A Remark on Wavelet Bases in Weighted 𝐿_𝑝 Spaces

Abstract

The paper deals with unconditional wavelet bases in weighted 𝐿_𝑝 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 𝐿_𝑝 spaces in dependence on 𝑝.

1. Introduction

Wavelet systems in weighted 𝐿_𝑝 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 𝐿_𝑝(ℝ^𝑛,𝑑𝜇), 1<𝑝<∞, if and only if 𝑑𝜇=𝑤𝑑𝑥, where 𝑤 is a weight belonging to the Muckenhoupt class 𝒜_𝑝. 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 𝐿_𝑝(𝑑𝜇) if and only if 𝑑𝜇=𝑤𝑑𝑥 with 𝑤∈𝒜_𝑝.

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 𝒜^∞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 in 𝐿_𝑝(𝑑𝜇) if and only if 𝑑𝜇=𝑤𝑑𝑥 with 𝑤∈𝒜^𝑝loc.

2. Classes of Weights

Let 𝑤 be a nonnegative and locally integrable function on ℝ^𝑛. These functions are called weights, and, for measurable set 𝐸, 𝑤(𝐸) denotes ∫_𝐸𝑤(𝑥)𝑑𝑥. Let 𝐿^𝑤𝑝(ℝ^𝑛) denote a space of 𝑝-integrable functions on ℝ^𝑛 with respect to the measure 𝑤𝑑𝑥.

2.1. Muckenhoupt Weights

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

Weight 𝑤∈𝒜_𝑝, 1<𝑝<∞ if𝐴_𝑝(𝑤)∶=sup_{𝑄⊂ℝ𝑛}1||𝑄||^𝑝_𝑄𝑤(𝑥)𝑑𝑥_𝑄𝑤^1−𝑝′(𝑥)𝑑𝑥^𝑝−1<∞,(2.1) and 𝑤∈𝒜₁ if𝐴₁(𝑤)∶=sup_{𝑄⊂ℝ𝑛}𝑤(𝑄)||𝑄||‖‖𝑤⁻¹‖‖_{𝐿∞(𝑄)}<∞,(2.2) where supremum is taken over all cubes 𝑄∈ℝ^𝑛.

As an example we can take𝑤(𝑥)=|𝑥|^𝛼∈𝒜_𝑝for1<𝑝<∞𝑝=1if−𝑛<𝛼<𝑛(𝑝−1),if−𝑛<𝛼≤0(2.3) or weights with logarithmic part𝑣(𝑥)=|𝑥|^𝛼log^−𝛽(2+|𝑥|).(2.4) Then𝑣∈𝒜₁if𝛽∈ℝ𝛽≥0and−𝑛<𝛼<0,𝑣∈𝒜_𝑝and𝛼=0,,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 𝒜^𝑝loc(1<𝑝<∞) to consist of all nonnegative locally integrable functions 𝑤 defined on ℝ^𝑛 for which 𝐴^𝑝loc(𝑤)∶=sup_||𝑄||≤11||𝑄||^𝑝_𝑄𝑤(𝑥)𝑑𝑥_𝑄𝑤^1−𝑝′(𝑥)𝑑𝑥^𝑝−1<∞(2.6) and 𝒜^1loc𝐴^1loc(𝑤)∶=sup_||𝑄||≤1𝑤(𝑄)||𝑄||‖‖𝑤⁻¹‖‖_{𝐿∞(𝑄)}<∞.(2.7)

It follows directly from definitions that 𝒜_𝑝⊂𝒜^𝑝loc and 𝐴^𝑝loc(𝑤)≤𝐴_𝑝(𝑤) for any 𝑤∈𝐴_𝑝, 1≤𝑝<∞.

Definition 2.2. We say that 𝑤∈𝒜^∞loc if for any 𝛼∈(0,1)sup_||𝑄||≤1sup_{||𝐹||||𝑄||𝐹⊂𝑄,≥𝛼}𝑤(𝑄)𝑤(𝐹)<∞,(2.8) where 𝐹 is taken over all measurable sets in ℝ^𝑛.

Remark 2.3. Any Muckenhoupt weight of the class 𝒜_𝑝 belongs to the class 𝒜^𝑝loc. But local Muckenhoupt weights cover also so-called admissible weights and locally regular weights, compare [5, 6, 11].

2.3. Properties of Classes 𝐴^𝑝loc

We would like to mention some important properties of classes 𝒜^𝑝loc.

Lemma 2.4 (see [5]). Let 1<𝑝₁<𝑝₂<∞. Then 𝒜^𝑝loc₁⊂𝒜^𝑝loc₂⊂𝐴^∞loc.
Conversely, if 𝑤∈𝒜^∞loc, then 𝑤∈𝒜^𝑝loc for some 𝑝<∞.

The last lemma implies that 𝒜^∞loc=⋃_𝑝≥1𝒜^𝑝loc. In consequence we can define for 𝑤∈𝒜^∞loc a positive number𝑟_𝑤=inf1≤𝑝<∞∶𝑤∈𝒜^𝑝loc.(2.9) Next lemma shows us an important relation between 𝒜_𝑝 and 𝒜^𝑝loc weights.

Lemma 2.5 (see [5]). Let 1≤𝑝<∞, 𝑤∈𝒜^𝑝loc and 𝑄 be a unit cube, that is, |𝑄|=1. Then there exists a 𝑤∈𝒜_𝑝, such that 𝑤=𝑤 on 𝑄 and 𝐴_𝑝𝑤≤𝑐𝐴^𝑝loc(𝑤),(2.10) where constant 𝑐 is independent of 𝑄.

Definition 2.6. Let 𝑓 be locally integrable. Operator 𝑀^loc𝑓(𝑥)=sup_𝑄∋𝑥1||𝑄||_𝑄||𝑓(𝑦)||𝑑𝑦,(2.11) where supremum is taken over all cubes in ℝ^𝑛 for which |𝑄|≤1, is called local maximal function.

The Fefferman-Stein maximal inequality holds for the operator 𝑀^loc and local Muckenhoupt weights.

Lemma 2.7 (see [5]). Let 1<𝑝<∞, 1<𝑞≤∞, and 𝑤∈𝒜^𝑝loc. Then for any sequence of measurable functions {𝑓^𝑗}, we have ‖‖𝑀^loc𝑓^𝑗∣𝐿^𝑤𝑝𝑙_𝑞‖‖≤𝑐‖‖𝑓^𝑗∣𝐿^𝑤𝑝𝑙_𝑞‖‖.(2.12)

Lemma 2.8 (see [5]). Let 𝑤∈𝒜^𝑝loc and 1<𝑝<∞. Then 𝑤(𝑡𝑄)≤exp𝑐_𝑤𝑡𝑤(𝑄)𝑡≥1,||𝑄||=1.(2.13)

It follows from the above lemma that classes 𝒜^𝑝loc 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 𝒮^𝑒, which is a topological dual to the following space: 𝒮_𝑒∶=𝜓∈𝐶^∞(ℝ^𝑛)∶𝑞_𝑁(𝜓)<∞∀𝑁∈ℕ,(3.1) where the seminorms 𝑞_𝑁 are given by𝑞_𝑁(𝜓)∶=sup_{𝛼∈ℕ0,|𝛼|≤𝑁}sup_{𝑥∈ℝ𝑛}𝑒^𝑁|𝑥|||𝐷^𝛼𝜓(𝑥)||.(3.2) We can identify the class 𝒮^𝑒 with the set of these distributions 𝑓∈𝒟^ for which the estimate||⟨𝑓,𝜓⟩||≤𝐴sup||𝐷^𝛼𝜓(𝑥)||exp(𝑁|𝑥|)∶𝑥∈ℝ^𝑛,|𝛼|≤𝑁∀𝜓∈𝐶^∞0(ℝ^𝑛),(3.3) is valid with some constants 𝐴,𝑁 depend on 𝑓. Such a distribution 𝑓 can be extended to a continuous functional on 𝒮_𝑒.

We take a function 𝜑₀∈𝒟 such that ∫_ℝ𝑛𝜑₀(𝑥)𝑑𝑥≠0 and ∫_ℝ𝑛𝑥^𝛽𝜑₀(𝑥)𝑑𝑥=0 for some 𝛽∈ℕ,0<|𝛽|≤𝐵. We put 𝜑(𝑥)=𝜑₀(𝑥)−2^−𝑛𝜑₀(𝑥/2) and 𝜑_𝑗(𝑥)=2^{(𝑗−1)𝑛}𝜑(2^𝑗−1𝑥) for 𝑗=1,2,…. Then ∫_ℝ𝑛𝜑_𝑗(𝑥)𝑥^𝛽𝑑𝑥=0 if |𝛽|≤𝐵. We will write 𝐵=−1 if no vanishing moment holds.

Definition 3.1 (see [5]). Let 0<𝑝<∞, 0<𝑞≤∞, 𝑠∈ℝ, and 𝑤∈𝒜^∞loc. Let function 𝜑₀∈𝒟(ℝ^𝑛) satisfies _ℝ𝑛𝜑₀(𝑥)𝑑𝑥≠0,_ℝ𝑛𝑥^𝛽𝜑₀(𝑥)𝑑𝑥=0,0<||𝛽||<𝐵,(3.4) where 𝐵≥[𝑠]. We define the weighted Triebel-Lizorkin space 𝐹^{𝑠,𝑤𝑝𝑞}(ℝ^𝑛) to be the set of all 𝑓∈𝑆^𝑒 for which the following quasinorm: ‖‖𝑓∣𝐹^{𝑠,𝑤𝑝𝑞}(ℝ^𝑛)‖‖_𝜑0=‖‖‖‖^∞_𝑗=02^𝑗𝑠𝑞||𝜑_𝑗∗𝑓||^𝑞^1/𝑞∣𝐿^𝑤𝑝(ℝ^𝑛)‖‖‖‖(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 𝑞≥1.

4. Characterization by Wavelets

We are going to deal with Daubechies wavelets on ℝ^𝑛. Let 𝜓^𝐹∈𝐶^𝑘(ℝ) be a Daubechies scaling function and 𝜓^𝑀∈𝐶^𝑘(ℝ) a Daubechies wavelet with ∫_ℝ𝜓(𝑥)𝑥^𝑣𝑑𝑥=0, 𝑘∈ℕ, 𝑣∈ℕ₀, and 𝑣<𝑘. We extend these wavelets from ℝ to ℝ^𝑛 by the usual tensor product procedureΨ^𝐺𝑗𝑚=2^{𝑛𝑗𝑛/2}_𝑟=1𝜓^𝐺𝑟2^𝑗𝑥_𝑟−𝑚_𝑟,(4.1) where 𝑗∈ℕ₀, 𝑚∈ℤ^𝑛, 𝐺=(𝐺₁,…,𝐺_𝑛)∈𝐺^𝑗, and 𝐺⁰={𝐹,𝑀}^𝑛 and for 𝑗>0   𝐺^𝑗={𝐹,𝑀}^𝑛∗, where ∗ indicates that at least one 𝐺_𝑟 must be an 𝑀. The familyΨ^𝐺𝑗𝑚∶𝑗∈ℕ₀,𝑚∈ℤ^𝑛,𝐺∈𝐺^𝑗(4.2) is an orthonormal basis in 𝐿₂(ℝ^𝑛), compare [12].

Definition 4.1. Let 𝑠∈ℝ, 0<𝑝<∞, 0<𝑞≤∞, and 𝑤∈𝒜^∞loc. Then 𝑓^{𝑠,𝑤𝑝𝑞} is the collection of all sequences 𝜆=𝜆_𝑗𝑚∈ℂ∶𝑗∈ℕ₀,𝑚∈ℤ^𝑛(4.3) such that ‖‖𝜆∣𝑓^{𝑠,𝑤𝑝𝑞}‖‖=‖‖‖‖_{𝑗,𝑚,𝐺}2^𝑗𝑠𝑞||𝜆^𝐺𝑗𝑚𝜒^{(𝑝)𝑗𝑚}||^𝑞^1/𝑞∣𝐿^𝑤𝑝‖‖‖‖<∞.(4.4)

For 𝑤∈𝒜^∞loc let us define𝜎_𝑝(𝑤)=𝑛𝑟^𝑤min𝑝,𝑟_𝑤−1+𝑟^𝑤𝜎−1𝑛,_𝑞=𝑛𝜎min(1,𝑞)−𝑛,_𝑝𝑞(𝑤)=max𝜎_𝑝(𝑤),𝜎_𝑞.(4.5)

The following theorem was obtained in [9].

Theorem 4.2. Let 0<𝑝<∞, 0<𝑞≤∞𝑠∈ℝ,   and 𝑤∈𝒜^∞loc. For wavelets defined in (4.1), we take 𝑘≥max0,[𝑠]+1,𝑛𝑟^𝑤𝑝−𝑛𝑝−𝑠+1,𝜎^𝑝𝑞(𝑤)−𝑠.(4.6) Let 𝑓∈𝒮^𝑒(ℝ^𝑛). Then 𝑓∈𝐹^{𝑠,𝑤𝑝𝑞}(ℝ^𝑛) if and only if it can be represented as 𝑓=_{𝑗,𝐺,𝑚}𝜆^𝐺𝑗𝑚2^{−𝑗𝑛/2}Ψ^𝐺𝑗𝑚,(4.7) where 𝜆∈𝑓^{𝑠,𝑤𝑝𝑞} and the series converges in 𝒮^𝑒(ℝ^𝑛). This representation is unique with 𝐼∶𝜆𝑓^𝐺𝑗𝑚↦=22^𝑗𝑛/2_𝑗𝑛/2𝑓,Ψ^𝐺𝑗𝑚,𝑓,Ψ^𝐺𝑗𝑚(4.8) is a linear isomorphism of 𝐹^{𝑠,𝑤𝑝𝑞}(ℝ^𝑛) onto 𝑓^{𝑠,𝑤𝑝𝑞}.
If 0<𝑝,𝑞<∞, then the system {Ψ^𝐺𝑗𝑚}_{𝑗,𝑚,𝐺} is an unconditional basis in 𝐹^{𝑠,𝑤𝑝𝑞}(ℝ^𝑛).

5. 𝐿_𝑝 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∈ℤ𝑛}𝜑(𝑥−𝑘)≠0 for all 𝑥∈ℝ^𝑛. Then 𝐹(𝑥,𝑦)=∑_{𝑘∈ℤ𝑛}𝜑(𝑥−𝑘)𝜑(𝑦−𝑘) satisfies (𝑥,𝑦)∈ℝ^2𝑛∶||𝑥−𝑦||<ℓ⊂(𝑥,𝑦)∈ℝ^2𝑛∶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∈ℤ𝑛}Ψ^𝐺𝐹(𝑥−𝑘)Ψ^𝐺𝐹(𝑦−𝑘) with 𝐺_𝐹=(𝐹,…,𝐹), where Ψ^𝐺𝐹 is a Daubechies scaling function. Now for a family {𝑃_𝑗(𝑥,𝑦)}_𝑗≥0={2^𝑗𝑛𝑃₀(2^𝑗𝑥,2^𝑗𝑦)}_𝑗≥0, we obtain that it satisfies conditions(𝑥,𝑦)∈ℝ^2𝑛∶||𝑥−𝑦||<ℓ_𝑗⊂(𝑥,y)∈ℝ^2𝑛∶𝑃_𝑗(𝑥,𝑦)>𝐶ℓ^{−𝑛𝑗+1}(5.2) for every 𝑗≥0 and a positive constant 𝐶>0, where {ℓ_𝑗}_𝑗≥0 is a decreasing sequence of positive real numbers and ℓ_𝑗→0 as 𝑗→∞.

Theorem 5.2. Let 1<𝑝<∞ and 𝜇 be a positive Borel measure on ℝ^𝑛 finite on compact sets. Let 𝑘≥max([𝑛−𝑛/𝑝]+1,𝑛(𝑝−1)). There exists an unconditional Daubechies wavelet basis in 𝐿_𝑝(ℝ^𝑛,𝑑𝜇) with smoothness 𝑘 if and only if 𝑑𝜇=𝑤(𝑥)𝑑𝑥 with 𝑤∈𝒜^𝑝loc.

Proof. Let 𝑤∈𝒜^𝑝loc. From Theorem 4.2, we have an unconditional basis in 𝐹^{𝑠,𝑤𝑝𝑞}(ℝ^𝑛). In [5] Rychkov shows a Littlewood-Paley characterization of spaces with local Muckenhoupt weights, and it means that 𝐹^0,𝑤𝑝,2(ℝ^𝑛)=𝐿^𝑤𝑝(ℝ^𝑛). Hence we have an unconditional basis in 𝐿^𝑤𝑝(ℝ^𝑛).
On the other side, Let {Ψ^𝐺𝑗𝑚∶𝑗∈ℕ₀,𝑚∈ℤ^𝑛,𝐺∈𝐺^𝑗} be a Daubechies wavelet system, which is an unconditional basis in 𝐿_𝑝(ℝ^𝑛,𝑑𝜇). So every 𝑓∈𝐿_𝑝(ℝ^𝑛,𝑑𝜇) has the representation 𝑓(𝑥)=_{𝑗,𝑘,𝐺}𝑓,Ψ^𝐺𝑗𝑘Ψ^𝐺𝑗𝑘(𝑥).(5.3) Operators 𝑃₀𝑓=_𝑘𝑃_𝑚𝑓,Ψ^{𝐺𝐹0,𝑘}Ψ^{𝐺𝐹0,𝑘},𝑓=_{0≤𝑗<𝑚,𝑘,𝐺}𝑓,Ψ^𝐺𝑗𝑘Ψ^𝐺𝑗𝑘,𝑚>0(5.4) are uniformly bounded on 𝐿_𝑝(ℝ^𝑛,𝑑𝜇). We can write 𝑃_𝑚𝑓(𝑥)=∫_ℝ𝑛𝑃_𝑚(𝑥,𝑦)𝑓(𝑦)𝑑𝑦, where 𝑃₀(𝑥,𝑦)=∑_𝑘Ψ^𝐺𝐹_0,𝑘(𝑥)Ψ^𝐺𝐹_0,𝑘(𝑦) and 𝑃_𝑚(𝑥,𝑦)=∑_{0≤𝑗<𝑚,𝑘,𝐺}Ψ^𝐺𝑗𝑘(𝑥)Ψ^𝐺𝑗𝑘(𝑦) if 𝑚>0, because wavelets have compact supports, and we deal with locally finite sums. Hence kernels 𝑃_𝑚(𝑥,𝑦) are bounded.
On the other hand, by the properties of the multiresolution analysis, the kernel 𝑃_𝑚(𝑥,𝑦) coincides with 𝑃_𝑚(𝑥,𝑦)=2^𝑚𝑛𝑃₀(2^𝑚𝑥,2^𝑚𝑦) 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 𝑃_𝑚(𝑥,𝑦)=𝑃_𝑚(𝑥,𝑦) almost everywhere. So the last equality is valid also for 𝐿_𝑝 spaces, and the kernels 𝑃_𝑚(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 𝑄_𝑖. From Lemma 5.1, we get that {𝑃_𝑗(𝑥,𝑦)}_𝑗≥0 satisfies the same conditions as weakly positive family. Let {ℓ_𝑗}_𝑗≥0 be a sequence connected with that family. Without, being generality, we assume that ℓ₀>𝑚𝑎𝑥(𝑑(𝑄_𝑖)), where 𝑑(𝑄) denote a diameter of 𝑄 and maximum is taken over all cubes 𝑄_𝑖 from decomposition of 𝐹. For fixed 𝑖 let 𝑗₀≥0 be the integer such that ℓ_𝑗0+1≤𝑑(𝑄_𝑖)<ℓ_𝑗0. If 𝑥,𝑦∈𝑄_𝑖, we get |𝑥−𝑦|<ℓ_𝑗0 and 𝑃_𝑗0(𝑥,𝑦)>𝐶ℓ^𝑗−𝑛₀₊₁. So for every 𝑥∈𝑄_𝑖, we have ||𝑃_𝑗0𝜒_{𝑄𝑖⧵𝐸}(𝑥)||=||||_{𝑄𝑖⧵𝐸}𝑃_𝑗0(𝑥,𝑦)𝑑𝑦||||>𝐶ℓ^𝑗−𝑛₀₊₁||𝑄_𝑖⧵𝐸||.(5.5) Therefore, |𝑃_𝑗0(𝜒_{𝑄𝑖⧵𝐸})(𝑥)|>𝑐_𝑛, for some constant depending only on 𝐶 and 𝑛. From weak type inequality for operators 𝑃_𝑗, we get 𝜇𝑄_𝑖≤𝜇𝑥∶||𝑃_𝑗0𝜒_{𝑄𝑖⧵𝐸}(𝑥)||>𝑐_𝑛≤𝐶𝑐^𝑛−𝑝𝜇𝑄_𝑖⧵𝐸.(5.6) Summing over 𝑖, we have 𝜇(𝐹)=_𝑖𝜇𝑄_𝑖≤𝐶𝑐^𝑛−𝑝_𝑖𝜇𝑄_𝑖⧵𝐸=𝐶𝑐^𝑛−𝑝𝜇(𝐹⧵𝐸)<𝐶𝑐^𝑛−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∈𝒜^𝑝loc. We pick a sequence {ℓ_𝑗}_𝑗≥0. Let 𝑄⊂ℝ^𝑛 be a cube with |𝑄|≤ℓ₀. We can find 𝑚₀≥0 with ℓ_𝑚0+1≤𝑑(𝑄)<ℓ_𝑚0. Inequalities ||𝑃_𝑚0𝜎_𝜀𝜒_𝑄(𝑥)||=||||_𝑄𝑃_𝑚0(𝑥,𝑦)𝜎_𝜀(𝑦)𝑑𝑦||||>𝐶ℓ^𝑚−𝑛₀₊₁_𝑄𝜎_𝜀≥𝑐_𝑛||𝑄||⁻¹_𝑄𝜎_𝜀≡𝜆,(5.8) where 𝜎_𝜀=(𝑤+𝜀)^{−1/(𝑝−1)},𝜀>0, holds for every 𝑥∈𝑄. Since operators 𝑃_𝑚 are of weak type (𝑝,𝑝), we get 𝑤(𝑄)≤𝑤𝑥∶||𝑃_𝑚0𝜎_𝜀𝜒_𝑄(𝑥)||>𝜆≤𝐶𝑐^𝑛−𝑝||𝑄||^𝑝_𝑄𝜎_𝜀^−𝑝_𝑄𝜎^𝑝𝜀𝑤.(5.9) Multiplying both sides by (∫_𝑄𝜎_𝜀)^𝑝(∫_𝑄𝜎^𝑝𝜀𝑤)⁻¹ and choosing 𝜀 close to zero, we get 𝑤(𝑄)_𝑄𝑤^{−1/(𝑝−1)}^𝑝_𝑄𝑤^{−1/(𝑝−1)}⁻¹≤𝐶||𝑄||^𝑝(5.10) for every 𝑄, |𝑄|<ℓ₀. From Lemma  1.4 in [5], we know that classes 𝒜^𝑝loc are independent of the upper bound for the cube size in their definition. So we get a condition for 𝐴^𝑝loc.

Following [5], we can state square-function characterization. Let us define𝑆(𝑓)(𝑥)=_𝑗||𝜑_𝑗∗𝑓(𝑥)||²^1/2,(5.11) where 𝜑₀∈𝒟 have nonzero integral and 𝜑=𝜑₀−2^−𝑛𝜑₀(⋅/2) and 𝜑_𝑗(𝑥)=2^𝑗𝑛𝜑(2^𝑗𝑥), 𝑗>0.

Corollary 5.3. Let 1<𝑝<∞ and 𝑤∈𝒜^∞loc. The following equivalence holds: ‖‖𝑆(𝑓)∣𝐿^𝑤𝑝‖‖∼‖‖𝑓∣𝐿^𝑤𝑝‖‖(5.12) if and only if 𝑤∈𝒜^𝑝loc.

Proof. Let 𝑤∈𝒜^𝑝loc. From [5] we have that 𝐹^0,𝑤𝑝,2(ℝ^𝑛)=𝐿^𝑤𝑝(ℝ^𝑛) with norm equivalence _ℝ𝑛_𝑗∈ℤ||𝜑_𝑗∗𝑓(𝑥)||²^𝑝/2𝑤(𝑥)𝑑𝑥=_ℝ𝑛||𝑓(𝑥)||^𝑝𝑤(𝑥)𝑑𝑥.(5.13) Conversely, if we assume that 𝑤∈𝒜^∞loc, then from Theorem 4.2 we get that in 𝐹^0,𝑤𝑝,2(ℝ^𝑛) there exists an unconditional basis. Hence also in 𝐿^𝑤𝑝(ℝ^𝑛), we have an unconditional basis. So from Theorem 5.2, we obtain that 𝑤∈𝒜^𝑝loc.

Remark 5.4. It is known that above statements are not true for general Muckenhoupt weights. Taking for example weight, 𝑤(𝑥)=|𝑥|^𝛼|𝑥|_𝛽for|𝑥|≤1,for|𝑥|>1,(5.14) for 𝛼,𝛽>−𝑛, For 𝛼<(𝑝₁−1)𝑛 we have 𝑤∈𝒜^𝑝loc₁ and 𝑟_𝑤=max(0,𝛼)/𝑛+1; for 𝛼,𝛽<(𝑝₂−1)𝑛 we have 𝑤∈𝒜_𝑝2 and 𝑟_𝑤=max(0,𝛼,𝛽)/𝑛+1. Taking 𝛽 big enough, we get that 𝑤 is in 𝒜^𝑝loc∩𝒜_∞, but not in 𝒜_𝑝.

For 𝜑₀∈𝐶^∞0(ℝ^𝑛) with ∫𝜑₀(𝑥)𝑑𝑥≠0 and 𝑓∈𝑆^′_𝑒, we introduce the “vertical” maximal function𝜑⁺⁰𝑓(𝑥)=sup_𝑗∈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 𝑤∈𝒜^∞loc. The following equivalence holds: ‖‖𝜑⁺⁰𝑓∣𝐿^𝑤𝑝‖‖∼‖‖𝑓∣𝐿^𝑤𝑝‖‖(5.16) if and only if 𝑤∈𝒜^𝑝loc.

Please note that it follows from the last corollary that if 1<𝑝<∞ and 𝑤∈𝒜^∞loc, then the weighted local Hardy spaceℎ^𝑤𝑝=𝑓∈𝑆^𝑒∶‖‖𝜑⁺⁰𝑓‖‖<∞(5.17) coincides with 𝐿^𝑤𝑝 if and only if 𝑤∈𝒜^𝑝loc.