Convergence and Characterization of Wavelets Associated with Dilation Matrix in Variable 𝐿^𝑝 Spaces

Abstract

We study the convergence of wavelet expansions associated with dilation matrix in the variable 𝐿^𝑝(⋅)(ℝ^𝑑) spaces using the approximate identity. Also, we obtain conditions for the wavelet characterizations of 𝐿^𝑝(⋅)(ℝ^𝑑) with respect to norm estimates. Moreover, the results of Izuki (2008) and Kumar (2009) have been extended.

1. Introduction

Various authors such as Cruz-Uribe et al. [1, 2], Diening [3, 4], Nekvinda [5], Izuki [6], and Kopaliani [7] studied the conditions for the boundedness of the Hardy-Littlewood maximal function 𝑀 on variable 𝐿^𝑝 spaces 𝐿^𝑝(⋅)(Ω). In the case Ω=ℝ^𝑑, Diening [4] has proved that the boundedness of 𝑀 on 𝐿^𝑝(⋅)(ℝ^𝑑) is equivalent to that on 𝐿^𝑝′(⋅)(ℝ^𝑑), where 𝑝^′(⋅) is the conjugate exponent function. Cruz-Uribe et al. [8] used the extrapolation technique to give a different proof of the result obtained by Izuki [6].

The convergence of the wavelet series has been studied by Meyer [9], Walter [10], and Kelly et al. [11, 12]. Meyer was amongst the first to study convergence results for wavelet expansions. Wavelet theory is based on analyzing signals to their components by using a set of bases functions. An important characteristic of wavelet bases functions is that they relate to each other by simple scaling and translation. Meyer [9] showed that the regular wavelet expansions converge in 𝐿^𝑝, 1≤𝑝≤∞, for expansions of uniformly continuous functions; the expansions of continuous functions converge everywhere. His results are based on the assumption of the regularity for basic wavelets and their derivatives. Walter [10] studied the results for regular wavelet expansions of continuous functions. Kelly et al. [11, 12] extended and obtained results analogous to those obtained by Carleson [13] and Hunt [14] for the Fourier series. The results of Kelly et al. [11, 12] only assumed that the wavelets are bounded by radial decreasing 𝐿¹-functions. The behavior of wavelet expansions outside the Lebesgue set is discussed by Reyes [15], for 1≤𝑝<∞, whereas Kelly et al. [11] proved that the wavelet expansion of a function in 𝐿^𝑝 spaces converges pointwise everywhere on the Lebesgue set of a given function. Tao [16] has extended the results of Meyer [9] and Kelly et al. [11, 12] and showed that the wavelet expansion of any 𝐿^𝑝-function converges pointwise almost everywhere under the wavelet projections, hard sampling, and soft sampling summation methods, for 1<𝑝<∞.

The pointwise convergence of wavelet expansions associated with a dilation matrix has been studied by Manchanda et al. [17]. For the applications point of view, sometimes it is reasonable to study the characterization and convergence of wavelets associated with a dilation matrix in variable 𝐿^𝑝(⋅)(ℝ^𝑑) spaces. The construction of wavelet bases in variable exponent spaces was first studied by Sharapudinov [18, 19] who gave a basis of 𝐿^𝑝(⋅)([0,1]) using the Haar system. In this paper, we study the convergence and characterization of wavelet expansions associated with dilation matrix in variable 𝐿^𝑝(⋅)(Ω) spaces where Ω⊆ℝ^𝑑.

2. Definitions and Auxiliary Results

Let 𝐴 denote a dilation matrix in ℤ^𝑑; that is, 𝐴 is a 𝑑×𝑑 matrix such that(i)𝐴(ℤ^𝑑)⊂ℤ^𝑑;(ii)all eigen values 𝜆 of 𝐴 satisfy |𝜆|>1.

Property (i) implies that 𝐴 has integer entries, and hence, |det𝐴| is an integer, and from property (ii) we know that |det𝐴|>1. Considering ℤ^𝑑 as an additive group, we find that 𝐴ℤ^𝑑 is a normal subgroup of ℤ^𝑑; so, we can form the cosets of 𝐴ℤ^𝑑 in ℤ^𝑑. It is a well-known fact that the number of distinct cosets of 𝐴ℤ^𝑑 in ℤ^𝑑 is equal to |det𝐴| [20].

2.1. Variable 𝐿^𝑝 Spaces and Hardy-Littlewood Maximal Function

First, we define variable 𝐿^𝑝 spaces on Ω, Ω⊆ℝ^𝑑, and introduce some known results on the boundedness of the Hardy-Littlewood maximal operator. Throughout the paper, we consider a measurable function 𝑝∶ℝ^𝑑→[1,∞), and 𝑝^′(⋅) means the conjugate exponent function of 𝑝(⋅); that is, 𝑝^′(⋅) satisfies 1/𝑝(𝑥)+1/𝑝^′(𝑥)=1(𝑥∈Ω). 𝜒_𝐸 denotes the characteristic function of a measurable set 𝐸⊂ℝ^𝑑.

Definition 1. The variable 𝐿^𝑝 space 𝐿^𝑝(⋅)(Ω) consists of all measurable functions 𝑓 defined on Ω such that 𝜌_𝑝(𝑓/𝜆)<∞ for some constant 𝜆>0, where 𝜌_𝑝(𝑓)=∫_Ω|𝑓(𝑥)|^𝑝(𝑥)𝑑𝑥.

𝐿^𝑝(⋅)(Ω) is a Banach space with the norm defined by 𝑓_{𝐿𝑝(⋅)(Ω)}=inf𝜆>0∶𝜌_𝑝𝑓𝜆≤1. (1) Now, we define two classes of exponent functions. We use the following notations: 𝑝₋=ess𝑝(𝑥),𝑝inf∶𝑥∈Ω₊=esssup𝑝(𝑥)∶𝑥∈Ω. (2)

Definition 2. The set ℘(Ω) consists of all 𝑝(⋅) satisfying 𝑝₋>1 and 𝑝₊<∞.

Given a function 𝑓∈𝐿¹_loc(Ω), the Hardy-Littlewood maximal operator 𝑀 is defined by 𝑀𝑓(𝑥)=sup_𝑥∈𝑄1|𝑄|_𝑄𝑓𝑦𝑑𝑦 (3)(𝑥∈Ω), where the supremum is taken over all cubes 𝑄 containing 𝑥.

Definition 3. 𝐵(Ω) is the set of 𝑝(⋅)∈℘(Ω) satisfying the condition that 𝑀 is bounded on 𝐿^𝑝(⋅)(Ω).

Cruz-Uribe et al. [1] and Nekvinda [5] proved the following sufficient conditions independently. Note that Nekvinda [5] gave a more general condition in place of (5).

Proposition 4 (see [1, 5]). Let 𝑝(⋅)∈℘(Ω). Suppose that 𝑝(𝑥)𝑦≤𝐶−𝑝≤1−log𝑥−𝑦𝑖𝑓𝑥−𝑦2,𝑝(𝑥)𝑦≤𝐶−𝑝(|𝑥𝑦≥|𝑥|,log𝑒+|)𝑖𝑓 (4)(5) where 𝐶>0 is a constant independent of 𝑥 and 𝑦. Then, one has 𝑝(⋅)∈𝐵(Ω). If 𝑝(⋅) satisfies (4) and (5), then so does 𝑝^′(⋅). Diening [4] proved the following equivalence.

Proposition 5 (see [4, Theorem 8.1]). Let 𝑝(⋅)∈℘(ℝ^𝑑). Then, the following four conditions are equivalent:(1)𝑝(⋅)∈𝐵(ℝ^𝑑);(2)𝑝^′(⋅)∈𝐵(ℝ^𝑑);(3)there exists a constant 𝑞∈(1,𝑝₋) such that 𝑝(⋅)/𝑞∈𝐵(ℝ^𝑑);(4)there exists a constant 𝑞∈(1,𝑝₋) such that (𝑝(⋅)/𝑞)^′∈𝐵(ℝ^𝑑).

Definition 6 (wavelets). Let {𝜑^𝑙∶𝑙=1,2,…,|det𝐴|−1} be a set of functions belonging to 𝐿²(ℝ^𝑑). Define 𝜑^𝑙_𝑗,𝑘(𝑥)=|det𝐴|^𝑗/2𝜑^𝑙𝐴^𝑗=||𝑥−𝑘det𝐴^𝑗/2𝜑^𝑙𝐴^𝑗𝑥₁−𝑘₁,…,𝐴^𝑗𝑥_𝑑−𝑘_𝑑,𝑥=𝑥₁,…,𝑥_𝑑∈ℝ^𝑑 (6) for each 𝑙=1,2,…,|det𝐴|−1, 𝑗∈ℤ, and 𝑘=(𝑘₁,…,𝑘_𝑑)∈ℤ^𝑑. The sequence {𝜑^𝑙∶𝑙=1,2,…,|det𝐴|−1} is a wavelet set if {𝜑^𝑙_𝑗,𝑘∶𝑙=1,2,…,|det𝐴|−1}, 𝑗∈ℤ, 𝑘∈ℤ^𝑑, forms an orthonormal basis in 𝐿²(ℝ^𝑑), and each 𝜑^𝑙 is a wavelet.

We generally need suitable smoothness or decay on wavelets in order to obtain wavelet characterizations of function spaces. Now, we define the class 𝑆(ℝ^𝑑) of rapidly decreasing 𝐶^∞-functions on ℝ^𝑑 as 𝑆ℝ^𝑑=𝑓∶ℝ^𝑑→ℝ^𝑑∶sup_{𝑥∈ℝ𝑑}𝑥^𝑛𝑑^𝑚𝑑𝑥^𝑚𝑓(𝑥)<∞,𝑛∈𝑁₀,𝑚∈𝑁^𝑑₀,𝑁₀=𝑁∪{0}. (7)

In this paper, we use a wavelet set which consists of wavelets in 𝑆(ℝ^𝑑). We can construct it by means of a multiresolution analysis (MRA) [9, 20, 21].

Definition 7. An MRA is a sequence {𝑉_𝑗}_𝑗∈ℤ of closed subspaces of 𝐿²(ℝ^𝑑) associated with dilation matrix 𝐴 such that the following conditions hold:(1)𝑉_𝑗⊂𝑉_𝑗+1 for all 𝑗∈ℤ;(2)∪_𝑗∈ℤ𝑉_𝑗 is dense in 𝐿²(ℝ^𝑑);(3)∩_𝑗∈ℤ𝑉_𝑗={0};(4)𝑓∈𝑉_𝑗 holds if and only if 𝑓(𝐴𝑥)∈𝑉_𝑗+1 for all 𝑗∈ℤ;(5)𝑓∈𝑉₀ holds if and only if 𝑓(𝑥−𝑘)∈𝑉₀ for every 𝑘∈ℤ^𝑑;(6)there exists a function 𝜙∈𝑉₀ such that the system {𝜙(𝑥−𝑘)}_𝑘∈ℤ^𝑑 is an orthonormal basis in 𝑉₀. 𝜙 is called a scaling function of {𝑉_𝑗}_𝑗∈ℤ.

Denoting the orthogonal complement of 𝑉_𝑗 in 𝑉_𝑗+1 by 𝑊_𝑗, it is easy to show that the integer translations {𝜙(𝑥−𝑘)∶𝑘∈ℤ^𝑑} of the mother wavelet 𝜙 form an orthonormal basis for 𝑊₀. Then, we can construct a wavelet set {𝜑^𝑙∶𝑙=1,2,…,|det𝐴|−1} associated with {𝑉_𝑗}_𝑗∈ℤ and satisfying the assumption that each wavelet 𝜑^𝑙 is in 𝑆(ℝ^𝑑), and {𝜑^𝑙_𝑗,𝑘(𝑥)=|det𝐴|^𝑗/2𝜑^𝑙(𝐴^𝑗𝑥−𝑘)∶𝑘∈ℤ^𝑑} forms an orthonormal basis in 𝑊_𝑗 for every 𝑗∈ℤ. Since ∪_𝑗∈ℤ𝑉_𝑗 is dense in 𝐿²(ℝ^𝑑), the collection {𝜑^𝑙_𝑗,𝑘} is an orthonormal basis of 𝐿²(ℝ^𝑑).

Given an MRA {𝑉_𝑗}_𝑗∈ℤ with scaling function 𝜙, let 𝑃_𝑗 and 𝑄_𝑗, respectively, be the orthogonal projections onto the spaces 𝑉_𝑗 and 𝑊_𝑗 with the kernels 𝑃_𝑗(𝑥,𝑦) and 𝑄_𝑗(𝑥,𝑦), defined as 𝑃_𝑗𝑥,𝑦=^{|det𝐴|−1}_{𝑙=1𝑘∈ℤ𝑑}𝜙^𝑙_𝑗,𝑘(𝑥)𝜙^𝑙_𝑗,𝑘𝑦, (8) where 𝜙^𝑙_𝑗,𝑘(𝑥)=|det𝐴|^𝑗/2𝜙^𝑙(𝐴^𝑗𝑥−𝑘) and 𝑄_𝑗𝑥,𝑦=^{|det𝐴|−1}_{𝑙=1𝑘∈ℤ𝑑}𝜑^𝑙_𝑗,𝑘(𝑥)𝜑^𝑙_𝑗,𝑘(𝑦). (9) In view of 𝑉_𝑗+1=𝑉_𝑗⊕𝑊_𝑗, 𝑃_𝑗(𝑥,𝑦) can be written as 𝑃_𝑗𝑥,𝑦=_𝑚<𝑗𝑄_𝑚=𝑥,𝑦_{|det𝐴|−1𝑙=1𝑚<𝑗𝑘∈ℤ𝑑}𝜑^𝑙_𝑚,𝑘(𝑥)𝜑^𝑙_𝑚,𝑘(𝑦). (10) The sequence of projections {𝑃_𝑗𝑓(𝑥)} is called the multiresolution expansion of 𝑓. The scaling expansion of any function 𝑓∈𝐿²(ℝ^𝑑) is defined as 𝑓(𝑥)=^{|det𝐴|−1}_{𝑙=1𝑘∈ℤ𝑑}𝑎_{𝑙,𝑗,𝑘}|det𝐴|^𝑗𝜙^𝑙𝐴^𝑗+𝑥−𝑘_{|det𝐴|−1𝑙=1𝑗≤𝑚𝑘∈ℤ𝑑}𝑏_{𝑙,𝑗,𝑘}𝜑^𝑙_𝑗,𝑘(𝑥), (11) where 𝑎_{𝑙,𝑗,𝑘}=_ℝ𝑑𝑓(𝑥)|det𝐴|^𝑗𝜙^𝑙𝐴^𝑗𝑏𝑥−𝑘𝑑𝑥,_{𝑙,𝑗,𝑘}=_ℝ𝑑𝑓(𝑥)𝜑^𝑙_𝑗,𝑘(𝑥)𝑑𝑥. (12) Therefore the wavelet expansion associated with dilation matrix 𝐴 of 𝑓 is given by 𝑓(𝑥)∼^{|det𝐴|−1}_{𝑙=1𝑗∈ℤ𝑘∈ℤ𝑑}𝑏_{𝑙,𝑗,𝑘}𝜑^𝑙_𝑗,𝑘(𝑥), (13) where the wavelet coefficients 𝑏_{𝑙,𝑗,𝑘}=⟨𝑓,𝜑^𝑙_𝑗,𝑘⟩ are defined as earlier.

By considering convergence in the sense of 𝐿²(ℝ^𝑑), we may write 𝑓(𝑥)=^{|det𝐴|−1}_{𝑙=1𝑗∈ℤ𝑘∈ℤ𝑑}𝑏_{𝑙,𝑗,𝑘}𝜑^𝑙_𝑗,𝑘(𝑥),𝑓(𝑥)=^{|det𝐴|−1}_{𝑙=1𝑘∈ℤ𝑑}𝑎_{𝑙,𝑗,𝑘}|det𝐴|^𝑗𝜙^𝑙𝐴^𝑗+𝑥−𝑘_{|det𝐴|−1𝑙=1𝑗≤𝑚𝑘∈ℤ𝑑}𝑏_{𝑙,𝑚,𝑘}𝜑^𝑙_𝑗,𝑘(𝑥)=𝑓^𝑙_𝑗(𝑥)+𝑟^𝑙_𝑚(𝑥). (14)

The function 𝑓^𝑙_𝑚(𝑥) is in the space 𝑉_𝑚. It is the projection of 𝑓 onto 𝑉_𝑚 and can be written as 𝑓^𝑙_𝑚(𝑥)=_ℝ𝑑𝑞^𝑙_𝑚(𝑥,𝑡)𝑓(𝑡)𝑑𝑡, (15) where 𝑞^𝑙_𝑚(𝑥,𝑡)=|det𝐴|^𝑚𝑞^𝑙₀𝐴^𝑚𝑥,𝐴^𝑚𝑡=𝜆^𝑑𝑞₀(𝜆𝑥,𝜆𝑡),𝜆=𝐴^𝑚=^{|det𝐴|−1}_{𝑙=1𝑘∈ℤ𝑑}𝜙^𝑙_𝑚,𝑘(𝑥)𝜙^𝑙_𝑚,𝑘(𝑡),𝑞^𝑙₀(𝑥,𝑡)=^{|det𝐴|−1}_{𝑙=1𝑘∈ℤ𝑑}𝜙^𝑙(𝑥−𝑘)𝜙^𝑙(𝑡−𝑘). (16)𝑞^𝑙_𝑚(𝑥,𝑡) will be called the reproducing kernel of 𝑉_𝑚. Now, (15) takes the form 𝑓^𝑙_𝑚(𝑥)=_ℝ𝑑𝜆^𝑑𝑞^𝑙₀(𝜆𝑥,𝜆𝑡)𝑓(𝑡)𝑑𝑡. (17) We will assume that the scaling function 𝜙 does not generate an orthonormal basis of 𝑉₀ or of 𝑊₀. This leads to the fact that condition (6) of MRA can be replaced by the weaker condition that {𝜙(𝑥−𝑘)∶𝑘∈ℤ^𝑑} is a Riesz basis of 𝑉₀. We also assume that Fourier transform of 𝜙 has a compact support. Since {𝜙(𝑥−𝑘)∶𝑘∈ℤ} is a Riesz basis of 𝑉₀, it has a biorthonormal basis {𝜙^∗_𝑘(𝑥)} such that for any function 𝑓∈𝑉₀, we have 𝑓(𝑥)=_{𝑘∈ℤ𝑑}𝑓,𝜙^∗_𝑘𝜙_0,𝑘(𝑥). (18) Similarly, for 𝑓∈𝑉_𝑚, we have 𝑓^𝑙_𝑚(𝑥)=^{|det𝐴|−1}_{𝑙=1𝑘∈ℤ𝑑}𝑓,𝜙^∗𝑙_𝑚,𝑘𝜙^𝑙_𝑚,𝑘(𝑥), (19) where {𝜙^∗𝑙_𝑚,𝑘(𝑥)}_𝑘∈ℤ^𝑑 is the biorthonormal basis of {𝜙^𝑙_𝑚,𝑘(𝑥)}_𝑘∈ℤ^𝑑 and {𝑙=1,2,…,|det𝐴|−1}.

Definition 8. If 𝜙∈𝐿¹(ℝ^𝑑) with 𝜙(0)=1 and one defines 𝜙_𝑛(𝑥)=𝑛^𝑑𝜙(𝑛𝑥), where 𝑛→∞, and 𝜙 denotes the Fourier transform of 𝜙, then the sequence of functions {𝜙_𝑛}^∞_𝑛=1 is an approximate identity if (1)∫_ℝ𝑑𝜙_𝑛(𝑥)𝑑𝑥=1 for all 𝑛, (2)sup_𝑛∫_ℝ𝑑|𝜙_𝑛(𝑥)|𝑑𝑥<∞, (3)lim_𝑛→∞∫_|𝑥|>𝛿|𝜙_𝑛(𝑥)|𝑑𝑥=0 for every 𝛿>0.

Remark 9. If 0≤𝜙(𝑥)∈𝑆(ℝ^𝑑), then 𝜙_𝑛(𝑥)=𝑛^𝑑𝜙(𝑛𝑥) is an approximate identity.

Lemma 10. If 𝑞^𝑙₀(𝑥,𝑡)∈𝐿¹(ℝ^𝑑) with ∫_ℝ𝑑𝑞^𝑙₀(𝑥,𝑡)𝑑𝑡=1, then 𝑞^𝑙_𝑚(𝑥,𝑡) is an approximate identity if (1)∫_ℝ𝑑𝑞^𝑙_𝑚(𝑥,𝑡)𝑑𝑡=1 for all 𝑚, (2)sup_𝑚∫_ℝ𝑑|𝑞^𝑙_𝑚(𝑥,𝑡)|𝑑𝑡<∞, (3)lim_𝑚→∞∫_{|𝑥−𝑡|>𝛿}|𝑞^𝑙_𝑚(𝑥,𝑡)|𝑑𝑡=0 for every 𝛿>0.

Proof. From [22, pages 64-65], we can easily obtain that 𝑞^𝑙₀(𝑥,𝑡)=𝑞^𝑙₀(|𝑥−𝑡|) almost everywhere for {𝑙=1,2,…,|det𝐴|−1}. Now, following the lines of proof of [22, Lemma 2.2], the proof of Lemma 10 is immediate.

Remark 11. Let 0≤𝑞^𝑙₀(𝑥,𝑡)∈𝑆(ℝ^𝑑). Then, 𝑞^𝑙_𝑚(𝑥,𝑡)=𝜆^𝑑𝑞₀(𝜆|𝑥−𝑡|) is an approximate identity.

Lemma 12. If 𝑓∈𝐿¹(ℝ^𝑑) and 𝑞^𝑙_𝑚(𝑥,𝑡)∈𝑆(ℝ^𝑑), then 𝑞^𝑙_𝑚(𝑥,𝑡)∗𝑓∈𝑆(ℝ^𝑑), where ∗ denotes the convolution operator.

Proof. The proof follows on proceeding on the lines of proof of [22, Lemma 1.3].

Lemma 13. Suppose that 𝑝₊<∞. Then, 𝑆(ℝ^𝑑) is dense in 𝐿^𝑝(⋅)(ℝ^𝑑).

Proof. Set 𝐶^∞_𝑐(ℝ^𝑑)={𝜙∈𝐶^∞(ℝ^𝑑)} such that supp(𝜙) is compact.

Obviously, 𝑆(ℝ^𝑑) is a vector space and 𝐶^∞_𝑐(ℝ^𝑑)⊆𝑆(ℝ^𝑑) and 𝑆(ℝ^𝑑) is dense in 𝐿^𝑝(ℝ^𝑑). So, we have 𝐶^∞_𝑐ℝ^𝑑⊆𝑆ℝ^𝑑⊆𝐿^𝑝ℝ^𝑑,1≤𝑝<∞. (20) It has been proved by Kováčik and Rákosník [23, Theorem 2.11] that if 𝑝₊<∞, then 𝐶^∞_𝑐(ℝ^𝑑) is dense in 𝐿^𝑝(⋅)(ℝ^𝑑). By virtue of (20), it follows that 𝑆(ℝ^𝑑) is dense in 𝐿^𝑝(⋅)(ℝ^𝑑).

Definition 14. Let 1<𝑝₀<∞ be a constant, and let 𝑤 be a weight on ℝ^𝑑 such that 𝑤^{−1/(𝑝0−1)} is locally integrable; 𝑤 is said to be an 𝐴_𝑝0 weight if 𝑤 satisfies 𝐴_𝑝0,𝑤=sup_𝑄∶cube1|𝑄|_𝑄𝑤(𝑥)×1𝑑𝑥|𝑄|_𝑄𝑤(𝑥)^{−1/(𝑝0−1)}𝑑𝑥^𝑝0−1<∞, (21) where |𝑄| means the Lebesgue measure of 𝑄. 𝐴_𝑝0 denotes the set of all 𝐴_𝑝0 weights.

Definition 15. (1) A weight 𝑤 is said to be an 𝐴_∞ weight if for all 0<𝛼<1, there exists a constant 0<𝛽<1 such that for every cube 𝑄 and measurable set 𝐸⊂𝑄 satisfying |𝐸|≥𝛼|𝑄|, 𝛽_𝑄𝑤(𝑥)𝑑𝑥≤_𝐸𝑤(𝑥)𝑑𝑥. (22)𝐴_∞ denotes the set of all 𝐴_∞ weights.
(2) A family of weights {𝑤_𝑡}_𝑡>0 is said to be a family of 𝐴_∞ weights uniformly in 𝑡 if each 𝑤_𝑡 is an 𝐴_∞ weight and the constant 𝛽 is independent of 𝑡. It is well known that 𝐴_∞=∪_1<𝑝0<∞𝐴_𝑝0 [24]. Lerner [24] gave the following characterization of families of 𝐴_∞ weights.

Lemma 16 (see [24, Lemma 2.1]). Let 𝑝(⋅) be a nonnegative and measurable function on ℝ^𝑑. The family of weights {𝑡^𝑝(⋅)}_𝑡>0 is a family of 𝐴_∞ weights uniformly in 𝑡 if and only if 𝑝(⋅)≡𝑝₀ for some constant 𝑝₀.

Definition 17. Let 1≤𝑝₀<∞ be a constant, and let 𝑤 be a weight on ℝ^𝑑. The weighted 𝐿^𝑝0 space 𝐿^𝑝0_𝑤(ℝ^𝑑) is the space of all measurable functions 𝑓 with 𝑓_𝐿^𝑝_{0𝑤(ℝ𝑑)}=_ℝ𝑑𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥^1/𝑝0<∞. (23)𝐿^𝑝0_𝑤(ℝ^𝑑) is a Banach space with the norm ‖.‖_𝐿^𝑝_{0𝑤(ℝ𝑑)}.

The following lemma also holds for variable 𝐿^𝑝 spaces on an open set of ℝ^𝑑.

Lemma 18 (see [25, Corollary 1.11]). Let 𝑝(⋅)∈℘(ℝ^𝑑), and let 𝐹 be a family of ordered pairs of non-negative and measurable functions (𝑓,𝑔). Suppose that 𝑝(⋅)𝑝₁^′∈𝐵ℝ^𝑑𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑝₁∈1,𝑝₋, (24) and that there exists a constant 𝑝₀∈(1,∞) such that for all 𝑤∈𝐴_𝑝0,𝑤 and all (𝑓,𝑔)∈𝐹 with 𝑓∈𝐿^𝑝𝑜_𝑤(𝑅), 𝑓_𝐿^𝑝_{𝑜𝑤(𝑅𝑛)}≤𝐶_𝑜𝑔_𝐿^𝑝_{𝑜𝑤(𝑅𝑛)}, (25) where 𝐶_𝑜 is a positive constant depending only on 𝑛, 𝑝_𝑜, and 𝐴_{𝑝𝑜,𝑤}. Then, it follows that for all (𝑓,𝑔)∈𝐹 such that 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑), 𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}≤𝐶₁𝑔_{𝐿𝑝(⋅)(ℝ𝑑)}, (26) where 𝐶₁ is a positive constant independent of (𝑓,𝑔).

2.2. Banach Function Spaces

Definition 19. Let 𝑀(ℝ^𝑑) be the set of all measurable functions on ℝ^𝑑. (1)A linear space 𝑋⊂𝑀(ℝ^𝑑) is said to be a Banach function space if there exists a functional ‖⋅‖∶𝑀(ℝ^𝑑)→ℝ which has the norm property and satisfies the following conditions:(a)𝑓∈𝑋 if and only if ‖𝑓‖<∞;(b)‖𝑓‖=‖|𝑓|‖ for all 𝑓∈𝑀(ℝ^𝑑);(c)if 0≤𝑓₁≤𝑓₂≤⋯ and lim_𝑛→∞𝑓_𝑛=𝑓, then ‖𝑓₁‖≤‖𝑓₂‖≤⋯ and lim_𝑛→∞‖𝑓_𝑛‖=‖𝑓‖;(d)for all 𝐸⊂ℝ^𝑑 with |𝐸|<∞, it follows that ‖𝜒_𝐸‖<∞;(e)for all 𝐸⊂ℝ^𝑑 with |𝐸|<∞, there exists a constant 𝐶_𝐸>0 such that ∫_𝐸|𝑓(𝑥)|𝑑𝑥≤𝐶_𝐸‖𝑓‖ for all 𝑓∈𝑋.(2)Let 𝑋 be a Banach function space with the norm ‖⋅‖. The norm ‖⋅‖ is said to be an absolutely continuous norm if lim_𝑗→∞‖𝑓𝜒_𝐸𝑗‖=0 for all 𝑓∈𝑋 and all sequences of measurable sets {𝐸_𝑗}_𝑗=1 such that lim_𝑗→∞𝜒_𝐸𝑗=0 almost everywhere.

The following two lemmas imply that 𝐿^𝑝(⋅)(ℝ^𝑑) is a Banach function space with the absolutely continuous norm ‖⋅‖_𝐿^𝑝(⋅)_(ℝ^𝑑₎ (cf. [26]).

Lemma 20 (see [27]). Let (𝑋,‖⋅‖) be a Banach function space. Then, 𝑋 is separable if and only if the norm ‖⋅‖ is absolutely continuous on 𝑋.

Lemma 21 (see [23]). Suppose that 𝑝₊<∞. Then, 𝐿^𝑝(⋅)(ℝ^𝑑) is separable.

3. Main Results

First, we prove the convergence of wavelet associated with dilation matrix in 𝐿^𝑝(⋅)(ℝ^𝑑) in reference of MRA.

Theorem 22. Let 𝑝(⋅)∈𝐵(ℝ^𝑑) and 𝜙, 𝜙^∗∈𝑆(ℝ^𝑑). Define 𝑃^∗_𝑗𝑓=_{𝑘∈ℤ𝑑}𝑓,𝜙_𝑗,𝑘𝜙^∗_𝑗,𝑘. (27) If {𝑞_𝑗(𝑥,𝑡)}^∞_𝑗=1 is an approximate identity, then for every 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑) and 𝑗∈ℤ, lim_𝑗→∞𝑃^∗_𝑗(𝑓)−𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}=0. (28)

Remark 23. (1) If 𝜙 is a scaling function of an MRA and 𝜙=𝜙^∗, then we see that 𝑃^∗_𝑗(𝑓)=𝑃_𝑗(𝑓), where 𝑃_𝑗 is the orthogonal projection for each 𝑗∈ℤ.
(2) In view of generalized Holder inequality [11, Theorem 2.1], we have _ℝ𝑑𝑓(𝑥)𝑔(𝑥)𝑑𝑥≤𝑟_𝑝𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}𝑔_{𝐿𝑝′(⋅)(ℝ𝑑)} (29) for all 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑) and 𝑔∈𝐿^𝑝′(⋅)(ℝ^𝑑), where 𝑟_𝑝=1+1/𝑝₋−1/𝑝₊. Hence, the previous 𝐿²-inner product ⟨𝑓,𝜙_𝑗,𝑘⟩ makes sense.

Proof of Theorem 22. The nonorthogonal projection of 𝑓∈𝐿²(ℝ^𝑑) on the space 𝑉_𝑗 is given by 𝑃^∗_𝑗𝑓=_{𝑘∈ℤ𝑑}𝑓,𝜙_𝑗,𝑘𝜙^∗_𝑗,𝑘=_{𝑘∈ℤ𝑑ℝ𝑑}𝑓(𝑡)𝜙_𝑗,𝑘(𝑥)𝜙^∗_𝑗,𝑘(𝑡)=𝑑𝑡_ℝ𝑑𝑓(𝑡)_{𝑘∈ℤ𝑑}𝜙_𝑗,𝑘(𝑥)𝜙^∗_𝑗,𝑘(𝑡)=𝑑𝑡_ℝ𝑑𝑓(𝑡)𝑞_𝑗()=𝑥,𝑡𝑑𝑡_ℝ𝑑𝑓(𝑡)𝑞_𝑗()=𝑥−𝑡𝑑𝑡𝑓∗𝑞_𝑗(𝑥)→𝑓(𝑥)byLemma12. (30) Thus, if 𝑓∈𝐿^𝑝(ℝ^𝑑) and 𝑞_𝑗(𝑥,𝑡)∈𝑆(ℝ^𝑑), then 𝑞_𝑗∗𝑓∈𝑆(ℝ^𝑑). Now, applying Lemma 18, we obtain that if 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑) and 𝑞_𝑗(𝑥,𝑡)∈𝑆(ℝ^𝑑), then 𝑞_𝑗∗𝑓∈𝑆(ℝ^𝑑). By virtue of Lemma 13 we get 𝑞_𝑗∗𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑). Hence, for 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑), 𝑞_𝑗∗𝑓→𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑) for 𝑗→∞. Hence, the proof is completed.

Corollary 24. Let 𝑝(⋅)∈𝐵(ℝ^𝑑) and 𝜙, 𝜙^∗∈𝑆(ℝ^𝑑). If {𝑞_𝑗(𝑥,𝑡)}^∞_𝑗=1 is an approximate identity, then for every 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑), the wavelet series 𝑓_𝑗(𝑥)=_{𝑘∈ℤ𝑑}𝑓,𝜙_𝑗,𝑘𝜙^∗_𝑗,𝑘(𝑥) (31) converges to 𝑓(𝑥) as 𝑗→∞ at every point of continuity of 𝑓(𝑥).

To characterize the wavelet associated with dilation matrix 𝐴 in 𝐿^𝑝(⋅)(ℝ^𝑑) with respect to the norm ‖⋅‖_𝐿ℝ𝑑^𝑝(⋅), we define a cube 𝐼_𝑗,𝑘=Π^𝑑_𝑖=1|det𝐴|^−𝑗𝑘_𝑖,det𝐴^−𝑗𝑘_𝑖+1 (32) and denote 𝜒_𝑗,𝑘=|det𝐴|^𝑗/2𝜒_{𝐼𝑗,𝑘} for 𝑗∈ℤ and 𝑘∈ℤ^𝑑. Given a wavelet set {𝜑^𝑙∶𝑙=1,2,…,|det𝐴|−1}, we consider the following two square functions in order to obtain wavelet characterizations: 𝑉^∗𝑓=^{|det𝐴|−1∞}_{𝑙=1𝑗=−∞𝑘∈ℤ𝑑}𝑓,𝜑^𝑙_𝑗,𝑘𝜑^𝑙_𝑗,𝑘^21/2,𝑊^∗𝑓=^{|det𝐴|−1∞}_{𝑙=1𝑗=−∞𝑘∈ℤ𝑑}𝑓,𝜑^𝑙_𝑗,𝑘𝜒^𝑙_𝑗,𝑘^21/2. (33)

Lemma 25. Let 𝑝₀∈(1,∞) be a constant, 𝑤∈𝐴_𝑝0, and {𝜑^𝑙} a wavelet set such that each wavelet 𝜑^𝑙∈𝑆(ℝ^𝑑). Then, there exist constants 𝑐,𝐶,𝑐^′,𝐶^′>0 depending only on 𝑑, 𝑝₀, {𝜑^𝑙}, and 𝐴_𝑝0,𝑤 such that for all 𝑓∈𝐿^𝑝0_𝑤(ℝ^𝑑), 𝑐𝑓_𝐿^𝑝_{0𝑤(ℝ𝑑)}≤𝑉^∗𝑓_𝐿^𝑝_{0𝑤(ℝ𝑑)}≤𝐶𝑓_𝐿^𝑝_{0𝑤(ℝ𝑑)},𝑐^′𝑓_𝐿^𝑝_{0𝑤(ℝ𝑑)}≤𝑊^∗𝑓_𝐿^𝑝_{0𝑤(ℝ𝑑)}≤𝐶^′𝑓_𝐿^𝑝_{0𝑤(ℝ𝑑)}. (34)

Proof. In view of [28, 29], the estimates (34) hold for 𝜑^𝑙∈ℝ⁰(ℝ^𝑑) [3]. The class ℝ⁰(ℝ^𝑑) is contained in 𝑆(ℝ^𝑑); therefore, estimates (34) also hold for 𝜑^𝑙∈𝑆(ℝ^𝑑).

Now, we prove the following.

Theorem 26. Let 𝑝(⋅)∈𝐵(ℝ^𝑑), and let {𝜑^𝑙∶𝑙=1,2,…,|det𝐴|−1} be a wavelet set such that wavelet 𝜑^𝑙∈𝑆(ℝ^𝑑). Then, there exist constants 𝑐,𝐶,𝑐^′,𝐶^′>0 such that for all 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑), 𝑐𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}≤𝑉^∗𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}≤𝐶𝑓_{𝐿𝑝(⋅)(ℝ𝑑)},𝑐^′𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}≤𝑊^∗𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}≤𝐶^′𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}. (35)

Proof. We see that 𝑝(⋅)∈𝐵(ℝ^𝑑) is equivalent to condition (24) in view of Proposition 5. Thus, Lemma 25 together with Lemma 18 implies that the estimates (35) hold for all 𝑓∈𝑆(ℝ^𝑑) and 𝜑^𝑙∈𝑆(ℝ^𝑑). By using Lemma 13, these inequalities also hold for 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑).

Applying the wavelet characterization (Theorem 26) and [27, Proposition 3.6, Ch. 1], we can construct unconditional basis of 𝐿^𝑝(⋅)(ℝ^𝑑) in terms of wavelets.

Theorem 27. Let 𝑝(⋅)∈𝐵(ℝ^𝑑), and let {𝜑^𝑙∶𝑙=1,2,…,|det𝐴|−1} be a wavelet set such that each wavelet 𝜑^𝑙 is in 𝑆(ℝ^𝑑). Then, the wavelet basis {𝜑^𝑙_𝑗,𝑘∶𝑙=1,2,…,|det𝐴|−1,𝑗∈ℤ,𝑘∈ℤ^𝑑} forms an unconditional basis on 𝐿^𝑝(⋅)(ℝ^𝑑).

Proof. Let us denote Λ={1,2,…,|det𝐴|−1}×ℤ×ℤ^𝑑 and 𝑆_𝐷𝑓=_{(𝑙,𝑗,𝑘)∈𝐷}𝑓,𝜑^𝑙_𝑗,𝑘𝜑^𝑙_𝑗,𝑘for𝐷⊂Λ. (36) In order to complete the proof, we have to check the following two conditions:(1)there exists a constant 𝐶>0 such that ‖𝑆_𝐷𝑓‖_𝐿^𝑝(⋅)_(ℝ^𝑑₎≤𝐶‖𝑓‖_𝐿^𝑝(⋅)_(ℝ^𝑑₎ for all 𝐷⊂Λ and all 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑);(2)span {𝜙^𝑙_𝑗,𝑘∶(𝑙,𝑗,𝑘)∈Λ} is dense in 𝐿^𝑝(⋅)(ℝ^𝑑).
To check condition (1), using Theorem 26 and the orthonormality, we obtain for all 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑), 𝑆_𝐷𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}≤𝐶₀𝑊^∗(𝑆_𝐷𝑓)_{𝐿𝑝(⋅)(ℝ𝑑)}≤𝐶₀𝑊^∗𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}≤𝐶₀𝐶₁𝑓_{𝐿𝑝(⋅)(ℝ𝑑)}, (37) where 𝐶₀,𝐶₁>0 are constants independent of 𝑓. Using monotone convergence theorem as finite sets 𝐷 exhaust Λ, we get that (37) holds for (𝑙,𝑗,𝑘)∈Λ.
To check condition (2), it suffices to show that lim_𝐷→Λ‖𝑓−𝑆_𝐷𝑓‖_𝐿^𝑝(⋅)_(ℝ^𝑑₎=0. By Lemmas 20 and 21, 𝐿^𝑝(⋅)(ℝ^𝑑) is a Banach function space with absolutely continuous norm ‖⋅‖_𝐿^𝑝(⋅)_(ℝ^𝑑₎. For any 𝑓∈𝐿^𝑝(⋅)(ℝ^𝑑)∩𝐿²(ℝ^𝑑), we have 𝑓=∑_{(𝑙,𝑗,𝑘)∈𝐷}⟨𝑓,𝜑^𝑙_𝑗,𝑘⟩𝜑^𝑙_𝑗,𝑘, with convergence in 𝐿²(ℝ^𝑑), and there exists a subsequence of partial sums ∑_{(𝑙,𝑗,𝑘𝑖)∈𝐷}⟨𝑓,𝜑^𝑙_{𝑗,𝑘𝑖}⟩𝜑^𝑙_{𝑗,𝑘𝑖} for 𝑖∈𝑁 that converge almost everywhere on ℝ^𝑑. Using Fatou’s lemma, we get 𝑓−^𝑁_{𝑖=1,(𝑙,𝑗,𝑘𝑖)∈𝐷}𝑓,𝜑^𝑙_{𝑗,𝑘𝑖}𝜑^𝑙_{𝑗,𝑘𝑖𝐿𝑝(⋅)(ℝ𝑑)}≤lim_𝑚→∞inf^𝑁𝑚_{𝑖=𝑁+1,(𝑙,𝑗,𝑘𝑖)∈𝐷}𝑓,𝜑^𝑙_{𝑗,𝑘𝑖}𝜑^𝑙_{𝑗,𝑘𝑖𝐿𝑝(⋅)(ℝ𝑑)},𝑚∈𝑁,≤𝐶_𝑝^∞_{𝑖=𝑁+1,𝑙,𝑗,𝑘𝑖∈𝐷}𝑓,𝜑^𝑙_{𝑗,𝑘𝑖}𝜑^𝑙_{𝑗,𝑘𝑖}^21/2_{𝐿𝑝(⋅)(ℝ𝑑)}. (38) The result [27, Proposition 3.6, Ch. 1] can be applied to 𝐿^𝑝(⋅)(ℝ^𝑑) since 𝐿^𝑝(⋅)(ℝ^𝑑) is a Banach function space with the absolutely continuous norm (cf. [26]). Using this result, the last term is zero, and by virtue of (37), ‖(∑_{(𝑙,𝑗,𝑘)∈Λ}|⟨𝑓,𝜑^𝑙_𝑗,𝑘⟩𝜑^𝑙_𝑗,𝑘|²)^1/2‖_𝐿^𝑝(⋅)_(ℝ^𝑑₎<∞; we can use the dominated convergence theorem to obtain 𝑓=^∞_{𝑖=1,(𝑙,𝑗,𝑘𝑖)∈𝐷}𝑓,𝜑^𝑙_{𝑗,𝑘𝑖}𝜑^𝑙_{𝑗,𝑘𝑖} (39) with convergence in 𝐿^𝑝(⋅)(ℝ^𝑑). Therefore, lim_𝐷→Λ‖𝑓−𝑆_𝐷𝑓‖_𝐿^𝑝(⋅)_(ℝ^𝑑₎=0.
Hence, the proof is completed.

Acknowledgments

The author is extremely thankful to the learned reviewers for taking pains and giving fruitful comments to improve the paper. This work is supported by University Grants Commission, New Delhi, India, under the Grant of Major Research Project no. 41-792/2012(SR).