The paper deals with unconditional wavelet bases in weighted 𝐿𝑝 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.

1. Introduction

Wavelet systems in weighted 𝐿𝑝 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 weights ğ’œâˆžloc 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 ğ’œâˆžloc, 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 𝐿𝑝(𝑑𝜇) if and only if 𝑑𝜇=𝑤𝑑𝑥 with 𝑤∈𝒜𝑝loc.

2. Classes of Weights

Let 𝑤 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 𝐿𝑤𝑝(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].

Weight 𝑤∈𝒜𝑝, 1<𝑝<∞ if𝐴𝑝(𝑤)∶=sup𝑄⊂ℝ𝑛1||𝑄||𝑝𝑄𝑤(𝑥)𝑑𝑥𝑄𝑤1−𝑝′(𝑥)𝑑𝑥𝑝−1<∞,(2.1) and 𝑤∈𝒜1 if𝐴1(𝑤)∶=sup𝑄⊂ℝ𝑛𝑤(𝑄)||𝑄||‖‖𝑤−1â€–â€–ğ¿âˆž(𝑄)<∞,(2.2) where supremum is taken over all cubes 𝑄∈ℝ𝑛.

As an example we can take𝑤(𝑥)=|𝑥|𝛼∈𝒜𝑝for1<𝑝<∞if−𝑛<𝛼<𝑛(𝑝−1),𝑝=1if−𝑛<𝛼≤0(2.3) or weights with logarithmic part𝑣(𝑥)=|𝑥|𝛼log−𝛽(2+|𝑥|).(2.4) Then𝑣∈𝒜1if𝛽∈ℝand−𝑛<𝛼<0,𝛽≥0and𝛼=0,𝑣∈𝒜𝑝,1<𝑝<∞if−𝑛<𝛼<𝑛(𝑝−1),𝛽∈ℝ.(2.5)

2.2. Local Muckenhoupt Weights

Definition 2.1 (see [5]). We define the class of weight 𝒜𝑝loc(1<𝑝<∞) to consist of all nonnegative locally integrable functions 𝑤 defined on ℝ𝑛 for which 𝐴𝑝loc(𝑤)∶=sup||𝑄||≤11||𝑄||𝑝𝑄𝑤(𝑥)𝑑𝑥𝑄𝑤1−𝑝′(𝑥)𝑑𝑥𝑝−1<∞(2.6) and 𝒜1loc𝐴1loc(𝑤)∶=sup||𝑄||≤1𝑤(𝑄)||𝑄||‖‖𝑤−1â€–â€–ğ¿âˆž(𝑄)<∞.(2.7)

It follows directly from definitions that 𝒜𝑝⊂𝒜𝑝loc and 𝐴𝑝loc(𝑤)≤𝐴𝑝(𝑤) for any 𝑤∈𝐴𝑝, 1≤𝑝<∞.

Definition 2.2. We say that ğ‘¤âˆˆğ’œâˆžloc if for any 𝛼∈(0,1)sup||𝑄||≤1sup||𝐹||||𝑄||𝐹⊂𝑄,≥𝛼𝑤(𝑄)𝑤(𝐹)<∞,(2.8) where 𝐹 is taken over all measurable sets in ℝ𝑛.

Remark 2.3. Any Muckenhoupt weight of the class 𝒜𝑝 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 Classes 𝐴𝑝loc

We would like to mention some important properties of classes 𝒜𝑝loc.

Lemma 2.4 (see [5]). Let 1<𝑝1<𝑝2<∞. Then 𝒜𝑝loc1⊂𝒜𝑝loc2âŠ‚ğ´âˆžloc.
Conversely, if ğ‘¤âˆˆğ’œâˆžloc, then 𝑤∈𝒜𝑝loc for some 𝑝<∞.

The last lemma implies that ğ’œâˆžloc=⋃𝑝≥1𝒜𝑝loc. In consequence we can define for ğ‘¤âˆˆğ’œâˆžloc a positive number𝑟𝑤=inf1≤𝑝<âˆžâˆ¶ğ‘¤âˆˆğ’œğ‘loc.(2.9) Next lemma shows us an important relation between 𝒜𝑝 and 𝒜𝑝loc weights.

Lemma 2.5 (see [5]). Let 1≤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 𝐴𝑝𝑤≤𝑐𝐴𝑝loc(𝑤),(2.10) where constant 𝑐 is independent of 𝑄.

Definition 2.6. Let 𝑓 be locally integrable. Operator 𝑀loc𝑓(𝑥)=sup𝑄∋𝑥1||𝑄||𝑄||||𝑓(𝑦)𝑑𝑦,(2.11) 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 operator 𝑀loc and local Muckenhoupt weights.

Lemma 2.7 (see [5]). Let 1<𝑝<∞, 1<ğ‘žâ‰¤âˆž, and 𝑤∈𝒜𝑝loc. Then for any sequence of measurable functions {𝑓𝑗}, we have ‖‖𝑀locğ‘“ğ‘—î€¾âˆ£ğ¿ğ‘¤ğ‘î€·ğ‘™ğ‘žî€¸â€–â€–â€–â€–î€½ğ‘“â‰¤ğ‘ğ‘—î€¾âˆ£ğ¿ğ‘¤ğ‘î€·ğ‘™ğ‘žî€¸â€–â€–.(2.12)

Lemma 2.8 (see [5]). Let 𝑤∈𝒜𝑝loc and 1<𝑝<∞. Then 𝑤𝑐(𝑡𝑄)≤exp𝑤𝑡𝑤||𝑄||(𝑄)𝑡≥1,=1.(2.13)

It follows from the above lemma that classes 𝒜𝑝loc 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 ğ’®î…ž is too narrow for this purpose we introduce a class ğ’®î…žğ‘’, which is a topological dual to the following space: 𝒮𝑒∶=ğœ“âˆˆğ¶âˆž(ℝ𝑛)âˆ¶ğ‘žğ‘î€¾,(𝜓)<âˆžâˆ€ğ‘âˆˆâ„•(3.1) where the seminorms ğ‘žğ‘ are given byğ‘žğ‘(𝜓)∶=sup𝛼∈ℕ0,|𝛼|≤𝑁sup𝑥∈ℝ𝑛𝑒𝑁|𝑥|||𝐷𝛼||.𝜓(𝑥)(3.2) We can identify the class ğ’®î…žğ‘’ with the set of these distributions ğ‘“âˆˆğ’Ÿî…ž for which the estimate||||||𝐷⟨𝑓,𝜓⟩≤𝐴sup𝛼𝜓||(𝑥)exp(𝑁|𝑥|)∶𝑥∈ℝ𝑛,|𝛼|â‰¤ğ‘âˆ€ğœ“âˆˆğ¶âˆž0(ℝ𝑛),(3.3) 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 function 𝜑0∈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]). Let 0<𝑝<∞, 0<ğ‘žâ‰¤âˆž, 𝑠∈ℝ, and ğ‘¤âˆˆğ’œâˆžloc. Let function 𝜑0∈𝒟(ℝ𝑛) satisfies ℝ𝑛𝜑0(𝑥)𝑑𝑥≠0,ℝ𝑛𝑥𝛽𝜑0||𝛽||(𝑥)𝑑𝑥=0,0<<𝐵,(3.4) where 𝐵≥[𝑠]. We define the weighted Triebel-Lizorkin space 𝐹𝑠,ğ‘¤ğ‘ğ‘ž(ℝ𝑛) to be the set of all ğ‘“âˆˆğ‘†î…žğ‘’ for which the following quasinorm: ‖‖𝑓∣𝐹𝑠,ğ‘¤ğ‘ğ‘ž(ℝ𝑛)‖‖𝜑0=â€–â€–â€–â€–îƒ©âˆžî“ğ‘—=02ğ‘—ğ‘ ğ‘ž||𝜑𝑗||âˆ—ğ‘“ğ‘žîƒª1/ğ‘žâˆ£ğ¿ğ‘¤ğ‘(ℝ𝑛)‖‖‖‖(3.5) is finite.

Remark 3.2. The definition of the above spaces is independent of choice of the function 𝜑0, up to the equivalence of quasinorms. The spaces are quasi-Banach, Banach spaces if 𝑝≥1 and ğ‘žâ‰¥1.

4. Characterization by Wavelets

We are going to deal with Daubechies wavelets on ℝ𝑛. 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 procedureΨ𝐺𝑗𝑚=2𝑛𝑗𝑛/2𝑟=1𝜓𝐺𝑟2𝑗𝑥𝑟−𝑚𝑟,(4.1) 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Ψ𝐺𝑗𝑚∶𝑗∈ℕ0,𝑚∈ℤ𝑛,𝐺∈𝐺𝑗(4.2) is an orthonormal basis in 𝐿2(ℝ𝑛), compare [12].

Definition 4.1. Let 𝑠∈ℝ, 0<𝑝<∞, 0<ğ‘žâ‰¤âˆž, and ğ‘¤âˆˆğ’œâˆžloc. Then 𝑓𝑠,ğ‘¤ğ‘ğ‘ž is the collection of all sequences 𝜆𝜆=𝑗𝑚∈ℂ∶𝑗∈ℕ0,𝑚∈ℤ𝑛(4.3) such that ‖‖𝜆∣𝑓𝑠,ğ‘¤ğ‘ğ‘žâ€–â€–=‖‖‖‖𝑗,𝑚,𝐺2ğ‘—ğ‘ ğ‘ž||𝜆𝐺𝑗𝑚𝜒(𝑝)𝑗𝑚||ğ‘žîƒª1/ğ‘žâˆ£ğ¿ğ‘¤ğ‘â€–â€–â€–â€–<∞.(4.4)

For ğ‘¤âˆˆğ’œâˆžloc let us defineğœŽğ‘îƒ©ğ‘Ÿ(𝑤)=𝑛𝑤min𝑝,𝑟𝑤+𝑟−1ğ‘¤î€¸ğœŽâˆ’1𝑛,ğ‘ž=ğ‘›ğœŽmin(1,ğ‘ž)−𝑛,ğ‘ğ‘ž(î€·ğœŽğ‘¤)=max𝑝(𝑤),ğœŽğ‘žî€¸.(4.5)

The following theorem was obtained in [9].

Theorem 4.2. Let 0<𝑝<∞, 0<ğ‘žâ‰¤âˆžğ‘ âˆˆâ„,   and ğ‘¤âˆˆğ’œâˆžloc. For wavelets defined in (4.1), we take [𝑠]𝑘≥max0,+1,ğ‘›ğ‘Ÿğ‘¤ğ‘âˆ’ğ‘›ğ‘î‚¹î€ºğœŽâˆ’ğ‘ +1,ğ‘ğ‘žî€»î‚¶.(𝑤)−𝑠(4.6) Let ğ‘“âˆˆğ’®î…žğ‘’(ℝ𝑛). Then 𝑓∈𝐹𝑠,ğ‘¤ğ‘ğ‘ž(ℝ𝑛) if and only if it can be represented as 𝑓=𝑗,𝐺,𝑚𝜆𝐺𝑗𝑚2−𝑗𝑛/2Ψ𝐺𝑗𝑚,(4.7) where 𝜆∈𝑓𝑠,ğ‘¤ğ‘ğ‘ž and the series converges in ğ’®î…žğ‘’(ℝ𝑛). This representation is unique with 𝜆𝐺𝑗𝑚=2𝑗𝑛/2𝑓,Ψ𝐺𝑗𝑚,2𝐼∶𝑓↦𝑗𝑛/2𝑓,Ψ𝐺𝑗𝑚(4.8) is a linear isomorphism of 𝐹𝑠,ğ‘¤ğ‘ğ‘ž(ℝ𝑛) onto 𝑓𝑠,ğ‘¤ğ‘ğ‘ž.
If 0<𝑝,ğ‘ž<∞, then the system {Ψ𝐺𝑗𝑚}𝑗,𝑚,𝐺 is an unconditional basis in 𝐹𝑠,ğ‘¤ğ‘ğ‘ž(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. Let 𝜑 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 (𝑥,𝑦)∈ℝ2𝑛∶||||⊂𝑥−𝑦<ℓ(𝑥,𝑦)∈ℝ2𝑛,∶𝐹(𝑥,𝑦)>𝛿(5.1) 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(𝑥,𝑦)∈ℝ2𝑛∶||||𝑥−𝑦<ℓ𝑗⊂(𝑥,y)∈ℝ2𝑛∶𝑃𝑗(𝑥,𝑦)>𝐶ℓ−𝑛𝑗+1(5.2) 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 ğ‘—â†’âˆž.

Theorem 5.2. Let 1<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. Let 𝑤∈𝒜𝑝loc. From Theorem 4.2, we have an unconditional basis in 𝐹𝑠,ğ‘¤ğ‘ğ‘ž(ℝ𝑛). In [5] Rychkov shows a Littlewood-Paley characterization of spaces with local Muckenhoupt weights, and it means that 𝐹0,𝑤𝑝,2(ℝ𝑛)=𝐿𝑤𝑝(ℝ𝑛). Hence we have an unconditional basis in 𝐿𝑤𝑝(ℝ𝑛).
On the other side, Let {Ψ𝐺𝑗𝑚∶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 𝑓(𝑥)=𝑗,𝑘,𝐺𝑓,Ψ𝐺𝑗𝑘Ψ𝐺𝑗𝑘(𝑥).(5.3) Operators 𝑃0𝑓=𝑘𝑓,Ψ𝐺𝐹0,𝑘Ψ𝐺𝐹0,𝑘,𝑃𝑚𝑓=0≤𝑗<𝑚,𝑘,𝐺𝑓,Ψ𝐺𝑗𝑘Ψ𝐺𝑗𝑘,𝑚>0(5.4) are uniformly bounded on 𝐿𝑝(ℝ𝑛,𝑑𝜇). We can write 𝑃𝑚∫𝑓(𝑥)=ℝ𝑛𝑃𝑚(𝑥,𝑦)𝑓(𝑦)𝑑𝑦, where 𝑃0∑(𝑥,𝑦)=𝑘Ψ𝐺𝐹0,𝑘(𝑥)Ψ𝐺𝐹0,𝑘(𝑦) and 𝑃𝑚∑(𝑥,𝑦)=0≤𝑗<𝑚,𝑘,𝐺Ψ𝐺𝑗𝑘(𝑥)Ψ𝐺𝑗𝑘(𝑦) if 𝑚>0, because wavelets have compact supports, and we deal with locally finite sums. Hence kernels 𝑃𝑚(𝑥,𝑦) are bounded.
On the other hand, by the properties of the multiresolution analysis, the kernel 𝑃𝑚(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𝑃(𝑥,𝑦)=𝑚(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 that 𝜇 is absolutely continuous. Let 𝐸 be a set such that |𝐸|=0. For every 𝜀>0, there exists an open set 𝐹 such that 𝐸⊂𝐹 and 𝜇(𝐹⧵𝐸)<𝜀. 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>ğ‘šğ‘Žğ‘¥(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)>𝐶ℓ𝑗−𝑛0+1. So for every 𝑥∈𝑄𝑖, we have ||𝑃𝑗0𝜒𝑄𝑖⧵𝐸||=||||(𝑥)𝑄𝑖⧵𝐸𝑃𝑗0||||(𝑥,𝑦)𝑑𝑦>𝐶ℓ𝑗−𝑛0+1||𝑄𝑖||.⧵𝐸(5.5) Therefore, |𝑃𝑗0(𝜒𝑄𝑖⧵𝐸)(𝑥)|>𝑐𝑛, for some constant depending only on 𝐶 and 𝑛. From weak type inequality for operators 𝑃𝑗, we get 𝜇𝑄𝑖||𝑃≤𝜇𝑥∶𝑗0𝜒𝑄𝑖⧵𝐸||(𝑥)>𝑐𝑛≤𝐶𝑐𝑛−𝑝𝜇𝑄𝑖.⧵𝐸(5.6) Summing over 𝑖, we have 𝜇(𝐹)=𝑖𝜇𝑄𝑖≤𝐶𝑐𝑛−𝑝𝑖𝜇𝑄𝑖⧵𝐸=𝐶𝑐𝑛−𝑝𝜇(𝐹⧵𝐸)<𝐶𝑐𝑛−𝑝𝜀(5.7) for every 𝜀>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 that 𝑤∈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 ||𝑃𝑚0î€·ğœŽğœ€ğœ’ğ‘„î€¸||=||||(𝑥)𝑄𝑃𝑚0(𝑥,𝑦)ğœŽğœ€||||(𝑦)𝑑𝑦>𝐶ℓ𝑚−𝑛0+1î€œğ‘„ğœŽğœ€â‰¥ğ‘ğ‘›||𝑄||−1î€œğ‘„ğœŽğœ€â‰¡ğœ†,(5.8) where ğœŽğœ€=(𝑤+𝜀)−1/(𝑝−1),𝜀>0, holds for every 𝑥∈𝑄. Since operators 𝑃𝑚 are of weak type (𝑝,𝑝), we get ||𝑃𝑤(𝑄)≤𝑤𝑥∶𝑚0î€·ğœŽğœ€ğœ’ğ‘„î€¸||(𝑥)>𝜆≤𝐶𝑐𝑛−𝑝||𝑄||ğ‘î‚µî€œğ‘„ğœŽğœ€î‚¶âˆ’ğ‘î€œğ‘„ğœŽğ‘ğœ€ğ‘¤.(5.9) Multiplying both sides by (âˆ«ğ‘„ğœŽğœ€)𝑝(âˆ«ğ‘„ğœŽğ‘ğœ€ğ‘¤)−1 and choosing 𝜀 close to zero, we get 𝑤(𝑄)𝑄𝑤−1/(𝑝−1)𝑝𝑄𝑤−1/(𝑝−1)−1||𝑄||≤𝐶𝑝(5.10) 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𝑆(𝑓)(𝑥)=𝑗||𝜑𝑗||∗𝑓(𝑥)21/2,(5.11) where 𝜑0∈𝒟 have nonzero integral and 𝜑=𝜑0−2−𝑛𝜑0(⋅/2) and 𝜑𝑗(𝑥)=2𝑗𝑛𝜑(2𝑗𝑥), 𝑗>0.

Corollary 5.3. Let 1<𝑝<∞ and ğ‘¤âˆˆğ’œâˆžloc. The following equivalence holds: ‖‖𝑆(𝑓)∣𝐿𝑤𝑝‖‖∼‖‖𝑓∣𝐿𝑤𝑝‖‖(5.12) if and only if 𝑤∈𝒜𝑝loc.

Proof. Let 𝑤∈𝒜𝑝loc. From [5] we have that 𝐹0,𝑤𝑝,2(ℝ𝑛)=𝐿𝑤𝑝(ℝ𝑛) with norm equivalence ℝ𝑛𝑗∈ℤ||𝜑𝑗||∗𝑓(𝑥)2𝑝/2𝑤(𝑥)𝑑𝑥=ℝ𝑛||||𝑓(𝑥)𝑝𝑤(𝑥)𝑑𝑥.(5.13) Conversely, if we assume that ğ‘¤âˆˆğ’œâˆžloc, then from Theorem 4.2 we get that in 𝐹0,𝑤𝑝,2(ℝ𝑛) there exists an unconditional basis. Hence also in 𝐿𝑤𝑝(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, 𝑤(𝑥)=|𝑥|𝛼for||𝑥|≤1,𝑥|𝛽for|𝑥|>1,(5.14) for 𝛼,𝛽>−𝑛, For 𝛼<(𝑝1−1)𝑛 we have 𝑤∈𝒜𝑝loc1 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 𝒜𝑝locâˆ©ğ’œâˆž, but not in 𝒜𝑝.

For 𝜑0âˆˆğ¶âˆž0(ℝ𝑛) with ∫𝜑0(𝑥)𝑑𝑥≠0 and 𝑓∈𝑆′𝑒, we introduce the “vertical” maximal function𝜑+0𝑓(𝑥)=sup𝑗∈ℕ|||𝜑0𝑗|||.∗𝑓(𝑥)(5.15) The following corollary follows from Theorem  2.25 in [5] and Corollary 5.3.

Corollary 5.5. Let 1<𝑝<∞ and ğ‘¤âˆˆğ’œâˆžloc. The following equivalence holds: ‖‖𝜑+0𝑓∣𝐿𝑤𝑝‖‖∼‖‖𝑓∣𝐿𝑤𝑝‖‖(5.16) if and only if 𝑤∈𝒜𝑝loc.

Please note that it follows from the last corollary that if 1<𝑝<∞ and ğ‘¤âˆˆğ’œâˆžloc, then the weighted local Hardy spaceâ„Žğ‘¤ğ‘=î€½ğ‘“âˆˆğ‘†î…žğ‘’âˆ¶â€–â€–ğœ‘+0𝑓‖‖<∞(5.17) coincides with 𝐿𝑤𝑝 if and only if 𝑤∈𝒜𝑝loc.