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Journal of Function Spaces and Applications
Volumeย 2012, Article IDย 328310, 10 pages
Research Article

A Remark on Wavelet Bases in Weighted ๐ฟ๐‘ Spaces

Faculty of Mathematics and Computer Science, A. Mickiewicz University, 61-614 Poznan, Poland

Received 9 May 2010; Accepted 20 May 2010

Academic Editor: Hansย Triebel

Copyright ยฉ 2012 Agnieszka Wojciechowska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper deals with unconditional wavelet bases in weighted ๐ฟ๐‘ 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 ๐ฟ๐‘ spaces in dependence on ๐‘.

1. Introduction

Wavelet systems in weighted ๐ฟ๐‘ spaces were investigated by several authors. Lemariรฉ-Rieusset considered one-dimensional case, compare [1]. He proved that the homogeneous wavelet system of Daubechies type is an unconditional basis in ๐ฟ๐‘(โ„๐‘›,๐‘‘๐œ‡), 1<๐‘<โˆž, if and only if ๐‘‘๐œ‡=๐‘ค๐‘‘๐‘ฅ, where ๐‘ค is a weight belonging to the Muckenhoupt class ๐’œ๐‘. 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 Garcรญa-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 ๐ฟ๐‘(๐‘‘๐œ‡) if and only if ๐‘‘๐œ‡=๐‘ค๐‘‘๐‘ฅ with ๐‘คโˆˆ๐’œ๐‘.

The aim of the paper is to prove the counterpart of Lemariรฉ-Rieusset result for multidimensional inhomogeneous wavelet systems. To formulate the necessary and sufficient condition we use the class of local Muckenhoupt weights ๐’œโˆžloc 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 ๐’œโˆžloc, 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 in ๐ฟ๐‘(๐‘‘๐œ‡) if and only if ๐‘‘๐œ‡=๐‘ค๐‘‘๐‘ฅ with ๐‘คโˆˆ๐’œ๐‘loc.

2. Classes of Weights

Let ๐‘ค be a nonnegative and locally integrable function on โ„๐‘›. These functions are called weights, and, for measurable set ๐ธ, ๐‘ค(๐ธ) denotes โˆซ๐ธ๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ. Let ๐ฟ๐‘ค๐‘(โ„๐‘›) denote a space of ๐‘-integrable functions on โ„๐‘› with respect to the measure ๐‘ค๐‘‘๐‘ฅ.

2.1. Muckenhoupt Weights

Let us recall the definition of Muckenhoupt weights, compare [10].

Weight ๐‘คโˆˆ๐’œ๐‘, 1<๐‘<โˆž if๐ด๐‘(๐‘ค)โˆถ=sup๐‘„โŠ‚โ„๐‘›1||๐‘„||๐‘๎€œ๐‘„๎‚ต๎€œ๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ๐‘„๐‘ค1โˆ’๐‘โ€ฒ๎‚ถ(๐‘ฅ)๐‘‘๐‘ฅ๐‘โˆ’1<โˆž,(2.1) and ๐‘คโˆˆ๐’œ1 if๐ด1(๐‘ค)โˆถ=sup๐‘„โŠ‚โ„๐‘›๐‘ค(๐‘„)||๐‘„||โ€–โ€–๐‘คโˆ’1โ€–โ€–๐ฟโˆž(๐‘„)<โˆž,(2.2) where supremum is taken over all cubes ๐‘„โˆˆโ„๐‘›.

As an example we can take๐‘ค(๐‘ฅ)=|๐‘ฅ|๐›ผโˆˆ๐’œ๐‘for๎‚ป1<๐‘<โˆžifโˆ’๐‘›<๐›ผ<๐‘›(๐‘โˆ’1),๐‘=1ifโˆ’๐‘›<๐›ผโ‰ค0(2.3) or weights with logarithmic part๐‘ฃ(๐‘ฅ)=|๐‘ฅ|๐›ผlogโˆ’๐›ฝ(2+|๐‘ฅ|).(2.4) Then๐‘ฃโˆˆ๐’œ1if๎‚ป๐›ฝโˆˆโ„andโˆ’๐‘›<๐›ผ<0,๐›ฝโ‰ฅ0and๐›ผ=0,๐‘ฃโˆˆ๐’œ๐‘,1<๐‘<โˆžifโˆ’๐‘›<๐›ผ<๐‘›(๐‘โˆ’1),๐›ฝโˆˆโ„.(2.5)

2.2. Local Muckenhoupt Weights

Definition 2.1 (see [5]). We define the class of weight ๐’œ๐‘loc(1<๐‘<โˆž) to consist of all nonnegative locally integrable functions ๐‘ค defined on โ„๐‘› for which ๐ด๐‘loc(๐‘ค)โˆถ=sup||๐‘„||โ‰ค11||๐‘„||๐‘๎€œ๐‘„๎‚ต๎€œ๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ๐‘„๐‘ค1โˆ’๐‘โ€ฒ๎‚ถ(๐‘ฅ)๐‘‘๐‘ฅ๐‘โˆ’1<โˆž(2.6) and ๐’œ1loc๐ด1loc(๐‘ค)โˆถ=sup||๐‘„||โ‰ค1๐‘ค(๐‘„)||๐‘„||โ€–โ€–๐‘คโˆ’1โ€–โ€–๐ฟโˆž(๐‘„)<โˆž.(2.7)

It follows directly from definitions that ๐’œ๐‘โŠ‚๐’œ๐‘loc and ๐ด๐‘loc(๐‘ค)โ‰ค๐ด๐‘(๐‘ค) for any ๐‘คโˆˆ๐ด๐‘, 1โ‰ค๐‘<โˆž.

Definition 2.2. We say that ๐‘คโˆˆ๐’œโˆžloc if for any ๐›ผโˆˆ(0,1)sup||๐‘„||โ‰ค1๎ƒฉsup||๐น||||๐‘„||๐นโŠ‚๐‘„,โ‰ฅ๐›ผ๐‘ค(๐‘„)๎ƒช๐‘ค(๐น)<โˆž,(2.8) where ๐น is taken over all measurable sets in โ„๐‘›.

Remark 2.3. Any Muckenhoupt weight of the class ๐’œ๐‘ belongs to the class ๐’œ๐‘loc. But local Muckenhoupt weights cover also so-called admissible weights and locally regular weights, compare [5, 6, 11].

2.3. Properties of Classes ๐ด๐‘loc

We would like to mention some important properties of classes ๐’œ๐‘loc.

Lemma 2.4 (see [5]). Let 1<๐‘1<๐‘2<โˆž. Then ๐’œ๐‘loc1โŠ‚๐’œ๐‘loc2โŠ‚๐ดโˆžloc.
Conversely, if ๐‘คโˆˆ๐’œโˆžloc, then ๐‘คโˆˆ๐’œ๐‘loc for some ๐‘<โˆž.

The last lemma implies that ๐’œโˆžloc=โ‹ƒ๐‘โ‰ฅ1๐’œ๐‘loc. In consequence we can define for ๐‘คโˆˆ๐’œโˆžloc a positive number๐‘Ÿ๐‘ค๎€ฝ=inf1โ‰ค๐‘<โˆžโˆถ๐‘คโˆˆ๐’œ๐‘loc๎€พ.(2.9) Next lemma shows us an important relation between ๐’œ๐‘ and ๐’œ๐‘loc weights.

Lemma 2.5 (see [5]). Let 1โ‰ค๐‘<โˆž, ๐‘คโˆˆ๐’œ๐‘loc and ๐‘„ be a unit cube, that is, |๐‘„|=1. Then there exists a ๐‘คโˆˆ๐’œ๐‘, such that ๐‘ค=๐‘ค on ๐‘„ and ๐ด๐‘๎€ท๐‘ค๎€ธโ‰ค๐‘๐ด๐‘loc(๐‘ค),(2.10) where constant ๐‘ is independent of ๐‘„.

Definition 2.6. Let ๐‘“ be locally integrable. Operator ๐‘€loc๐‘“(๐‘ฅ)=sup๐‘„โˆ‹๐‘ฅ1||๐‘„||๎€œ๐‘„||||๐‘“(๐‘ฆ)๐‘‘๐‘ฆ,(2.11) where supremum is taken over all cubes in โ„๐‘› for which |๐‘„|โ‰ค1, is called local maximal function.

The Fefferman-Stein maximal inequality holds for the operator ๐‘€loc and local Muckenhoupt weights.

Lemma 2.7 (see [5]). Let 1<๐‘<โˆž, 1<๐‘žโ‰คโˆž, and ๐‘คโˆˆ๐’œ๐‘loc. Then for any sequence of measurable functions {๐‘“๐‘—}, we have โ€–โ€–๎€ฝ๐‘€loc๐‘“๐‘—๎€พโˆฃ๐ฟ๐‘ค๐‘๎€ท๐‘™๐‘ž๎€ธโ€–โ€–โ€–โ€–๎€ฝ๐‘“โ‰ค๐‘๐‘—๎€พโˆฃ๐ฟ๐‘ค๐‘๎€ท๐‘™๐‘ž๎€ธโ€–โ€–.(2.12)

Lemma 2.8 (see [5]). Let ๐‘คโˆˆ๐’œ๐‘loc and 1<๐‘<โˆž. Then ๐‘ค๎€ท๐‘(๐‘ก๐‘„)โ‰คexp๐‘ค๐‘ก๎€ธ๐‘ค||๐‘„||(๐‘„)๐‘กโ‰ฅ1,=1.(2.13)

It follows from the above lemma that classes ๐’œ๐‘loc are independent of the upper bound for the cube size used in their definition; that is, for any ๐ถ>0, we could have replaced |๐‘„|โ‰ค1 by |๐‘„|โ‰ค๐ถ 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 ๐’ฎ๎…ž is too narrow for this purpose we introduce a class ๐’ฎ๎…ž๐‘’, which is a topological dual to the following space: ๐’ฎ๐‘’๎€ฝโˆถ=๐œ“โˆˆ๐ถโˆž(โ„๐‘›)โˆถ๐‘ž๐‘๎€พ,(๐œ“)<โˆžโˆ€๐‘โˆˆโ„•(3.1) where the seminorms ๐‘ž๐‘ are given by๐‘ž๐‘(๐œ“)โˆถ=sup๐›ผโˆˆโ„•0,|๐›ผ|โ‰ค๐‘๎‚ตsup๐‘ฅโˆˆโ„๐‘›๐‘’๐‘|๐‘ฅ|||๐ท๐›ผ||๎‚ถ.๐œ“(๐‘ฅ)(3.2) We can identify the class ๐’ฎ๎…ž๐‘’ with the set of these distributions ๐‘“โˆˆ๐’Ÿ๎…ž for which the estimate||||๎€ฝ||๐ทโŸจ๐‘“,๐œ“โŸฉโ‰ค๐ดsup๐›ผ๐œ“||(๐‘ฅ)exp(๐‘|๐‘ฅ|)โˆถ๐‘ฅโˆˆโ„๐‘›,๎€พ|๐›ผ|โ‰ค๐‘โˆ€๐œ“โˆˆ๐ถโˆž0(โ„๐‘›),(3.3) is valid with some constants ๐ด,๐‘ depend on ๐‘“. Such a distribution ๐‘“ can be extended to a continuous functional on ๐’ฎ๐‘’.

We take a function ๐œ‘0โˆˆ๐’Ÿ such that โˆซโ„๐‘›๐œ‘0(๐‘ฅ)๐‘‘๐‘ฅโ‰ 0 and โˆซโ„๐‘›๐‘ฅ๐›ฝ๐œ‘0(๐‘ฅ)๐‘‘๐‘ฅ=0 for some ๐›ฝโˆˆโ„•,0<|๐›ฝ|โ‰ค๐ต. We put ๐œ‘(๐‘ฅ)=๐œ‘0(๐‘ฅ)โˆ’2โˆ’๐‘›๐œ‘0(๐‘ฅ/2) and ๐œ‘๐‘—(๐‘ฅ)=2(๐‘—โˆ’1)๐‘›๐œ‘(2๐‘—โˆ’1๐‘ฅ) for ๐‘—=1,2,โ€ฆ. Then โˆซโ„๐‘›๐œ‘๐‘—(๐‘ฅ)๐‘ฅ๐›ฝ๐‘‘๐‘ฅ=0 if |๐›ฝ|โ‰ค๐ต. We will write ๐ต=โˆ’1 if no vanishing moment holds.

Definition 3.1 (see [5]). Let 0<๐‘<โˆž, 0<๐‘žโ‰คโˆž, ๐‘ โˆˆโ„, and ๐‘คโˆˆ๐’œโˆžloc. Let function ๐œ‘0โˆˆ๐’Ÿ(โ„๐‘›) satisfies ๎€œโ„๐‘›๐œ‘0(๎€œ๐‘ฅ)๐‘‘๐‘ฅโ‰ 0,โ„๐‘›๐‘ฅ๐›ฝ๐œ‘0||๐›ฝ||(๐‘ฅ)๐‘‘๐‘ฅ=0,0<<๐ต,(3.4) where ๐ตโ‰ฅ[๐‘ ]. We define the weighted Triebel-Lizorkin space ๐น๐‘ ,๐‘ค๐‘๐‘ž(โ„๐‘›) to be the set of all ๐‘“โˆˆ๐‘†๎…ž๐‘’ for which the following quasinorm: โ€–โ€–๐‘“โˆฃ๐น๐‘ ,๐‘ค๐‘๐‘ž(โ„๐‘›)โ€–โ€–๐œ‘0=โ€–โ€–โ€–โ€–๎ƒฉโˆž๎“๐‘—=02๐‘—๐‘ ๐‘ž||๐œ‘๐‘—||โˆ—๐‘“๐‘ž๎ƒช1/๐‘žโˆฃ๐ฟ๐‘ค๐‘(โ„๐‘›)โ€–โ€–โ€–โ€–(3.5) is finite.

Remark 3.2. The definition of the above spaces is independent of choice of the function ๐œ‘0, up to the equivalence of quasinorms. The spaces are quasi-Banach, Banach spaces if ๐‘โ‰ฅ1 and ๐‘žโ‰ฅ1.

4. Characterization by Wavelets

We are going to deal with Daubechies wavelets on โ„๐‘›. Let ๐œ“๐นโˆˆ๐ถ๐‘˜(โ„) be a Daubechies scaling function and ๐œ“๐‘€โˆˆ๐ถ๐‘˜(โ„) a Daubechies wavelet with โˆซโ„๐œ“(๐‘ฅ)๐‘ฅ๐‘ฃ๐‘‘๐‘ฅ=0, ๐‘˜โˆˆโ„•, ๐‘ฃโˆˆโ„•0, and ๐‘ฃ<๐‘˜. We extend these wavelets from โ„ to โ„๐‘› by the usual tensor product procedureฮจ๐บ๐‘—๐‘š=2๐‘›๐‘—๐‘›/2๎‘๐‘Ÿ=1๐œ“๐บ๐‘Ÿ๎€ท2๐‘—๐‘ฅ๐‘Ÿโˆ’๐‘š๐‘Ÿ๎€ธ,(4.1) where ๐‘—โˆˆโ„•0, ๐‘šโˆˆโ„ค๐‘›, ๐บ=(๐บ1,โ€ฆ,๐บ๐‘›)โˆˆ๐บ๐‘—, and ๐บ0={๐น,๐‘€}๐‘› and for ๐‘—>0 โ€‰ ๐บ๐‘—={๐น,๐‘€}๐‘›โˆ—, where โˆ— indicates that at least one ๐บ๐‘Ÿ must be an ๐‘€. The family๎€ฝฮจ๐บ๐‘—๐‘šโˆถ๐‘—โˆˆโ„•0,๐‘šโˆˆโ„ค๐‘›,๐บโˆˆ๐บ๐‘—๎€พ(4.2) is an orthonormal basis in ๐ฟ2(โ„๐‘›), compare [12].

Definition 4.1. Let ๐‘ โˆˆโ„, 0<๐‘<โˆž, 0<๐‘žโ‰คโˆž, and ๐‘คโˆˆ๐’œโˆžloc. Then ๐‘“๐‘ ,๐‘ค๐‘๐‘ž is the collection of all sequences ๎€ฝ๐œ†๐œ†=๐‘—๐‘šโˆˆโ„‚โˆถ๐‘—โˆˆโ„•0,๐‘šโˆˆโ„ค๐‘›๎€พ(4.3) such that โ€–โ€–๐œ†โˆฃ๐‘“๐‘ ,๐‘ค๐‘๐‘žโ€–โ€–=โ€–โ€–โ€–โ€–๎ƒฉ๎“๐‘—,๐‘š,๐บ2๐‘—๐‘ ๐‘ž||๐œ†๐บ๐‘—๐‘š๐œ’(๐‘)๐‘—๐‘š||๐‘ž๎ƒช1/๐‘žโˆฃ๐ฟ๐‘ค๐‘โ€–โ€–โ€–โ€–<โˆž.(4.4)

For ๐‘คโˆˆ๐’œโˆžloc let us define๐œŽ๐‘๎ƒฉ๐‘Ÿ(๐‘ค)=๐‘›๐‘ค๎€ทmin๐‘,๐‘Ÿ๐‘ค๎€ธ๎ƒช+๎€ท๐‘Ÿโˆ’1๐‘ค๎€ธ๐œŽโˆ’1๐‘›,๐‘ž=๐‘›๐œŽmin(1,๐‘ž)โˆ’๐‘›,๐‘๐‘ž(๎€ท๐œŽ๐‘ค)=max๐‘(๐‘ค),๐œŽ๐‘ž๎€ธ.(4.5)

The following theorem was obtained in [9].

Theorem 4.2. Let 0<๐‘<โˆž, 0<๐‘žโ‰คโˆž๐‘ โˆˆโ„, โ€‰ and ๐‘คโˆˆ๐’œโˆžloc. For wavelets defined in (4.1), we take ๎‚ต[๐‘ ]๎‚ธ๐‘˜โ‰ฅmax0,+1,๐‘›๐‘Ÿ๐‘ค๐‘โˆ’๐‘›๐‘๎‚น๎€บ๐œŽโˆ’๐‘ +1,๐‘๐‘ž๎€ป๎‚ถ.(๐‘ค)โˆ’๐‘ (4.6) Let ๐‘“โˆˆ๐’ฎ๎…ž๐‘’(โ„๐‘›). Then ๐‘“โˆˆ๐น๐‘ ,๐‘ค๐‘๐‘ž(โ„๐‘›) if and only if it can be represented as ๎“๐‘“=๐‘—,๐บ,๐‘š๐œ†๐บ๐‘—๐‘š2โˆ’๐‘—๐‘›/2ฮจ๐บ๐‘—๐‘š,(4.7) where ๐œ†โˆˆ๐‘“๐‘ ,๐‘ค๐‘๐‘ž and the series converges in ๐’ฎ๎…ž๐‘’(โ„๐‘›). This representation is unique with ๐œ†๐บ๐‘—๐‘š=2๐‘—๐‘›/2๎ซ๐‘“,ฮจ๐บ๐‘—๐‘š๎ฌ,๎€ฝ2๐ผโˆถ๐‘“โ†ฆ๐‘—๐‘›/2๎ซ๐‘“,ฮจ๐บ๐‘—๐‘š๎ฌ๎€พ(4.8) is a linear isomorphism of ๐น๐‘ ,๐‘ค๐‘๐‘ž(โ„๐‘›) onto ๐‘“๐‘ ,๐‘ค๐‘๐‘ž.
If 0<๐‘,๐‘ž<โˆž, then the system {ฮจ๐บ๐‘—๐‘š}๐‘—,๐‘š,๐บ is an unconditional basis in ๐น๐‘ ,๐‘ค๐‘๐‘ž(โ„๐‘›).

5. ๐ฟ๐‘ 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. Let ๐œ‘ be a continuous function absolutely bounded by an ๐ฟ1 radial decreasing function such that โˆ‘๐‘˜โˆˆโ„ค๐‘›๐œ‘(๐‘ฅโˆ’๐‘˜)โ‰ 0 for all ๐‘ฅโˆˆโ„๐‘›. Then โˆ‘๐น(๐‘ฅ,๐‘ฆ)=๐‘˜โˆˆโ„ค๐‘›๐œ‘(๐‘ฅโˆ’๐‘˜)๐œ‘(๐‘ฆโˆ’๐‘˜) satisfies ๎€ฝ(๐‘ฅ,๐‘ฆ)โˆˆโ„2๐‘›โˆถ||||๎€พโŠ‚๎€ฝ๐‘ฅโˆ’๐‘ฆ<โ„“(๐‘ฅ,๐‘ฆ)โˆˆโ„2๐‘›๎€พ,โˆถ๐น(๐‘ฅ,๐‘ฆ)>๐›ฟ(5.1) for some positive real numbers โ„“ and ๐›ฟ.

A proof of the above lemma can be found in [4]. Following [4] we can find that lemma applies to ๐‘ƒ0โˆ‘(๐‘ฅ,๐‘ฆ)=๐‘˜โˆˆโ„ค๐‘›ฮจ๐บ๐น(๐‘ฅโˆ’๐‘˜)ฮจ๐บ๐น(๐‘ฆโˆ’๐‘˜) with ๐บ๐น=(๐น,โ€ฆ,๐น), where ฮจ๐บ๐น is a Daubechies scaling function. Now for a family {๐‘ƒ๐‘—(๐‘ฅ,๐‘ฆ)}๐‘—โ‰ฅ0={2๐‘—๐‘›๐‘ƒ0(2๐‘—๐‘ฅ,2๐‘—๐‘ฆ)}๐‘—โ‰ฅ0, we obtain that it satisfies conditions๎€ฝ(๐‘ฅ,๐‘ฆ)โˆˆโ„2๐‘›โˆถ||||๐‘ฅโˆ’๐‘ฆ<โ„“๐‘—๎€พโŠ‚๎‚†(๐‘ฅ,y)โˆˆโ„2๐‘›โˆถ๐‘ƒ๐‘—(๐‘ฅ,๐‘ฆ)>๐ถโ„“โˆ’๐‘›๐‘—+1๎‚‡(5.2) for every ๐‘—โ‰ฅ0 and a positive constant ๐ถ>0, where {โ„“๐‘—}๐‘—โ‰ฅ0 is a decreasing sequence of positive real numbers and โ„“๐‘—โ†’0 as ๐‘—โ†’โˆž.

Theorem 5.2. Let 1<๐‘<โˆž and ๐œ‡ be a positive Borel measure on โ„๐‘› finite on compact sets. Let ๐‘˜โ‰ฅmax([๐‘›โˆ’๐‘›/๐‘]+1,๐‘›(๐‘โˆ’1)). There exists an unconditional Daubechies wavelet basis in ๐ฟ๐‘(โ„๐‘›,๐‘‘๐œ‡) with smoothness ๐‘˜ if and only if ๐‘‘๐œ‡=๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ with ๐‘คโˆˆ๐’œ๐‘loc.

Proof. Let ๐‘คโˆˆ๐’œ๐‘loc. From Theorem 4.2, we have an unconditional basis in ๐น๐‘ ,๐‘ค๐‘๐‘ž(โ„๐‘›). In [5] Rychkov shows a Littlewood-Paley characterization of spaces with local Muckenhoupt weights, and it means that ๐น0,๐‘ค๐‘,2(โ„๐‘›)=๐ฟ๐‘ค๐‘(โ„๐‘›). Hence we have an unconditional basis in ๐ฟ๐‘ค๐‘(โ„๐‘›).
On the other side, Let {ฮจ๐บ๐‘—๐‘šโˆถ๐‘—โˆˆโ„•0,๐‘šโˆˆโ„ค๐‘›,๐บโˆˆ๐บ๐‘—} be a Daubechies wavelet system, which is an unconditional basis in ๐ฟ๐‘(โ„๐‘›,๐‘‘๐œ‡). So every ๐‘“โˆˆ๐ฟ๐‘(โ„๐‘›,๐‘‘๐œ‡) has the representation ๎“๐‘“(๐‘ฅ)=๐‘—,๐‘˜,๐บ๎‚ฌ๐‘“,ฮจ๐บ๐‘—๐‘˜๎‚ญฮจ๐บ๐‘—๐‘˜(๐‘ฅ).(5.3) Operators ๎‚๐‘ƒ0๎“๐‘“=๐‘˜๎‚ฌ๐‘“,ฮจ๐บ๐น0,๐‘˜๎‚ญฮจ๐บ๐น0,๐‘˜,๎‚๐‘ƒ๐‘š๎“๐‘“=0โ‰ค๐‘—<๐‘š,๐‘˜,๐บ๎‚ฌ๐‘“,ฮจ๐บ๐‘—๐‘˜๎‚ญฮจ๐บ๐‘—๐‘˜,๐‘š>0(5.4) are uniformly bounded on ๐ฟ๐‘(โ„๐‘›,๐‘‘๐œ‡). We can write ๎‚๐‘ƒ๐‘šโˆซ๐‘“(๐‘ฅ)=โ„๐‘›๎‚๐‘ƒ๐‘š(๐‘ฅ,๐‘ฆ)๐‘“(๐‘ฆ)๐‘‘๐‘ฆ, where ๎‚๐‘ƒ0โˆ‘(๐‘ฅ,๐‘ฆ)=๐‘˜ฮจ๐บ๐น0,๐‘˜(๐‘ฅ)ฮจ๐บ๐น0,๐‘˜(๐‘ฆ) and ๎‚๐‘ƒ๐‘šโˆ‘(๐‘ฅ,๐‘ฆ)=0โ‰ค๐‘—<๐‘š,๐‘˜,๐บฮจ๐บ๐‘—๐‘˜(๐‘ฅ)ฮจ๐บ๐‘—๐‘˜(๐‘ฆ) if ๐‘š>0, because wavelets have compact supports, and we deal with locally finite sums. Hence kernels ๎‚๐‘ƒ๐‘š(๐‘ฅ,๐‘ฆ) are bounded.
On the other hand, by the properties of the multiresolution analysis, the kernel ๎‚๐‘ƒ๐‘š(๐‘ฅ,๐‘ฆ) coincides with ๐‘ƒ๐‘š(๐‘ฅ,๐‘ฆ)=2๐‘š๐‘›๐‘ƒ0(2๐‘š๐‘ฅ,2๐‘š๐‘ฆ) if it is regarded as a kernel of projection in ๐ฟ2. Again, because wavelets and the scaling function are compactly supported, we deal with locally finite sums and get ๐‘ƒ๐‘š๎‚๐‘ƒ(๐‘ฅ,๐‘ฆ)=๐‘š(๐‘ฅ,๐‘ฆ) almost everywhere. So the last equality is valid also for ๐ฟ๐‘ spaces, and the kernels ๐‘ƒ๐‘š(๐‘ฅ,๐‘ฆ) are bounded.
We are going to show that ๐œ‡ is absolutely continuous. Let ๐ธ be a set such that |๐ธ|=0. For every ๐œ€>0, there exists an open set ๐น such that ๐ธโŠ‚๐น and ๐œ‡(๐นโงต๐ธ)<๐œ€. Set ๐น can be decomposed into a countable union of disjoint and dyadic cubes ๐‘„๐‘–. From Lemma 5.1, we get that {๐‘ƒ๐‘—(๐‘ฅ,๐‘ฆ)}๐‘—โ‰ฅ0 satisfies the same conditions as weakly positive family. Let {โ„“๐‘—}๐‘—โ‰ฅ0 be a sequence connected with that family. Without, being generality, we assume that โ„“0>๐‘š๐‘Ž๐‘ฅ(๐‘‘(๐‘„๐‘–)), where ๐‘‘(๐‘„) denote a diameter of ๐‘„ and maximum is taken over all cubes ๐‘„๐‘– from decomposition of ๐น. For fixed ๐‘– let ๐‘—0โ‰ฅ0 be the integer such that โ„“๐‘—0+1โ‰ค๐‘‘(๐‘„๐‘–)<โ„“๐‘—0. If ๐‘ฅ,๐‘ฆโˆˆ๐‘„๐‘–, we get |๐‘ฅโˆ’๐‘ฆ|<โ„“๐‘—0 and ๐‘ƒ๐‘—0(๐‘ฅ,๐‘ฆ)>๐ถโ„“๐‘—โˆ’๐‘›0+1. So for every ๐‘ฅโˆˆ๐‘„๐‘–, we have ||๐‘ƒ๐‘—0๎€ท๐œ’๐‘„๐‘–โงต๐ธ๎€ธ||=||||๎€œ(๐‘ฅ)๐‘„๐‘–โงต๐ธ๐‘ƒ๐‘—0||||(๐‘ฅ,๐‘ฆ)๐‘‘๐‘ฆ>๐ถโ„“๐‘—โˆ’๐‘›0+1||๐‘„๐‘–||.โงต๐ธ(5.5) Therefore, |๐‘ƒ๐‘—0(๐œ’๐‘„๐‘–โงต๐ธ)(๐‘ฅ)|>๐‘๐‘›, for some constant depending only on ๐ถ and ๐‘›. From weak type inequality for operators ๐‘ƒ๐‘—, we get ๐œ‡๎€ท๐‘„๐‘–๎€ธ||๐‘ƒโ‰ค๐œ‡๎€ท๎€ฝ๐‘ฅโˆถ๐‘—0๎€ท๐œ’๐‘„๐‘–โงต๐ธ๎€ธ||(๐‘ฅ)>๐‘๐‘›๎€พ๎€ธโ‰ค๐ถ๐‘๐‘›โˆ’๐‘๐œ‡๎€ท๐‘„๐‘–๎€ธ.โงต๐ธ(5.6) Summing over ๐‘–, we have ๎“๐œ‡(๐น)=๐‘–๐œ‡๎€ท๐‘„๐‘–๎€ธโ‰ค๐ถ๐‘๐‘›โˆ’๐‘๎“๐‘–๐œ‡๎€ท๐‘„๐‘–๎€ธโงต๐ธ=๐ถ๐‘๐‘›โˆ’๐‘๐œ‡(๐นโงต๐ธ)<๐ถ๐‘๐‘›โˆ’๐‘๐œ€(5.7) for every ๐œ€>0. Hence ๐œ‡(๐ธ)=0. From Radon-Nikodym Theorem, we get that there exists locally integrable function ๐‘ค such that ๐‘‘๐œ‡=๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ.
Now we can show that ๐‘คโˆˆ๐’œ๐‘loc. We pick a sequence {โ„“๐‘—}๐‘—โ‰ฅ0. Let ๐‘„โŠ‚โ„๐‘› be a cube with |๐‘„|โ‰คโ„“0. We can find ๐‘š0โ‰ฅ0 with โ„“๐‘š0+1โ‰ค๐‘‘(๐‘„)<โ„“๐‘š0. Inequalities ||๐‘ƒ๐‘š0๎€ท๐œŽ๐œ€๐œ’๐‘„๎€ธ||=||||๎€œ(๐‘ฅ)๐‘„๐‘ƒ๐‘š0(๐‘ฅ,๐‘ฆ)๐œŽ๐œ€||||(๐‘ฆ)๐‘‘๐‘ฆ>๐ถโ„“๐‘šโˆ’๐‘›0+1๎€œ๐‘„๐œŽ๐œ€โ‰ฅ๐‘๐‘›||๐‘„||โˆ’1๎€œ๐‘„๐œŽ๐œ€โ‰ก๐œ†,(5.8) where ๐œŽ๐œ€=(๐‘ค+๐œ€)โˆ’1/(๐‘โˆ’1),๐œ€>0, holds for every ๐‘ฅโˆˆ๐‘„. Since operators ๐‘ƒ๐‘š are of weak type (๐‘,๐‘), we get ||๐‘ƒ๐‘ค(๐‘„)โ‰ค๐‘ค๎€ท๎€ฝ๐‘ฅโˆถ๐‘š0๎€ท๐œŽ๐œ€๐œ’๐‘„๎€ธ||(๐‘ฅ)>๐œ†๎€พ๎€ธโ‰ค๐ถ๐‘๐‘›โˆ’๐‘||๐‘„||๐‘๎‚ต๎€œ๐‘„๐œŽ๐œ€๎‚ถโˆ’๐‘๎€œ๐‘„๐œŽ๐‘๐œ€๐‘ค.(5.9) Multiplying both sides by (โˆซ๐‘„๐œŽ๐œ€)๐‘(โˆซ๐‘„๐œŽ๐‘๐œ€๐‘ค)โˆ’1 and choosing ๐œ€ close to zero, we get ๎‚ต๎€œ๐‘ค(๐‘„)๐‘„๐‘คโˆ’1/(๐‘โˆ’1)๎‚ถ๐‘๎‚ต๎€œ๐‘„๐‘คโˆ’1/(๐‘โˆ’1)๎‚ถโˆ’1||๐‘„||โ‰ค๐ถ๐‘(5.10) for every ๐‘„, |๐‘„|<โ„“0. From Lemmaโ€‰โ€‰1.4 in [5], we know that classes ๐’œ๐‘loc are independent of the upper bound for the cube size in their definition. So we get a condition for ๐ด๐‘loc.

Following [5], we can state square-function characterization. Let us define๎ƒฉ๎“๐‘†(๐‘“)(๐‘ฅ)=๐‘—||๐œ‘๐‘—||โˆ—๐‘“(๐‘ฅ)2๎ƒช1/2,(5.11) where ๐œ‘0โˆˆ๐’Ÿ have nonzero integral and ๐œ‘=๐œ‘0โˆ’2โˆ’๐‘›๐œ‘0(โ‹…/2) and ๐œ‘๐‘—(๐‘ฅ)=2๐‘—๐‘›๐œ‘(2๐‘—๐‘ฅ), ๐‘—>0.

Corollary 5.3. Let 1<๐‘<โˆž and ๐‘คโˆˆ๐’œโˆžloc. The following equivalence holds: โ€–โ€–๐‘†(๐‘“)โˆฃ๐ฟ๐‘ค๐‘โ€–โ€–โˆผโ€–โ€–๐‘“โˆฃ๐ฟ๐‘ค๐‘โ€–โ€–(5.12) if and only if ๐‘คโˆˆ๐’œ๐‘loc.

Proof. Let ๐‘คโˆˆ๐’œ๐‘loc. From [5] we have that ๐น0,๐‘ค๐‘,2(โ„๐‘›)=๐ฟ๐‘ค๐‘(โ„๐‘›) with norm equivalence ๎€œโ„๐‘›๎ƒฉ๎“๐‘—โˆˆโ„ค||๐œ‘๐‘—||โˆ—๐‘“(๐‘ฅ)2๎ƒช๐‘/2๎€œ๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ=โ„๐‘›||||๐‘“(๐‘ฅ)๐‘๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ.(5.13) Conversely, if we assume that ๐‘คโˆˆ๐’œโˆžloc, then from Theorem 4.2 we get that in ๐น0,๐‘ค๐‘,2(โ„๐‘›) there exists an unconditional basis. Hence also in ๐ฟ๐‘ค๐‘(โ„๐‘›), we have an unconditional basis. So from Theorem 5.2, we obtain that ๐‘คโˆˆ๐’œ๐‘loc.

Remark 5.4. It is known that above statements are not true for general Muckenhoupt weights. Taking for example weight, ๎‚ป๐‘ค(๐‘ฅ)=|๐‘ฅ|๐›ผfor||๐‘ฅ|โ‰ค1,๐‘ฅ|๐›ฝfor|๐‘ฅ|>1,(5.14) for ๐›ผ,๐›ฝ>โˆ’๐‘›, For ๐›ผ<(๐‘1โˆ’1)๐‘› we have ๐‘คโˆˆ๐’œ๐‘loc1 and ๐‘Ÿ๐‘ค=max(0,๐›ผ)/๐‘›+1; for ๐›ผ,๐›ฝ<(๐‘2โˆ’1)๐‘› we have ๐‘คโˆˆ๐’œ๐‘2 and ๐‘Ÿ๐‘ค=max(0,๐›ผ,๐›ฝ)/๐‘›+1. Taking ๐›ฝ big enough, we get that ๐‘ค is in ๐’œ๐‘locโˆฉ๐’œโˆž, but not in ๐’œ๐‘.

For ๐œ‘0โˆˆ๐ถโˆž0(โ„๐‘›) with โˆซ๐œ‘0(๐‘ฅ)๐‘‘๐‘ฅโ‰ 0 and ๐‘“โˆˆ๐‘†โ€ฒ๐‘’, we introduce the โ€œverticalโ€ maximal function๐œ‘+0๐‘“(๐‘ฅ)=sup๐‘—โˆˆโ„•|||๎€ท๐œ‘0๎€ธ๐‘—|||.โˆ—๐‘“(๐‘ฅ)(5.15) The following corollary follows from Theoremโ€‰โ€‰2.25 in [5] and Corollary 5.3.

Corollary 5.5. Let 1<๐‘<โˆž and ๐‘คโˆˆ๐’œโˆžloc. The following equivalence holds: โ€–โ€–๐œ‘+0๐‘“โˆฃ๐ฟ๐‘ค๐‘โ€–โ€–โˆผโ€–โ€–๐‘“โˆฃ๐ฟ๐‘ค๐‘โ€–โ€–(5.16) if and only if ๐‘คโˆˆ๐’œ๐‘loc.

Please note that it follows from the last corollary that if 1<๐‘<โˆž and ๐‘คโˆˆ๐’œโˆžloc, then the weighted local Hardy spaceโ„Ž๐‘ค๐‘=๎€ฝ๐‘“โˆˆ๐‘†๎…ž๐‘’โˆถโ€–โ€–๐œ‘+0๐‘“โ€–โ€–๎€พ<โˆž(5.17) coincides with ๐ฟ๐‘ค๐‘ if and only if ๐‘คโˆˆ๐’œ๐‘loc.


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