The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents

Abstract In this article, the authors establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on ℝ n {{\mathbb{R}}}^{n} in terms of molecular decompositions. Using the molecular decompositions, the authors obtain the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents.


Introduction
The theory of function spaces with variable exponents has rapidly made progress in the past 20 years since some elementary properties were established by Kováčik and Rákosník [1]. Lebesgue and Sobolev spaces with variable exponents have been extensively investigated, see [2] and the references therein. In 2012, Almeida and Drihem [3] introduced the Herz spaces with two variable exponents and obtain the boundedness of some sublinear operators on those spaces. In the same year, Wang and Liu [4] introduced the Herz-type Hardy spaces with variable exponents HK̇p α q n , ( ) (⋅) and HK p α q n , ( ) (⋅) , which is a generalization of the classical Herz-type Hardy spaces. In 2015, Dong and Xu [5] introduced the Herz-type Hardy spaces with two variable exponents HK̇p . Recently, extending classical function spaces arising in harmonic analysis of Euclidean spaces to other domains and non-isotropic settings is an important topic. In 2003, Bownik [6] introduced the anisotropic Hardy spaces H A p n ( ) associated with very general discrete groups of dilations. This formulation includes the classical isotropic Hardy space theory established by Fefferman and Stein [7] and the parabolic Hardy space theory established by Calderón and Torchinsky [8,9]. In 2008, Ding et al. [10] introduced the anisotropic Herz-type and established their atomic and molecular decompositions. Using these decompositions, they gave some applications. In 2019, Wang and Guo [12] introduced the variable anisotropic Herz-type Hardy spaces and established their atomic decomposition and some applications.
Inspired by the previous study, we would like to declare that the goal of this study is to establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on n in terms of molecular decompositions and obtain the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents.
First, we recall some standard notations in variable function spaces. A measurable function p : 0 , n (⋅) → ( ∞) is called a variable exponent. Let f be a measurable function on n and p (⋅) ∈ . Then, the modular function (or, for simplicity, the modular) ϱ p (⋅) , associated with p(·), is defined by setting Moreover, the variable Lebesgue space L p (⋅) is defined to the set of all measurable functions f satisfying that f ϱ p ( ) < ∞ (⋅) , equipped with the quasi-norm f L p ∥ ∥ (⋅). For any variable exponent p(·), let p p x p p x ess inf and ess sup .
Denote by the set of all variable exponents p(·) satisfying p − > 1 and p + < ∞. We call p′(·) the conjugate exponent to p(·), that is, p Let is the set of p (⋅) ∈ , such that the Hardy-Littlewood maximal operator M is bounded on L p(·) . It is well known that if p (⋅) ∈ and satisfies the following global log-Hölder continuous, then p (⋅) ∈ .
Definition 1.1. Let α(·) be a real function on n .
(i) α(·) is called log-Hölder continuous on n if there exists C > 0, such that for all x y , n ∈ and x y 1 2 | − | < . (ii) α(·) is called log-Hölder continuous at origin (or has a log decay at the origin), if there exists C > 0, such that for all x n ∈ . (iii) α(·) is called log-Hölder continuous at infinity (or has a log decay at the infinity), if there exist some α n ∈ ∞ and C > 0, such that By n 0 ( ) and n ( ) ∞ , we denote the class of all exponents p n ∈ ( ), which are locally log-Hölder continuous at the origin and at the infinity, respectively.
Next, we will recall the notion of expansive dilations on n ; see [6, p. 5]. A real n × n matrix A is called an expansive dilation, if all eigenvalues λ of A satisfy |λ| > 1. Suppose λ 1 …λ n are eigenvalues of A (taken according to the multiplicity), so that 1 < |λ 1 | ≤…≤|λ n |. A set Δ n ∈ is said to be an ellipsoid if Δ x P x : 1 n = { ∈ | | < }, for some nondegenerate n × n matrix P, where |·| denotes the Euclidean norm in n . For a dilation A, there exists an ellipsoid Δ and r > 1, such that Δ ⊂ rΔ ⊂ AΔ, where |Δ|, the Lebesgue measure of Δ, equals 1. Let B k = A k Δ for k ∈ , then we have B k ⊂ rB k ⊂ B k+1 , and B k = b k , where b = |det A| > 1. Let w be the smallest integer, so that 2B 0 ⊂ A w B 0 = B w . A homogeneous quasi-norm associated with an expansive matrix A is a measurable mapping ρ : 0 , for , , where C A is a positive constant. It was proved, in [6, p. 6, Lemma 2.4], that all homogeneous quasi-norms associated with a given dilation A are equivalent. Define the step homogeneous quasi-norm ρ on n induced by dilation A as Then, for any x y , n ∈ , ρ(x + y) ≤ b w (ρ(x) + ρ(y)). In the following we denote , where χ Ck is the characteristic function of C k . Throughout this paper, we denote by C a constant, which is independent of the main parameters and whose value may vary. associated with the dilation A is defined by The nonhomogeneous anisotropic Herz space K A; associated with the dilation A is defined by Here, the usual modifications are made when q = ∞.
In variable L p spaces, there are some important lemmas as follows. , then fg is integrable on n and where r p = 1 + 1/p − − 1/p + .
The following lemmas are from [13].
Then, there exists a constant C > 0, such that for all balls B in n , < < depending only on p(·) and n, such that for all measurable subsets S ⊂ B, and HK A; , which were obtained by Wang and Guo [12].
A C ∞ complex-valued function φ is said to belong to the Schwartz class , if for every integer The dual space of , namely, the space of all tempered distributions on n equipped with the weak-* topology, is denoted by ′. For any For any given N ∈ , the non-tangential grand maximal function M N (f) of f ∈ ′ is defined by setting, for any x n ∈ , are defined, respectively, by setting, and where (1) An anisotropic central (α(·), p(·), s)-atom is a measurable function a on n satisfying (2) An anisotropic central (α(·), p(·), s)-atom of restricted type is a measurable function a on n satisfying In this section, we first give the definitions of the molecules of the anisotropic Herz-type Hardy spaces with variable exponents. Before stating our results, we first give the notations of molecules.
Proof. We only prove (i). (ii) can be proved in the similar way.
Let M be a (α(·), p(·), s)-atom with support on a ball B k , then we get Now, we give the molecular decompositions of anisotropic Herz-type Hardy spaces with two variable exponents.
(i) f HK Ȧ ; . Moreover, where the infimum is taken over all above decompositions of f. where the infimum is taken over all above decompositions of f.

By theorem 3.2 of [12]
and Lemma 2.3, we see that Theorems 2.4 and 2.5 can be obtained from the following lemma.
respectively. (ii) There exists a constant C, such that for any dyadic central (α(·), p(·); s, ε) l -molecule M l , l ∈ , and any dyadic central (α(·), p(·); s, ε) l -molecule of restricted type M l , l 0 ∈ , and denote by σ r , the unique integer satisfying b r b We claim that (a) There is a positive constant C and a sequence of numbers {λ k } k , such that where each a k is a (α(·), p(·), 0)-atom; has a (α(·), p(·), 0)-atom decomposition, then our desired conclusion can be deduced directly.
We first show (a). Without loss of generality, we can suppose that M 1 p ( ) = (⋅) , which implies that and for k ∈ , Thus, for any k 0 ∈ ∪ { }, there is a constant C independent of k, such that Noting that m 0 = 0, summing by parts, we have where the a 2,k are central (α(·), p(·), 0)-atoms. Furthermore, where C is independent of M. The conclusion (b) then holds. Hence, the proof of Lemma 2.6 is completed. □

Applications
In this section, we give an application of the molecular decomposition theory established in Section 2.
We study the boundedness of the central δ-Calderón-Zygmund operators from HK Ȧ ; The central δ-Calderón-Zygmund operators, which are more general than the classical Calderón-Zygmund operators, were introduced by Lu and Yang [14] in the isotropic setting of n . Moreover, Ding et al. [10] extended them to the following non-isotropic setting of n associated with the dilation A.  , we can prove the following theorem: ln ln , ln ln  .
where C is independent of a. Case 2. For 1 < q < ∞. By a proof similar to that of [3, Proposition 3.8], we easily obtain an important lemma as follows. □ We now proceed with the proof of Theorem 3.2. Let a be a central (α(·), p(·), 0)-atom with its support in B k for some k ∈ . We write From the L p (⋅) boundedness of M N , the size condition of a, and the Hölder inequality, we conclude that