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

Qingdong Guo; Wenhua Wang

doi:10.1515/math-2020-0031

Article Open Access

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

Qingdong Guo and Wenhua Wang

Published/Copyright: May 28, 2020

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$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

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 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.

Keywords: anisotropic Herz-type Hardy space; two variable exponents; molecular decomposition; central δ-Calderón-Zygmund operator; boundedness

MSC 2010: Primary: 42B30; Secondary: 42B35; 46E30

1 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 H K ̇ p ( ⋅ ) α , q ( ℝ n ) and H K p ( ⋅ ) α , q ( ℝ 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 H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( ℝ n ) and H K p ( ⋅ ) α ( ⋅ ) , q ( ℝ n ) .

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 ( ℝ 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 Hardy spaces H K ̇ p α , q ( A ; ℝ n ) and H K p α , q ( A ; ℝ n ) and established their atomic and molecular decompositions. In 2018, Zhao and Zhou [11] introduced the variable anisotropic Herz-type Hardy spaces H K ̇ p ( ⋅ ) α , q ( A ; ℝ n ) and H K p ( ⋅ ) α , q ( A ; ℝ n ) 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 H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) and H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) 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 ℝ 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 ( ⋅ ) : ℝ n → ( 0 , ∞ ) is called a variable exponent. Let f be a measurable function on ℝ n and p ( ⋅ ) ∈ P . Then, the modular function (or, for simplicity, the modular) ϱ p ( ⋅ ) , associated with p(·), is defined by setting

ϱ p ( ⋅ ) ( f ) ≔ ∫ ℝ n | f ( x ) | p ( x ) d x

and the Luxemburg (also called Luxemburg–Nakano) quasi-norm ∥ f ∥ L p ( ⋅ ) by

∥ f ∥ L p ( ⋅ ) ≔ inf { λ ∈ ( 0 , ∞ ) : ϱ p ( ⋅ ) ( f / λ ) ≤ 1 } .

Moreover, the variable Lebesgue space L p ( ⋅ ) is defined to the set of all measurable functions f satisfying that ϱ p ( ⋅ ) ( f ) < ∞ , equipped with the quasi-norm ∥ f ∥ L p ( ⋅ ) . For any variable exponent p(·), let

p − ≔ ess inf x ∈ ℝ n p ( x ) and p + ≔ ess sup x ∈ ℝ n p ( x ) .

Denote by P the set of all variable exponents p(·) satisfying p ₋ > 1 and p ₊ < ∞. We call p′(·) the conjugate exponent to p(·), that is, p ′ ( ⋅ ) = p ( ⋅ ) p ( ⋅ ) − 1 . Let ℬ is the set of p ( ⋅ ) ∈ P , such that the Hardy–Littlewood maximal operator M is bounded on L ^p(·). It is well known that if p ( ⋅ ) ∈ P and satisfies the following global log-Hölder continuous, then p ( ⋅ ) ∈ ℬ .

Definition 1.1

Let α(·) be a real function on ℝ n .

α(·) is called log-Hölder continuous on ℝ n if there exists C > 0, such that

| α ( x ) − α ( y ) | ≤ C log ( e + 1 / | x − y | )

for all x , y ∈ ℝ n and | x − y | < 1 2 .

α(·) is called log-Hölder continuous at origin (or has a log decay at the origin), if there exists C > 0, such that
| α ( x ) − α (0)| ≤ C log ( e + 1/| x | )
for all x ∈ ℝ n .
α(·) is called log-Hölder continuous at infinity (or has a log decay at the infinity), if there exist some α ∞ ∈ ℝ n and C > 0, such that

| α ( x ) − α ∞ | ≤ C log ( e + | x | )

for all x ∈ ℝ n . By P 0 ( ℝ n ) and P ∞ ( ℝ n ) , we denote the class of all exponents p ∈ P ( ℝ 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 ℝ 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 ∈ ℝ n is said to be an ellipsoid if Δ = { x ∈ ℝ n : | P x | < 1 } , for some nondegenerate n × n matrix P, where |·| denotes the Euclidean norm in ℝ 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 ∈ Z , 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 A : ℝ n → [ 0 , ∞ ) satisfying

ρ A ( x ) > 0 for x ≠ 0 , ρ A ( A x ) = | det A | ρ A ( x ) for x ∈ ℝ n , ρ ( x + y ) ≤ C A ( ρ A ( x ) + ρ A ) ( y ) for x , y ∈ ℝ n ,

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 ℝ n induced by dilation A as

ρ ( x ) ≔ { b j , if x ∈ B j + 1 \ B j , 0, if x = 0 .

Then, for any x , y ∈ ℝ n , ρ(x + y) ≤ b ^w(ρ(x) + ρ(y)).

In the following we denote C _k = B _k\B _k−1 for k ∈ Z . Let ℕ ≔ { 1 , 2 , … } and Z + ≔ { 0 } ∪ ℕ , denote χ k = χ C k for k ∈ Z , χ ˜ k = χ k if k ∈ Z + and χ ˜ 0 = χ B 0 , where χ C k 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.

Definition 1.2

Let α ( ⋅ ) : ℝ n → ℝ with α ( ⋅ ) ∈ L ∞ ( ℝ n ) , 0 < q ≤ ∞ and p ( ⋅ ) ∈ P ( ℝ n ) . The homogeneous anisotropic Herz space K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) associated with the dilation A is defined by

K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) = { f ∈ L loc p ( ⋅ ) ( ℝ n \ { 0 } ) : ∥ f ∥ K ̇ p ( ⋅ ) α ( ⋅ ) , q < ∞ } ,

where

∥ f ∥ K ̇ p ( ⋅ ) α ( ⋅ ) , q = { ∑ k = − ∞ ∞ ∥ b k α ( ⋅ ) f χ k ∥ L p ( ⋅ ) ( ℝ n ) q } 1 / q .

The nonhomogeneous anisotropic Herz space K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) associated with the dilation A is defined by

K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) = { f ∈ L loc p ( ⋅ ) ( ℝ n ) : ∥ f ∥ K p ( ⋅ ) α ( ⋅ ) , q < ∞ } ,

where

∥ f ∥ K p ( ⋅ ) α ( ⋅ ) , q = { ∑ k = 0 ∞ ∥ b k α ( ⋅ ) f χ ˜ k ∥ L p ( ⋅ ) ( ℝ n ) q } 1 / q .

Here, the usual modifications are made when q = ∞.

In variable L ^p spaces, there are some important lemmas as follows.

Lemma 1.3

[1]. Let p ( ⋅ ) ∈ P ( ℝ n ) . If f ∈ L p ( ⋅ ) ( ℝ n ) and g ∈ L p ′ ( ⋅ ) ( ℝ n ) , then fg is integrable on ℝ n and

∫ ℝ n | f ( x ) g ( x ) | d x ≤ r p ∥ f ∥ L p ( ⋅ ) ( ℝ n ) ∥ g ∥ L p ′ ( ⋅ ) ( ℝ n ) ,

where r _p = 1 + 1/p ⁻ − 1/p ⁺.

The following lemmas are from [13].

Lemma 1.4

Suppose p ( ⋅ ) ∈ ℬ ( ℝ n ) . Then, there exists a constant C > 0, such that for all balls B in ℝ n ,

1 | B | ∥ χ B ∥ L p ( ⋅ ) ( ℝ n ) ∥ χ B ∥ L p ′ ( ⋅ ) ( ℝ n ) ≤ C .

Lemma 1.5

Let p ( ⋅ ) ∈ P ( ℝ n ) . Then, there exist 0 < δ 1 , δ 2 < 1 depending only on p(·) and n, such that for all measurable subsets S ⊂ B,

∥ χ B ∥ L p ( ⋅ ) ∥ χ S ∥ L p ( ⋅ ) ≲ | B | | S | , ∥ χ S ∥ L p ( ⋅ ) ∥ χ B ∥ L p ( ⋅ ) ≲ ( | S | | B | ) δ 1 , ∥ χ S ∥ L p ′ ( ⋅ ) ∥ χ B ∥ L p ′ ( ⋅ ) ≲ ( | S | | B | ) δ 2 .

Next, we introduce the definition of homogeneous anisotropic Herz-type Hardy space with two variable exponents H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) and the nonhomogeneous anisotropic Herz-type Hardy space with two variable exponents H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) and the atomic characterization of H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) and H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) , which were obtained by Wang and Guo [12].

A C ^∞ complex-valued function φ is said to belong to the Schwartz class S , if for every integer ℓ ∈ Z + and multi-index α, ∥ φ ∥ α , ℓ ≔ sup x ∈ ℝ n [ ρ ( x ) ] ℓ | ∂ α φ ( x ) | < ∞ . The dual space of S , namely, the space of all tempered distributions on ℝ n equipped with the weak-* topology, is denoted by S ′ . For any N ∈ Z + , let

S N ≔ { φ ∈ S : ∥ φ ∥ α , ℓ ≤ 1, | α | ≤ N , ℓ ≤ N } .

For φ ∈ S , k ∈ Z and x ∈ ℝ n , let φ _k(x) ≔ b ^−k φ(A ^−k x).

Let f ∈ S ′ . The non-tangential maximal function M _φ(f) with respect to φ is defined by setting, for any x ∈ ℝ n ,

M φ ( f ) ( x ) ≔ sup { | f ⋆ φ k ( y ) | : x − y ∈ B k , k ∈ Z } .

For any given N ∈ ℕ , the non-tangential grand maximal function M _N(f) of f ∈ S ′ is defined by setting, for any x ∈ ℝ n ,

M N ( f ) ( x ) ≔ sup φ ∈ S N M φ ( f ) ( x ) .

For 0 < q < ∞, we denote

N q ≔ { [ ( 1 / q − 1 ) ln b / ln λ − ] + 2 , 0 < q ≤ 1 , 2 , q > 1 .

Definition 1.6

Let α ( ⋅ ) ∈ L ∞ , 0 < q ≤ ∞ , p ( ⋅ ) ∈ P , and N > N _q. The homogeneous anisotropic Herz-type Hardy space with variable exponents H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) and the nonhomogeneous anisotropic Herz-type Hardy space with variable exponents H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) are defined, respectively, by setting,

H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ≔ { f ∈ S ′ : M N ( f ) ∈ K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) }

and

H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ≔ { f ∈ S ′ : M N ( f ) ∈ K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) } ,

where

∥ f ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) = ∥ M N ( f ) ∥ K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) and ∥ f ∥ H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) = ∥ M N ( f ) ∥ K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) .

Definition 1.7

Let p ( ⋅ ) ∈ P , α ( ⋅ ) ∈ L ∞ ∩ P 0 log ∩ P ∞ log , and nonnegative integer s ∈ [(α _l − δ ₂) ln b/ln λ ₋,∞) with δ ₂ as in Lemma 1.5. Here, α _l = α ₀, if l < 0, α _l = α _∞, if l > 0.

An anisotropic central (α(·), p(·), s)-atom is a measurable function a on ℝ n satisfying

supp a ⊂ B _l, for some l ∈ Z ;
∥ a ∥ L p ( ⋅ ) ≤ | b | − k α l ;
∫ ℝ n a ( x ) x β d x = 0 for any β ∈ Z + n with | β | ≤ s .

An anisotropic central (α(·), p(·), s)-atom of restricted type is a measurable function a on ℝ n satisfying

supp a ⊂ B _l, l ≥ 0, for some l ∈ Z ;
∥ a ∥ L p ( ⋅ ) ≤ | b | − k α ∞ ;
∫ ℝ n a ( x ) x β d x = 0 for any β ∈ Z + n with |β| ≤ s.

2 Molecular decompositions of H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n )

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.

Definition 2.1

Let 0 < q < ∞, p ( ⋅ ) ∈ P ( ℝ n ) , α ( ⋅ ) ∈ L ∞ ∩ P 0 ∩ P ∞ , and nonnegative integer s ≥ max{[(α(0) − δ ₁) ln b/ln λ ₋], [(α _∞ − δ ₁) ln b/ln λ ₋]}, where δ ₁ as in lemma 1.5. Set ε > max{s,(α(0) + δ ₁ − 1) ln b/ln λ ₋, (α _∞ + δ ₁ − 1) ln b/ln λ ₋} and d = 1 − δ ₁ + ε. Moreover, for any l ∈ Z , when l < 0, α _l := α(0) and a := 1 − δ ₁ − α(0) + ε; when l ≥ 0, α _l := α _∞ and a := 1 − δ ₁ − α _∞ + ε.

A function M _l ∈ L ^p(·) with l ∈ Z is said to be a dyadic central (α(·), p(·); s, ε)_l-molecule if it satisfies
1. ∥ M l ∥ L p ( ⋅ ) ≤ b − l α l ;
2. ℛ p ( ⋅ ) ( M l ) = ∥ M l ∥ L p ( ⋅ ) a / d ∥ ( ρ ( ⋅ ) ) d M l ∥ L p ( ⋅ ) 1 − a / d < ∞ ;
3. ∫ ℝ n M l ( x ) x β d x = 0 , for any β with |β| ≤ s.
A function M _l ∈ L ^p(·) with l ∈ ℕ 0 is said to be a dyadic central (α(·), p(·); s, ε)_l-molecule of restricted type if it satisfies (ii), (iii) and

∥ M l ∥ L p ( ⋅ ) ≤ b − l α l .

Definition 2.2

Let 0 < q < ∞, p ( ⋅ ) ∈ P ( ℝ n ) , α ( ⋅ ) ∈ L ∞ ∩ P 0 ∩ P ∞ , and nonnegative integer s ≥ max{[(α(0) − δ ₁)ln b/ln λ ₋], [(α _∞ − δ ₁) ln b/ln λ ₋]}, where δ ₁ as in Lemma 1.5. Set ε > max{s,(α(0) + δ ₁ − 1) ln b/ln λ ₋, (α _∞ + δ ₁ − 1)ln b/ln λ ₋} and d = 1 − δ ₁ + ε. Moreover, for any function M ∈ L ^p(·), when ∥ M ∥ L p ( ⋅ ) > 1 , a := 1 − δ ₁ − α(0) + ε; when ∥ M ∥ L p ( ⋅ ) ≤ 1 , a := 1 − δ ₁ − α _∞ + ε.

A function M ∈ L ^p(·) is said to be a central (α(·), p(·); s, ε)-molecule if it satisfies
1. ℛ p ( ⋅ ) ( M ) = ∥ M ∥ L p ( ⋅ ) a / d ∥ ( ρ ( ⋅ ) ) d M ∥ L p ( ⋅ ) 1 − a / d < ∞ ;
2. ∫ ℝ n M ( x ) x β d x = 0 , for any β with |β| ≤ s.
A function M ∈ L ^p(·) is said to be a central (α(·), p(·); s, ε)-molecule of restricted type if it satisfies (i), (ii) and

(i′) ∥ M ∥ L p ( ⋅ ) ≤ 1 .

The following lemma shows that a central (α(·), p(·); s, ε)-molecule is a generalization of the central (α(·), p(·), s)-atom.

Lemma 2.3

Let 0 < q < ∞, p ( ⋅ ) ∈ ℬ ( ℝ n ) , α ∈ L ∞ ∩ P 0 ∩ P ∞ , and nonnegative integer s ≥ max{[(α(0) − δ ₁) lnb/ln λ ₋], [(α _∞ − δ ₁) ln b/ln λ ₋], [(α(0) − δ ₂) ln b/ln λ ₋], [(α _∞ − δ ₂) ln b/ ln λ ₋]}, where max{δ ₁,δ ₂} ≤ α(0), α _∞ < ∞ and δ ₁, δ ₂ as in Lemma 1.5. Set ε > max{s,(α(0) + δ ₁ − 1) ln b/ln λ ₋, (α _∞ + δ ₁ − 1) ln b/ln λ ₋} and d = 1 − δ ₁ + ε. Moreover, for any function M ∈ L ^p(·), when ∥ M ∥ L p ( ⋅ ) > 1 , a := 1 − δ ₁ − α(0) + ε; when ∥ M ∥ L p ( ⋅ ) ≤ 1 , a≔ 1 − δ ₁ − α _∞ + ε.

If M is a central (α(·), p(·), s)-atom, then M is a central (α(·), p(·); s, ε)-molecule, such that ℛ p ( ⋅ ) ( M ) < C with C independent of M.
If M is a central (α(·), p(·), s)-atom of restricted type, then M is a central (α(·), p(·); s, ε)-molecule of restricted type, such that ℛ p ( ⋅ ) ( M ) < C with C independent of M.

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

∥ M ∥ L p ( ⋅ ) a / d ∥ ( ρ ( ⋅ ) ) d M ∥ L p ( ⋅ ) 1 − a / d ≤ b ( k − 1 ) d ( 1 − a / d ) ∥ M ∥ L p ( ⋅ ) ≤ C . □

Now, we give the molecular decompositions of anisotropic Herz-type Hardy spaces with two variable exponents.

Theorem 2.4

Let 0 < q < ∞, p ( ⋅ ) ∈ ℬ ( ℝ n ) , α ∈ L ∞ ∩ P 0 ∩ P ∞ , and nonnegative integer s ≥ max{[(α(0) − δ ₁)ln b/ln λ ₋], [(α _∞ − δ ₁) ln b/ln λ ₋], [(α(0) −δ ₂) ln b/ln λ ₋], [(α _∞ − δ ₂) ln b/ln λ ₋]}, where max{δ ₁,δ ₂} ≤ α(0), α _∞ < ∞ and δ ₁, δ ₂ as in Lemma 1.5. Set ε > max{s,(α(0) + δ ₁ − 1) ln b/ln λ ₋, (α _∞ + δ ₁ − 1) ln b/ln λ ₋} and d = 1 − δ ₁ + ε. Moreover, for any function M ∈ L ^p(·) , when ∥ M ∥ L p ( ⋅ ) > 1 , a := 1 − δ ₁ − α(0) + ε; when ∥ M ∥ L p ( ⋅ ) ≤ 1 , a := 1 − δ ₁ − α _∞ + ε.

f ∈ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) if and only if can be represented as
f = ∑ k = − ∞ ∞ λ k M k , in S ′ ,
where each M _k is a dyadic central (α(·), p(·); s, ε)_k -molecule and ∑ k = − ∞ ∞ | λ k | q < ∞ . Moreover,
∥ f ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ∼ inf ( ∑ k = − ∞ ∞ | λ k | q ) 1 / q ,
where the infimum is taken over all above decompositions of f.
f ∈ H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) if and only if can be represented as

f = ∑ k = 0 ∞ λ k M k , in S ′ ,

where each M _k is a dyadic central (α(·), p(·); s, ε)_k -molecule of restricted type and ∑ k = 0 ∞ | λ k | q < ∞ . Moreover,

∥ f ∥ H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ∼ inf ( ∑ k = 0 ∞ | λ k | q ) 1 / q ,

where the infimum is taken over all above decompositions of f.

Theorem 2.5

Let 0 < q ≤ 1, p ( ⋅ ) ∈ ℬ ( ℝ n ) , α ∈ L ∞ ∩ P 0 ∩ P ∞ , and nonnegative integer s ≥ max{[(α(0) − δ ₁) ln b/ln λ ₋], [(α _∞ − δ ₁) ln b/ln λ ₋], [(α(0) − δ ₂) ln b/ln λ ₋], [(α _∞ − δ ₂) ln b/ln λ ₋]}, where max{δ ₁,δ ₂} ≤ α(0), α _∞ < ∞ and δ ₁, δ ₂ as in Lemma 1.5. Set ε > max{s,(α(0) + δ ₁ − 1) ln b/ln λ ₋, (α _∞ + δ ₁ − 1) ln b/ln λ ₋} and d = 1 − δ ₁ + ε. Moreover, for any function M ∈ L ^p(·) , when ∥ M ∥ L p ( ⋅ ) > 1 , a := 1 − δ ₁ − α(0) + ε; when ∥ M ∥ L p ( ⋅ ) ≤ 1 , a := 1 − δ ₁ − α _∞ + ε.

f ∈ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) if and only if can be represented as
f = ∑ k = − ∞ ∞ λ k M k , in S ′ ,
where each M _k is a central (α(·), p(·); s, ε)-molecule and ∑ k = − ∞ ∞ | λ k | q < ∞ . Moreover,
∥ f ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ∼ inf ( ∑ k = − ∞ ∞ | λ k | q ) 1 / q ,
where the infimum is taken over all above decompositions of f.
f ∈ H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) if and only if can be represented as

f = ∑ k = 0 ∞ λ k M k , in S ′ ,

where each M _k is a central (α(·), p(·); s, ε)-molecule of restricted type and ∑ k = 0 ∞ | λ k | q < ∞ . Moreover,

∥ f ∥ H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ∼ inf ( ∑ k = 0 ∞ | λ k | q ) 1 / q ,

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.

Lemma 2.6

Let 0 < q < ∞, p ( ⋅ ) ∈ ℬ ( ℝ n ) , α ∈ L ∞ ∩ P 0 ∩ P ∞ , and nonnegative integer s ≥ max{[(α(0) − δ ₁)ln b/ln λ ₋], [(α _∞ − δ ₁) ln b/ln λ ₋], [(α(0) − δ ₂) ln b/ln λ ₋], [(α _∞ − δ ₂) ln b/ln λ ₋]}, where max{δ ₁,δ ₂} ≤ α(0), α _∞ < ∞ and δ ₁, δ ₂ as in Lemma 1.5. Set ε > max{s,(α(0) + δ ₁ − 1) ln b/ln λ ₋, (α _∞ + δ ₁ − 1) ln b/ln λ ₋} and d = 1 − δ ₁ + ε. Moreover, for any function M ∈ L p ( ⋅ ) and l ∈ Z , when ∥ M ∥ L p ( ⋅ ) > 1 or l < 0, let α _l := α(0) and a := 1 − δ ₁ − α(0) + ε; when ∥ M ∥ L p ( ⋅ ) ≤ 1 or l ≥ 0, let α _l := α _∞ and a := 1 − δ ₁ − α _∞ + ε.

If 0 < q ≤ 1, there exists a constant C, such that for any central (α(·), p(·); s, ε)-molecule M and any central (α(·), p(·); s, ε)-molecule of restricted type M,
∥ M ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ≤ C and ∥ M ∥ H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ≤ C ,
respectively.
There exists a constant C, such that for any dyadic central (α(·), p(·); s, ε)_l -molecule M _l, l ∈ Z , and any dyadic central (α(·), p(·); s, ε)_l -molecule of restricted type M _l, l ∈ ℕ 0 ,

∥ M l ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ≤ C and ∥ M l ∥ H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ≤ C ,

respectively.

Proof

We only prove (i) for the homogeneous case, the proof of the nonhomogeneous case and (ii) are similar.

Suppose that M is a central (α(·), p(·); s, ε)-molecule. Taking

r ≔ { ∥ M ∥ L p ( ⋅ ) − 1 / α ( 0 ) , ∥ M ∥ L p ( ⋅ ) > 1 , ∥ M ∥ L p ( ⋅ ) − 1 / α ∞ , ∥ M ∥ L p ( ⋅ ) ≤ 1 ,

and denote by σ _r, the unique integer satisfying b σ r < r ≤ b σ r + 1 . Denote E 0 = B σ r and E k = B σ r + k \ B σ r + k − 1 for k ∈ ℕ . Set

M ( x ) χ E k ( x ) − χ E k ( x ) | E k | ∫ ℝ n M ( y ) χ E k ( y ) d y ≔ H k ( x ) − F k ( x ) .

It follows that

M ( x ) = ∑ k = 0 ∞ ( H k ( x ) − F k ( x ) ) + ∑ k = 0 ∞ χ E k ( x ) | E k | ∫ ℝ n M ( y ) χ E k ( y ) d y .

Obviously, supp ( H k ( x ) − F k ( x ) ) ⊂ B σ r + k and ∫ ℝ n ( H k ( x ) − F k ( x ) ) d x = 0 . We claim that

There is a positive constant C and a sequence of numbers {λ _k}_k, such that
∑ k =0 ∞ | λ k | q < ∞ , H k − F k = λ k a k ,
where each a _k is a (α(·), p(·), 0)-atom;
∑ k =0 ∞ F k 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 ℛ p ( ⋅ ) ( M ) = 1 , which implies that

∥ ( ρ ( ⋅ ) ) d M ∥ L p ( ⋅ ) = ∥ M ∥ L p ( ⋅ ) − a / ( d − a ) = r a .

For k = 0, we have

∥ H 0 ( x ) − F 0 ( x ) ∥ L p ( ⋅ ) ≤ ∥ M ∥ L p ( ⋅ ) + ∥ χ B σ r ∥ L p ( ⋅ ) | B σ r | ∫ ℝ n | M ( y ) χ B σ r | d y ≤ C ∥ M ∥ L p ( ⋅ ) = C r − α l ≤ C | B σ r | − α l ,

and for k ∈ ℕ ,

∥ H k ( x ) − F k ( x ) ∥ L p ( ⋅ ) ≤ ∥ H k ( x ) ∥ L p ( ⋅ ) + ∥ F k ( x ) ∥ L p ( ⋅ ) ≤ ∥ H k ( x ) ∥ L p ( ⋅ ) + C | E k | ∥ H k ( x ) ∥ L p ( ⋅ ) ∥ χ E k ∥ L p ′ ( ⋅ ) ∥ χ E k ∥ L p ( ⋅ ) ≤ C ∥ H k ( x ) ∥ L p ( ⋅ ) ≤ C ∥ ( ρ ( ⋅ ) ) d M ∥ L p ( ⋅ ) ( b σ r + k − 1 ) − d = C r a ( b σ r + k − 1 ) − d ≤ C b − k a | B σ r + k | − α l .

Thus, for any k ∈ ℕ ∪ { 0 } , there is a constant C independent of k, such that

∥ H k ( x ) − F k ( x ) ∥ L p ( ⋅ ) ≤ C b − k a | B σ r + k | − α l .

If we denote λ 1 , k = C b − k a and a 1 , k = ( H k ( x ) − F k ( x ) ) / λ 1 , k , then the a _1,k are central (α(·), p(·), 0)-atoms and ∑ k = 0 ∞ ( H k ( x ) − F k ( x ) ) ( x ) = ∑ k = 0 ∞ λ 1 , k a 1 , k ( x ) . Moreover,

∑ k = 0 ∞ | λ k | q ≤ C ∑ k = 0 ∞ b − k a q ≤ C ,

where C is independent of M.

Next, we will show (b). Set

m k = ∑ i = k ∞ ∫ ℝ n M ( x ) χ E i ( x ) d x , φ k ( x ) = χ E k ( x ) | E k | .

Noting that m ₀ = 0, summing by parts, we have

∑ i = k ∞ F k ( x ) = ∑ i = k ∞ ( m k − m k + 1 ) φ k ( x ) = ∑ i = k ∞ m k + 1 ( φ k + 1 ( x ) − φ k ( x ) ) .

Clearly,

∫ ℝ n m k + 1 ( φ k + 1 ( x ) − φ k ( x ) ) d x = 0, supp { m k + 1 ( φ k + 1 − φ k ) } ⊂ B σ r + k + 1 ,

and

m k = m 0 − ∫ B σ r + k − 1 M ( x ) d x .

Hence, we obtain

∥ m k + 1 ( φ k + 1 − φ k ) ∥ L p ( ⋅ ) = ∥ − ∫ B σ r + k M ( x ) χ E i ( x ) d x ( φ k + 1 − φ k ) ∥ L p ( ⋅ ) ≤ ( ∥ χ E k ∥ L p ( ⋅ ) | E k + 1 | ∥ χ E k + 1 ∥ L p ( ⋅ ) ∥ χ E k ∥ L p ( ⋅ ) + ∥ χ E k ∥ L p ( ⋅ ) | E k + 1 | | E k + 1 | | E k | ) ∫ B σ r + k | M ( x ) | d x ≤ C b ∥ χ E k ∥ L p ( ⋅ ) | E k + 1 | ∥ ( ρ ( ⋅ ) ) d M ( ⋅ ) ∥ L p ( ⋅ ) ∥ ( ρ ( ⋅ ) ) − d χ B σ r + k ∥ L p ′ ( ⋅ ) ≤ C b ( b − 1)/ b r a b − ( σ r + k − 1) d | B σ r + k | − 1 ∥ χ E k ∥ L p ( ⋅ ) ∥ χ B σ r + k ∥ L p ′ ( ⋅ ) ≤ C b ( b − 1)/ b b σ r + 1 b − ( σ r + k − 1) d ≤ C b 2(1 − δ 1 + ε ) + 1 ( b − 1)/ b b − k a b − ( σ r + k + 1) α l ≤ C b − k a | B σ r + k + 1 | − α l ,

where C is independent of k. Setting

λ 2, k = C b − k a and a 2, k = m k + 1 ( φ k + 1 − φ k ) / λ 2, k ,

we have

∑ k =0 ∞ χ E k ( x ) | E k | ∫ ℝ n M ( y ) χ E k ( y ) d y = ∑ k =0 ∞ λ 2, k a 2, k ,

where the a _2,k are central (α(·), p(·), 0)-atoms. Furthermore,

∑ k =0 ∞ | λ 2, k | q ≤ C ∑ k =0 ∞ b − k a q ≤ C ,

where C is independent of M. The conclusion (b) then holds. Hence, the proof of Lemma 2.6 is completed.□

3 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 H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) to H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) . The following condition is necessary for our discussion on the central δ-Calderón-Zygmund operators on the H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) spaces:

(3.1) f = ∑ i ∈ ℕ λ i a i in S ′ ⇒ T f = ∑ i ∈ ℕ λ i T a i in S ′ .

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 ℝ n . Moreover, Ding et al. [10] extended them to the following non-isotropic setting of ℝ n associated with the dilation A.

Definition 3.1

Let 0 < δ < 1 and 1 < p < ∞. Let T : S ( ℝ n ) → S ' ( ℝ n ) be a linear continuous operator. If there exists K ( x , y ) ∈ S ′ ( ℝ n × ℝ n ) , being continuous away from the diagonal in ℝ 2 n and satisfying:

| K ( x , 0 ) | + | K ( 0 , x ) | ≤ C ( ρ ( x ) ) − 1 , for all x ≠ 0 ;
| K ( x , 0 ) − K ( x , y ) | + | K ( 0 , x ) − K ( y , x ) | ≤ C ( ρ ( y ) ) δ ( ρ ( x ) ) − 1 − δ , when ρ ( x ) ≥ b 2 w ρ ( y ) ;
〈 T ( f ) , g 〉 = ∫ ∫ ℝ n × ℝ n K ( x , y ) f ( y ) g ( x ) d y d x , for f , g ∈ S ( ℝ n ) with disjoint supports, and if T can be extended to a bounded operator on L p ( ℝ n ) , then we say T is a central δ-Calderón-Zygmund operator in L p ( ℝ n ) .

Using the molecular theory of H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) , we can prove the following theorem:

Theorem 3.2

Let 0 < δ < 1, 0 < q < ∞, p ( ⋅ ) ∈ ℬ ( ℝ n ) , α ∈ L ∞ ∩ P 0 ∩ P ∞ and max { δ 2 , δ 1 } ≤ α l < min { ( δ 1 + δ ) ln λ − / ln b , ( δ 2 + δ ) ln λ − / ln b } with α ₁ as in Definition 1.7. Suppose that T is a central δ-Calderón-Zygmund operator and is bounded on L p ( ⋅ ) ( ℝ n ) . If T satisfies (3.1) for every central atomic decomposition and ∫ ℝ n T a ( x ) d x = 0 for each central (α(·), p(·), 0)-atom a(x), then T can be extended to a bounded operator from H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) to H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) .

Remark 3.3

The boundedness of the central δ-Calderón-Zygmund operators on the homogeneous anisotropic Herz-type Hardy space H K ̇ p ( ⋅ ) α , q ( A ; ℝ n ) is contained by Theorem 3.2, and the results also hold for the nonhomogeneous anisotropic Herz-type Hardy space H K p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) .

Proof

Case 1. For 0 < q ≤ 1 . Let a be a central (α(·), p(·), 0)-atom with its support in B _k for some k ∈ Z . If Ta is a central (α(·), p(·); 0, ε)-molecule for some δ + δ ₁ > ε > δ ₁ + α _l − 1 and by condition (3.1) and Lemma 2.6, we have ∥ T a ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ≤ C . Then, for homogeneous case

∥ T f ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) q ≤ ∑ k = − ∞ ∞ | λ k | q ∥ T a ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) q ≤ ∞ .

Thus, T is a bounded operator on H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) by taking supremum of the above formula. It suffices to show Ta is a central (α(·), p(·), 0, ε)-molecule for some δ + δ ₁ > ε > δ ₁ + α _l − 1. To this aim, let a = 1 − δ ₁ − a ₁ + ε, d = 1 − δ ₁ + ε. Obviously, we only need to verify the size condition of molecules, that is,

(3.2) ℛ p ( ⋅ ) ( T a ) = ∥ T a ∥ L p ( ⋅ ) a / d ∥ ( ρ ( ⋅ ) ) d T a ( ⋅ ) ∥ L p ( ⋅ ) 1 − a / d ≤ C ,

with C independent of a. From the hypothesis of the theorem, we need to show only (3.2). To do this, we first estimate ∥ ( ρ ( ⋅ ) ) d T a ( ⋅ ) ∥ L p ( ⋅ ) . By the boundedness of T on the L ^p(·), we have

∥ ( ρ ( ⋅ ) ) d T a ( ⋅ ) ∥ L p ( ⋅ ) ( B k + 2 w ) ≤ C b k d ∥ T a ∥ L p ( ⋅ ) ( B k + 2 w ) ≤ C b k d ∥ a ∥ L p ( ⋅ ) ( B k + 2 w ) ≤ C b k ( d − α l ) .

On the other hand, if x ∈ B k + 2 w c , from the condition(ii) of definition 3.1 and ∫ B k a ( x ) d x = 0 , we can get

| T ( a ) | = | ∫ B k ( K ( x , y ) − K ( x , 0 ) ) a ( y ) d y | ≤ C ∫ B k ρ ( y ) δ ρ ( x ) 1 + δ | a ( y ) | d y ≤ C b k ( 1 + δ ) ρ ( x ) − ( 1 + δ ) 1 | B k | ∫ B k | a ( y ) | d y ≤ C b k ( 1 + δ ) ρ ( x ) − ( 1 + δ ) M ( a ) ( x ) .

Therefore, noting that ε < δ + δ ₁, we have

∥ ( ρ ( ⋅ ) ) d T a ( ⋅ ) ∥ L p ( ⋅ ) ( B k + 2 w c ) ≤ C b k ( 1 + δ ) ∥ ( ρ ( ⋅ ) ) d − 1 − δ M a ( ⋅ ) ∥ L p ( ⋅ ) ( B k + 2 w c ) ≤ C b k ( 1 + δ ) + k ( d − 1 − δ ) ∥ M a ( ⋅ ) ∥ L p ( ⋅ ) ( ℝ n ) ≤ C b k d ∥ a ( ⋅ ) ∥ L p ( ⋅ ) ( ℝ n ) ≤ C b k ( d − α l ) .

That is, ∥ ( ρ ( ⋅ ) ) d T a ( ⋅ ) ∥ L p ( ⋅ ) ≤ C b k ( d − α l ) . Thus,

ℛ p ( ⋅ ) ( T a ) = ∥ T a ∥ L p ( ⋅ ) a / d ∥ ( ρ ( ⋅ ) ) d T a ( ⋅ ) ∥ L p ( ⋅ ) 1 − a / d ≤ C ∥ a ∥ L p ( ⋅ ) a / d b k ( d − α l )(1 − a / d ) ≤ C b − ( k α l ) a / d b k ( d − α l )(1 − a / d ) = C < ∞ ,

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.□

Lemma 3.4

Let α ( ⋅ ) ∈ L ∞ ∩ P 0 ∩ P ∞ , p ( ⋅ ) ∈ P and q ∈ ( 0 , ∞ ) , then

∥ f ∥ K ̇ p ( ⋅ ) α , q ( A ; ℝ n ) ≈ ( ∑ k = ∞ − 1 b α 0 k q ∥ f χ k ∥ L p ( ⋅ ) q ) 1 / q + ( ∑ k = 0 + ∞ b α ∞ k q ∥ f χ k ∥ L p ( ⋅ ) q ) 1 / q .

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 ∈ Z . We write

∥ M N T ( f ) ∥ K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) q ≈ ∑ k = ∞ − 1 b α 0 k q ∥ M N T ( f ) χ k ∥ L p ( ⋅ ) q + ∑ k = 0 + ∞ b α ∞ k q ∥ M N T ( f ) χ k ∥ L p ( ⋅ ) q ≤ C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = − ∞ + ∞ | λ j | ∥ M N T ( a j ) χ k ∥ L p ( ⋅ ) ) q + C ∑ k = 0 + ∞ b α ∞ k q ( ∑ j = − ∞ + ∞ | λ j | ∥ M N T ( a j ) χ k ∥ L p ( ⋅ ) ) q ≕ I 3 + I 4 .

For I ₃, it is easy to see that

I 3 ≤ C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = − ∞ k − σ − 1 | λ j | ∥ M N T ( a j ) χ k ∥ L p ( ⋅ ) ) q + C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = k − σ + ∞ | λ j | ∥ M N T ( a j ) χ k ∥ L p ( ⋅ ) ) q ≕ I 31 + I 32 .

We first estimate M _N T(a) on C _k for k ≥ k 0 + σ + 1 . For any x ∈ C k , φ ∈ S N , j ∈ Z and a polynomial P _s of degree ≤ s , by a proof similar to those of [10, p. 1454], we have

(3.3) M N T ( a ) ( x ) ≤ C b − k 0 α 0 − k 0 ∥ χ B k 0 ∥ L p ' ( ⋅ ) ( b λ − s + 1 ) − m ,

where m = k − k ₀ − 1 − σ. By the Hölder inequality, | b α 0 − δ 2 λ − − s − 1 | < 1 and (3.3), we obtain

I 31 ≤ C ∑ k = − ∞ − 1 b k q α 0 ( ∑ j = − ∞ k − σ − 1 | λ j | b − j ( α 0 + 1 ) + k b ( j − k ) δ 2 ( b λ − s + 1 ) j + σ + 1 − k ) q ≤ C ∑ k = − ∞ − 1 b k q ( α 0 − δ 2 ) λ − − k q ( s + 1 ) [ ∑ j = − ∞ k − σ − 1 | λ j | ( b α 0 − δ 2 λ − − s − 1 ) − j ] q ≤ C ∑ k = − ∞ − 1 b k q ( α 0 − δ 2 ) λ − − k q ( s + 1 ) ∑ j = − ∞ k − σ − 1 | λ j | q ( b α 0 − δ 2 λ − − s − 1 ) − q j / 2 [ ∑ j = − ∞ k − σ − 1 ( b α 0 − δ 2 λ − − s − 1 ) − q ' j / 2 ] q / q ' ≤ C ∑ k = − ∞ − 1 ( b α 0 − δ 2 λ − − s − 1 ) k q / 2 ∑ j = − ∞ k − σ − 1 | λ j | q ( b α 0 − δ 2 λ − − s − 1 ) − q j / 2 ≤ C ∑ j = − ∞ + ∞ | λ j | q ( b α 0 − δ 2 λ − − s − 1 ) − j q / 2 ∑ k = j + σ + 1 + ∞ ( b α 0 − δ 2 λ − − s − 1 ) k q / 2 ≤ C ∑ j ∈ Z | λ j | q .

From the L p ( ⋅ ) boundedness of M N , the size condition of a, and the Hölder inequality, we conclude that

I 32 ≤ C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = k − σ + ∞ | λ j | ∥ T ( a j ) χ k ∥ L p ( ⋅ ) ) q ≤ C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = k − σ + ∞ | λ j | | B j | − α j ) q ≤ C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = k − σ − 1 | λ j | | B j | − α 0 ) q + C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = 0 + ∞ | λ j | | B j | − α ∞ ) q ≤ C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = k − σ − 1 | λ j | q | B j | − α 0 q / 2 ) ( ∑ j = k − σ − 1 | B j | − α 0 q ′ / 2 ) q / q ' + C ∑ k = − ∞ − 1 b α 0 k q ( ∑ j = 0 + ∞ | λ j | q | B j | − α ∞ q / 2 ) ( ∑ j = 0 + ∞ | B j | − α ∞ q ′ / 2 ) q / q ' ≤ C ∑ j = − ∞ − 1 ∑ k = − ∞ j | λ j | q b ( k − j ) α 0 q / 2 + C ∑ j = 0 + ∞ | λ j | q ≤ C ∑ j = − ∞ − 1 | λ j | q + ∑ j = 0 + ∞ | λ j | q ≤ C ∑ j ∈ Z | λ j | q .

The proof of I ₄ is similar to I ₃, we are omitting it there. From the I ₃, I ₄, we can get

∥ M N T ( f ) ∥ K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) q ≤ C ∑ j ∈ Z | λ j | q .

Thus,

∥ T ( f ) ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) ≤ C ∥ f ∥ H K ̇ p ( ⋅ ) α ( ⋅ ) , q ( A ; ℝ n ) .

Therefore, we finish the proof of Theorem 3.2.□

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Received: 2019-04-04

Revised: 2019-11-02

Accepted: 2020-01-18

Published Online: 2020-05-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2020-0031

Keywords for this article

anisotropic Herz-type Hardy space; two variable exponents; molecular decomposition; central δ-Calderón-Zygmund operator; boundedness

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