes ∫Ew(x)dx. Let Lpw(ℝn) denote a space of p-integrable functions on ℝn with respect to the measure wdx.


### 2.1. Muckenhoupt Weights

Let us recall the definition of Muckenhoupt weights, compare [10].Weightw∈𝒜p,  1<p<∞ if(2.1)Ap(w):=sup⁡Q⊂Rn1|Q|p∫Qw(x)dx(∫Qw1-p′(x)dx)p-1<∞,
and w∈𝒜1 if(2.2)A1(w):=sup⁡Q⊂Rnw(Q)|Q|‖w-1‖L∞(Q)<∞,
where supremum is taken over all cubes Q∈ℝn.As an example we can take(2.3)w(x)=|x|α∈Apfor{1<p<∞if-n<α<n(p-1),p=1if-n<α≤0
or weights with logarithmic part(2.4)v(x)=|x|αlog⁡-β(2+|x|).
Then(2.5)v∈A1if{β∈Rand-n<α<0,β≥0andα=0,v∈Ap,1<p<∞if-n<α<n(p-1),β∈R.


### 2.2. Local Muckenhoupt Weights

Definition 2.1 (see [5]).
We define the class of weight𝒜ploc(1<p<∞) to consist of all nonnegative locally integrable functions w defined on ℝn for which
(2.6)Aploc⁡(w):=sup⁡|Q|≤11|Q|p∫Qw(x)dx(∫Qw1-p′(x)dx)p-1<∞
and 𝒜1loc(2.7)A1loc⁡(w):=sup⁡|Q|≤1w(Q)|Q|‖w-1‖L∞(Q)<∞.It follows directly from definitions that𝒜p⊂𝒜ploc and Aploc(w)≤Ap(w) for any w∈Ap,  1≤p<∞.Definition 2.2.
We say thatw∈𝒜∞loc if for any α∈(0,1)(2.8)sup⁡|Q|≤1(sup⁡F⊂Q,|F|≥α|Q|w(Q)w(F))<∞,
where F is taken over all measurable sets in ℝn.Remark 2.3.
Any Muckenhoupt weight of the class𝒜p belongs to the class 𝒜ploc. But local Muckenhoupt weights cover also so-called admissible weights and locally regular weights, compare [5, 6, 11].


### 2.3. Properties of ClassesAploc

We would like to mention some important properties of classes𝒜ploc.Lemma 2.4 (see [5]).
Let1<p1<p2<∞. Then 𝒜p1loc⊂𝒜p2loc⊂A∞loc.
Conversely, ifw∈𝒜∞loc, then w∈𝒜ploc for some p<∞.The last lemma implies that𝒜∞loc=⋃p≥1𝒜ploc. In consequence we can define for w∈𝒜∞loc a positive number(2.9)rw=inf⁡{1≤p<∞:w∈Aploc⁡}.
Next lemma shows us an important relation between 𝒜p and 𝒜ploc weights.Lemma 2.5 (see [5]).
Let1≤p<∞, w∈𝒜ploc and Q be a unit cube, that is, |Q|=1. Then there exists a w¯∈𝒜p, such that w¯=w on Q and
(2.10)Ap(w¯)≤cAploc⁡(w),
where constant c is independent of Q.Definition 2.6.
Letf be locally integrable. Operator
(2.11)Mloc⁡f(x)=sup⁡Q∋x1|Q|∫Q|f(y)|dy,
where supremum is taken over all cubes in ℝn for which |Q|≤1, is called local maximal function.The Fefferman-Stein maximal inequality holds for the operatorMloc and local Muckenhoupt weights.Lemma 2.7 (see [5]).
Let1<p<∞,  1<q≤∞, and w∈𝒜ploc. Then for any sequence of measurable functions {fj}, we have
(2.12)‖{Mloc⁡fj}∣Lpw(lq)‖≤c‖{fj}∣Lpw(lq)‖.Lemma 2.8 (see [5]).
Letw∈𝒜ploc and 1<p<∞. Then
(2.13)w(tQ)≤exp⁡(cwt)w(Q)t≥1,|Q|=1.It follows from the above lemma that classes𝒜ploc are independent of the upper bound for the cube size used in their definition; that is, for any C>0, we could have replaced |Q|≤1 by |Q|≤C in Definition 2.1.


## 2.1. Muckenhoupt Weights

Let us recall the definition of Muckenhoupt weights, compare [10].Weightw∈𝒜p,  1<p<∞ if(2.1)Ap(w):=sup⁡Q⊂Rn1|Q|p∫Qw(x)dx(∫Qw1-p′(x)dx)p-1<∞,
and w∈𝒜1 if(2.2)A1(w):=sup⁡Q⊂Rnw(Q)|Q|‖w-1‖L∞(Q)<∞,
where supremum is taken over all cubes Q∈ℝn.As an example we can take(2.3)w(x)=|x|α∈Apfor{1<p<∞if-n<α<n(p-1),p=1if-n<α≤0
or weights with logarithmic part(2.4)v(x)=|x|αlog⁡-β(2+|x|).
Then(2.5)v∈A1if{β∈Rand-n<α<0,β≥0andα=0,v∈Ap,1<p<∞if-n<α<n(p-1),β∈R.


## 2.2. Local Muckenhoupt Weights

Definition 2.1 (see [5]).
We define the class of weight𝒜ploc(1<p<∞) to consist of all nonnegative locally integrable functions w defined on ℝn for which
(2.6)Aploc⁡(w):=sup⁡|Q|≤11|Q|p∫Qw(x)dx(∫Qw1-p′(x)dx)p-1<∞
and 𝒜1loc(2.7)A1loc⁡(w):=sup⁡|Q|≤1w(Q)|Q|‖w-1‖L∞(Q)<∞.It follows directly from definitions that𝒜p⊂𝒜ploc and Aploc(w)≤Ap(w) for any w∈Ap,  1≤p<∞.Definition 2.2.
We say thatw∈𝒜∞loc if for any α∈(0,1)(2.8)sup⁡|Q|≤1(sup⁡F⊂Q,|F|≥α|Q|w(Q)w(F))<∞,
where F is taken over all measurable sets in ℝn.Remark 2.3.
Any Muckenhoupt weight of the class𝒜p belongs to the class 𝒜ploc. But local Muckenhoupt weights cover also so-called admissible weights and locally regular weights, compare [5, 6, 11].


## 2.3. Properties of ClassesAploc

We would like to mention some important properties of classes𝒜ploc.Lemma 2.4 (see [5]).
Let1<p1<p2<∞. Then 𝒜p1loc⊂𝒜p2loc⊂A∞loc.
Conversely, ifw∈𝒜∞loc, then w∈𝒜ploc for some p<∞.The last lemma implies that𝒜∞loc=⋃p≥1𝒜ploc. In consequence we can define for w∈𝒜∞loc a positive number(2.9)rw=inf⁡{1≤p<∞:w∈Aploc⁡}.
Next lemma shows us an important relation between 𝒜p and 𝒜ploc weights.Lemma 2.5 (see [5]).
Let1≤p<∞, w∈𝒜ploc and Q be a unit cube, that is, |Q|=1. Then there exists a w¯∈𝒜p, such that w¯=w on Q and
(2.10)Ap(w¯)≤cAploc⁡(w),
where constant c is independent of Q.Definition 2.6.
Letf be locally integrable. Operator
(2.11)Mloc⁡f(x)=sup⁡Q∋x1|Q|∫Q|f(y)|dy,
where supremum is taken over all cubes in ℝn for which |Q|≤1, is called local maximal function.The Fefferman-Stein maximal inequality holds for the operatorMloc and local Muckenhoupt weights.Lemma 2.7 (see [5]).
Let1<p<∞,  1<q≤∞, and w∈𝒜ploc. Then for any sequence of measurable functions {fj}, we have
(2.12)‖{Mloc⁡fj}∣Lpw(lq)‖≤c‖{fj}∣Lpw(lq)‖.Lemma 2.8 (see [5]).
Letw∈𝒜ploc and 1<p<∞. Then
(2.13)w(tQ)≤exp⁡(cwt)w(Q)t≥1,|Q|=1.It follows from the above lemma that classes𝒜ploc are independent of the upper bound for the cube size used in their definition; that is, for any C>0, we could have replaced |Q|≤1 by |Q|≤C in Definition 2.1.


## 3. Weighted Function Spaces

Following Rychkov we define Triebel-Lizorkin spaces with local Muckenhoupt weights, compare [5]. Because the class of tempered distributions 𝒮′ is too narrow for this purpose we introduce a class 𝒮e′, which is a topological dual to the following space: (3.1)Se:={ψ∈C∞(Rn):qN(ψ)<∞∀N∈N},
where the seminorms qN are given by(3.2)qN(ψ):=sup⁡α∈N0,|α|≤N(sup⁡x∈RneN|x||Dαψ(x)|).
We can identify the class 𝒮e′ with the set of these distributions f∈𝒟′ for which the estimate(3.3)|〈f,ψ〉|≤Asup⁡{|Dαψ(x)|exp⁡(N|x|):x∈Rn,|α|≤N}∀ψ∈C0∞(Rn),
is valid with some constants A,N depend on f. Such a distribution f can be extended to a continuous functional on 𝒮e.We take a functionφ0∈𝒟 such that ∫ℝnφ0(x)dx≠0 and ∫ℝnxβφ0(x)dx=0 for some  β∈ℕ,0<|β|≤B. We put  φ(x)=φ0(x)-2-nφ0(x/2) and φj(x)=2(j-1)nφ(2j-1x) for  j=1,2,…. Then ∫ℝnφj(x)xβdx=0  if |β|≤B. We will write B=-1 if no vanishing moment holds.Definition 3.1 (see [5]).
Let0<p<∞,  0<q≤∞, s∈ℝ, and w∈𝒜∞loc. Let function φ0∈𝒟(ℝn) satisfies
(3.4)∫Rnφ0(x)dx≠0,∫Rnxβφ0(x)dx=0,0<|β|<B,
where B≥[s]. We define the weighted Triebel-Lizorkin space Fpqs,w(ℝn) to be the set of all f∈Se′ for which the following quasinorm:
(3.5)‖f∣Fpqs,w(Rn)‖φ0=‖(∑j=0∞2jsq|φj*f|q)1/q∣Lpw(Rn)‖
is finite.Remark 3.2.
The definition of the above spaces is independent of choice of the functionφ0, up to the equivalence of quasinorms. The spaces are quasi-Banach, Banach spaces if p≥1 and q≥1.


## 4. Characterization by Wavelets

We are going to deal with Daubechies wavelets onℝn. Let ψF∈Ck(ℝ) be a Daubechies scaling function and ψM∈Ck(ℝ) a Daubechies wavelet with ∫ℝψ(x)xvdx=0, k∈ℕ, v∈ℕ0, and v<k. We extend these wavelets from ℝ to ℝn by the usual tensor product procedure(4.1)ΨjmG=2jn/2∏r=1nψGr(2jxr-mr),
where j∈ℕ0, m∈ℤn, G=(G1,…,Gn)∈Gj, and G0={F,M}n and for j>0   Gj={F,M}n*, where * indicates that at least one Gr must be an M. The family(4.2){ΨjmG:j∈N0,m∈Zn,G∈Gj}
is an orthonormal basis in L2(ℝn), compare [12].Definition 4.1.
Lets∈ℝ, 0<p<∞, 0<q≤∞, and w∈𝒜∞loc. Then fpqs,w is the collection of all sequences
(4.3)λ={λjm∈C:j∈N0,m∈Zn}
such that
(4.4)‖λ∣fpqs,w‖=‖(∑j,m,G2jsq|λjmGχjm(p)|q)1/q∣Lpw‖<∞.Forw∈𝒜∞loc let us define(4.5)σp(w)=n(rwmin⁡(p,rw)-1)+(rw-1)n,σq=nmin⁡(1,q)-n,σpq(w)=max⁡(σp(w),σq).The following theorem was obtained in [9].Theorem 4.2.
Let0<p<∞, 0<q≤∞s∈ℝ,   and w∈𝒜∞loc. For wavelets defined in (4.1), we take
(4.6)k≥max⁡(0,[s]+1,[nrwp-np-s]+1,[σpq(w)-s]).
Let f∈𝒮e′(ℝn). Then f∈Fpqs,w(ℝn) if and only if it can be represented as
(4.7)f=∑j,G,mλjmG2-jn/2ΨjmG,
where λ∈fpqs,w and the series converges in 𝒮e′(ℝn). This representation is unique with
(4.8)λjmG=2jn/2〈f,ΨjmG〉,I:f↦{2jn/2〈f,ΨjmG〉}
is a linear isomorphism of Fpqs,w(ℝn) onto fpqs,w.
If0<p,q<∞, then the system {ΨjmG}j,m,G is an unconditional basis in Fpqs,w(ℝn).


## 5.Lp Spaces with Local Muckenhoupt Weights

In this section we prove our main result. We use the fact that wavelet projection operators satisfy condition (5.2) below. Furthermore we rely on the wavelet characterizations according to Theorem 4.2. We follow the main idea of Aimar et al. [4]. On the other side, we have the wavelet characterization theorem stated in [9].Lemma 5.1.
Letφ be a continuous function absolutely bounded by an L1 radial decreasing function such that ∑k∈ℤnφ(x-k)≠0 for all x∈ℝn. Then F(x,y)=∑k∈ℤnφ(x-k)φ(y-k)¯ satisfies
(5.1){(x,y)∈R2n:|x-y|<l}⊂{(x,y)∈R2n:F(x,y)>δ},
for some positive real numbers ℓ and δ.A proof of the above lemma can be found in [4]. Following [4] we can find that lemma applies to P0(x,y)=∑k∈ℤnΨGF(x-k)ΨGF(y-k) with GF=(F,…,F), where ΨGF is a Daubechies scaling function. Now for a family {Pj(x,y)}j≥0={2jnP0(2jx,2jy)}j≥0, we obtain that it satisfies conditions(5.2){(x,y)∈R2n:|x-y|<lj}⊂{(x,y)∈R2n:Pj(x,y)>Clj+1-n}
for every j≥0 and a positive constant C>0, where {ℓj}j≥0 is a decreasing sequence of positive real numbers and ℓj→0 as j→∞.Theorem 5.2.
Let1<p<∞ and μ be a positive Borel measure on ℝn finite on compact sets. Let k≥max([n-n/p]+1,n(p-1)). There exists an unconditional Daubechies wavelet basis in Lp(ℝn,dμ) with smoothness k if and only if dμ=w(x)dx with w∈𝒜ploc.Proof.
Letw∈𝒜ploc. From Theorem 4.2, we have an unconditional basis in Fpqs,w(ℝn). In [5] Rychkov shows a Littlewood-Paley characterization of spaces with local Muckenhoupt weights, and it means that  Fp,20,w(ℝn)=Lpw(ℝn). Hence we have an unconditional basis in Lpw(ℝn).
On the other side, Let{ΨjmG:j∈ℕ0,m∈ℤn,G∈Gj} be a Daubechies wavelet system, which is an unconditional basis in Lp(ℝn,dμ). So every f∈Lp(ℝn,dμ) has the representation
(5.3)f(x)=∑j,k,G〈f,ΨjkG〉ΨjkG(x).
Operators
(5.4)P̃0f=∑k〈f,Ψ0,kGF〉Ψ0,kGF,P̃mf=∑0≤j<m,k,G〈f,ΨjkG〉ΨjkG,m>0
are uniformly bounded on Lp(ℝn,dμ). We can write P̃mf(x)=∫ℝnP̃m(x,y)f(y)dy, where P̃0(x,y)=∑kΨ0,kGF(x)Ψ0,kGF(y) and P̃m(x,y)=∑0≤j<m,k,GΨjkG(x)ΨjkG(y) if m>0, because wavelets have compact supports, and we deal with locally finite sums. Hence kernels P̃m(x,y) are bounded.
On the other hand, by the properties of the multiresolution analysis, the kernelP̃m(x,y) coincides with Pm(x,y)=2mnP0(2mx,2my) if it is regarded as a kernel of projection in L2. Again, because wavelets and the scaling function are compactly supported, we deal with locally finite sums and get Pm(x,y)=P̃m(x,y) almost everywhere. So the last equality is valid also for Lp spaces, and the kernels Pm(x,y) are bounded.
We are going to show thatμ is absolutely continuous. Let E be a set such that |E|=0. For every ɛ>0, there exists an open set F such that E⊂F and μ(F∖E)<ɛ. Set F can be decomposed into a countable union of disjoint and dyadic cubes Qi. From Lemma 5.1, we get that {Pj(x,y)}j≥0 satisfies the same conditions as weakly positive family. Let {ℓj}j≥0 be a sequence connected with that family. Without, being generality, we assume that ℓ0>max(d(Qi)), where d(Q) denote a diameter of Q and maximum is taken over all cubes Qi from decomposition of F. For fixed i let j0≥0 be the integer such that ℓj0+1≤d(Qi)<ℓj0. If x,y∈Qi, we get |x-y|<ℓj0 and Pj0(x,y)>Cℓj0+1-n. So for every x∈Qi, we have
(5.5)|Pj0(χQi∖E)(x)|=|∫Qi∖EPj0(x,y)dy|>Clj0+1-n|Qi∖E|.
Therefore, |Pj0(χQi∖E)(x)|>cn, for some constant depending only on C and n. From weak type inequality for operators Pj, we get
(5.6)μ(Qi)≤μ({x:|Pj0(χQi∖E)(x)|>cn})≤Ccn-pμ(Qi∖E).
Summing over i, we have
(5.7)μ(F)=∑iμ(Qi)≤Ccn-p∑iμ(Qi∖E)=Ccn-pμ(F∖E)<Ccn-pɛ
for every ɛ>0. Hence μ(E)=0. From Radon-Nikodym Theorem, we get that there exists locally integrable function w such that dμ=w(x)dx.
Now we can show thatw∈𝒜ploc. We pick a sequence {ℓj}j≥0. Let Q⊂ℝn be a cube with |Q|≤ℓ0. We can find m0≥0 with ℓm0+1≤d(Q)<ℓm0. Inequalities
(5.8)|Pm0(σɛχQ)(x)|=|∫QPm0(x,y)σɛ(y)dy|>Clm0+1-n∫Qσɛ≥cn|Q|-1∫Qσɛ≡λ,
where σɛ=(w+ɛ)-1/(p-1),ɛ>0, holds for every x∈Q. Since operators Pm are of weak type (p,p), we get
(5.9)w(Q)≤w({x:|Pm0(σɛχQ)(x)|>λ})≤Ccn-p|Q|p(∫Qσɛ)-p∫Qσɛpw.
Multiplying both sides by (∫Qσɛ)p(∫Qσɛpw)-1 and choosing ɛ close to zero, we get
(5.10)w(Q)(∫Qw-1/(p-1))p(∫Qw-1/(p-1))-1≤C|Q|p
for every Q, |Q|<ℓ0. From Lemma  1.4 in [5], we know that classes 𝒜ploc are independent of the upper bound for the cube size in their definition. So we get a condition for Aploc.Following [5], we can state square-function characterization. Let us define(5.11)S(f)(x)=(∑j|φj*f(x)|2)1/2,
where φ0∈𝒟 have nonzero integral and φ=φ0-2-nφ0(·/2) and φj(x)=2jnφ(2jx), j>0.Corollary 5.3.
Let1<p<∞ and w∈𝒜∞loc. The following equivalence holds:
(5.12)‖S(f)∣Lpw‖~‖f∣Lpw‖
if and only if w∈𝒜ploc.Proof.
Letw∈𝒜ploc. From [5] we have that Fp,20,w(ℝn)=Lpw(ℝn) with norm equivalence
(5.13)∫Rn(∑j∈Z|φj*f(x)|2)p/2w(x)dx=∫Rn|f(x)|pw(x)dx.
Conversely, if we assume that w∈𝒜∞loc, then from Theorem 4.2 we get that in Fp,20,w(ℝn) there exists an unconditional basis. Hence also in Lpw(ℝn), we have an unconditional basis. So from Theorem 5.2, we obtain that w∈𝒜ploc.Remark 5.4.
It is known that above statements are not true for general Muckenhoupt weights. Taking for example weight,(5.14)w(x)={|x|αfor|x|≤1,|x|βfor|x|>1,
for α,β>-n, For α<(p1-1)n we have w∈𝒜p1loc and rw=max(0,α)/n+1; for α,β<(p2-1)n we have w∈𝒜p2 and rw=max(0,α,β)/n+1. Taking β big enough, we get that w is in 𝒜ploc∩𝒜∞, but not in 𝒜p.Forφ0∈C0∞(ℝn) with ∫φ0(x)dx≠0 and f∈Se′, we introduce the “vertical” maximal function(5.15)φ0+f(x)=sup⁡j∈N|(φ0)j*f(x)|.
The following corollary follows from Theorem  2.25 in [5] and Corollary 5.3.Corollary 5.5.
Let1<p<∞ and w∈𝒜∞loc. The following equivalence holds:
(5.16)‖φ0+f∣Lpw‖~‖f∣Lpw‖
if and only if w∈𝒜ploc.Please note that it follows from the last corollary that if1<p<∞ and w∈𝒜∞loc, then the weighted local Hardy space(5.17)hpw={f∈Se′:‖φ0+f‖<∞}
coincides with Lpw if and only if w∈𝒜ploc.



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