Startseite Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
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Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents

  • Zhenzhen Yang , Wanjing Zhang und Jing Zhang EMAIL logo
Veröffentlicht/Copyright: 3. Juni 2025
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 23 Heft 1

Abstract

This article aims to delve deeper into the weighted grand variable Herz-Morrey spaces, and try to establish the boundedness of fractional sublinear operators and their multilinear commutators within this framework. The results are still new even in the unweighted setting, extending previous research. As applications, the corresponding boundedness estimates for the commutators of maximal operator and Riesz potential operator are established.

Keywords: fractional sublinear operator; multilinear commutators; grand Herz-Morrey space; weight
MSC 2010: 42B25; 42B35; 46E30

1 Introduction

The main purpose of this article is to introduce and investigate the weighted grand variable Herz-Morrey spaces. As a class of classical operators in harmonic analysis, sublinear operators only need to meet size conditions compared with other operators. The boundedness of sublinear operators can be applied to several important operators, such as the Hardy-Littlewood maximal operators and Calderón-Zygmund operators.

On the Euclidean space R n , let f ∈ L 1 ( R n ) , and define the sublinear operator T by

T f ( x ) = ∫ R n K ( x , y ) f ( y ) d y , x ∉ supp f .

The commutator generated by a locally integrable function b and the operator T is defined by

[ b , T ] ( f ) ≔ b T ( f ) − T ( b f ) ,

for suitable functions f . It is well known that the commutator plays an important role in harmonic analysis and has important applications in the estimates of the solutions for the elliptic equations and some nonlinear partial differential equations (PDEs).

Let b = ( b 1 , b 2 , … , b m ) , b j ∈ BMO ( R n ) , j ∈ { 1 , 2 , … , m } , m ∈ N , x ∉ supp f, and define the multilinear commutators of sublinear operators T b by

T b ( f ) ( x ) = ∫ R n ∏ j = 1 m ∣ b j ( x ) − b j ( y ) ∣ K ( x , y ) f ( y ) d y ,

where K ( x , y ) is the integral kernel of the operator T [1].

As we all know, the variable function spaces have found their applications in fluid dynamics, image processing, PDEs, variational calculus, and harmonic analysis [2–6]. Due to the celebrated work of Kováçik and Rákosník [7] on Lebesgue spaces with variable exponent, there has been an increasing interest in extending Lebesgue spaces with variable exponent.

In 2009, Izuki [8] introduced Herz-Morrey spaces with variable exponent and obtained the boundedness of vector-valued sublinear operators satisfying a size condition on Herz-Morrey spaces with variable exponent. In 2010, Izuki [9] defined Herz spaces with variable exponent and application to wavelet characterization. In the same year, Izuki [10] established the boundedness of sublinear operators and their commutators on Herz spaces with variable exponent. Based on these works, in 2016, Izuki and Noi [11] introduced weighted Herz spaces with variable exponent and established the boundedness of fractional integrals on weighted Herz spaces with variable exponent. It is precisely because of Izuki’s significant contribution to the weighted variable exponent Herz-Morrey space that there is growing interest in extending the Herz-Morrey space using variable exponents.

In 2018, Wang and Shu [12] generalized Izuki’s result for the weighted Herz-Morrey space with variable exponent M K . p , q ( ⋅ ) α , λ ( ω ) and proved the boundedness of some sublinear operators. In recent years, the introduction and discussion of the grand function spaces have further enriched the theoretical study of the function space. The grand function spaces is a generalization of the classical function space and have wide applications in the field of PDE. Recently, Zhang et al. [13] introduced the weighted grand Herz-Morrey spaces with variable exponents M K . p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) , and they obtained the boundedness of sublinear operators.

Inspired by Wang and Shu [12] and Zhang et al. [13], the aim of this article is to introduce the weighted grand variable Herz-Morrey spaces with variable exponents and establish the boundedness of fractional sublinear operators and their multilinear commutators on the weighted grand variable Herz-Morrey spaces with variable exponents.

To be precise, this article is organized as follows.

In Section 2, we recall some notions, notations, and basic properties on the variable exponent Lebesgue spaces and the weighted variable exponent Lebesgue space L q ( ⋅ ) ( ω ) . Then, we recall the definition of the homogeneous weighted grand of Herz-Morrey spaces with variable exponent M K . p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) and show some related properties. Finally, the main results of this article are given. As applications, the corresponding boundedness estimates for the commutators of maximal operator and Riesz potential operator are established.

In Section 3, via borrowing some ideas from those used in the proofs of [12, Theorem 3.1] and [13, Theorem 4.2], we show that fractional sublinear operators are bounded on homogeneous weighted grand Herz-Morrey spaces with variable exponents. Then, based on the theory of variable exponent and the generalization of bounded mean oscillation (BMO) norm, we show that the boundedness of multilinear commutators of fractional sublinear operators T γ b from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) (see Theorem 2.6).

Finally, we make some conventions on notation. Throughout the whole article, we denote χ k = χ R k , R k = B k \ B k − 1 , and B k = x ∈ R n : ∣ x ∣ ≤ 2 k for k ∈ Z . We denote by C a positive constant, which is independent of the main parameters, but it may vary from line to line. (More precisely, let C be a variable independent constant, that can represent different values on different rows.) The symbol D ≲ F means that D ≤ C F . If D ≲ F and F ≲ D , we then write D ≈ F .

2 Preliminary and main results

In this section, we recall some notions, notations, and basic properties on the variable exponent Lebesgue spaces and the weighted variable exponent Lebesgue space L q ( ⋅ ) ( ω ) .

To begin with, we recall some basic definitions and results on the variable exponent Lebesgue spaces. We can refer to the monographs [14,15] for more information. Let q ( ⋅ ) : R n → [ 1 , ∞ ) be a measurable function, and the variable Lebesgue space L q ( ⋅ ) ( R n ) denotes the set of measurable functions f on R n such that, for some λ > 0 ,

ϱ q ( ⋅ ) ( f ⁄ λ ) ≔ ∫ R n ∣ f ( x ) ∣ λ q ( x ) d x < ∞ .

This set becomes a Banach function space when equipped with the Luxemburg-Nakano norm

∣ ∣ f ∣ ∣ L q ( ⋅ ) ( R n ) ≔ inf { λ > 0 : ϱ q ( ⋅ ) ( f ⁄ λ ) ≤ 1 } .

Given an open set Ω ⊆ R n , the space L loc q ( ⋅ ) ( Ω ) is defined by

L loc q ( ⋅ ) ( Ω ) = { f : f ∈ L q ( ⋅ ) ( K ) , for all compact subsets K ⊂ Ω } .

For conciseness, we denote by P ( R n ) the set of all measurable functions q ( x ) on R n with range in [ 1 , ∞ ) such that

(2.1) 1 < q − ≔ essinf x ∈ R n q ( x ) ≤ q ( x ) ≤ q + ≔ ess sup x ∈ R n q ( x ) < ∞ .

An important tool we need is the boundedness of the Hardy-Littlewood maximal operator on variable exponent function spaces. Let ℬ ( R n ) denote the set of all functions q ( ⋅ ) that satisfy q ( ⋅ ) ∈ P ( R n ) and make the Hardy-Littlewood maximal operator M bounded on L q ( ⋅ ) ( R n ) , namely,

ℬ ( R n ) ≔ { q ( ⋅ ) ∈ P ( R n ) : M is bounded on L q ( ⋅ ) ( R n ) } ,

where M is the Hardy-Littlewood maximal operator defined by

M f ( x ) = sup x ∈ R n , r > 0 r − n ∫ B ( x , r ) ∣ f ( y ) ∣ d y .

A real-valued measurable function q ( ⋅ ) : R n → ( 0 , ∞ ) is called locally log-Hölder continuous if there exists a constant C log > 0 such that

(2.2) ∣ q ( x ) − q ( y ) ∣ ≤ C log − log ( ∣ x − y ∣ ) , ∀ x , y ∈ R n , ∣ x − y ∣ < 1 2 .

If, for some q ∞ ∈ ( 0 , ∞ ) and C log > 0 , there holds

(2.3) ∣ q ( x ) − q ( 0 ) ∣ ≤ C log log ( e + 1 ∣ x ∣ ) , ∀ x ∈ R n ,

(2.4) ∣ q ( x ) − q ∞ ∣ ≤ C log log ( e + ∣ x ∣ ) , ∀ x ∈ R n ,

then we say q ( ⋅ ) is log-Hölder continuous at the origin (or has a log decay at the origin) and at infinity (or has a log decay at infinity ), respectively. The function q ( ⋅ ) is global log-Hölder continuous if q ( ⋅ ) are both locally log-Hölder continuous (2.2) and log-Hölder continuous at infinity (2.4).

The set P 0 ( R n ) consists of all measurable function q ( ⋅ ) satisfying q − > 0 and q + < ∞ . By P 0 log ( R n ) and P ∞ log ( R n ) , we denote the class of exponents q ( ⋅ ) ∈ P ( R n ) , which satisfy conditions (2.3) and (2.4), respectively. P log ( R n ) is the set of functions q ( ⋅ ) ∈ P ( R n ) satisfying conditions (2.3) and (2.4), with q ∞ ≔ lim ∣ x ∣ → ∞ q ( x ) . In particular, we note that if q ( ⋅ ) ∈ P log ( R n ) with 1 < q − ≤ q + < ∞ , then the Hardy-Littlewood maximal operator M is bounded on L q ( ⋅ ) ( R n ) , namely, q ( ⋅ ) ∈ ℬ ( R n ) [15–17].

Lemma 1

([7, Theorem 2.1]) (generalized Hölder’s inequality) Let q ( ⋅ ) ∈ P ( R n ) , f ∈ L q ( ⋅ ) ( R n ) , and g ∈ L q ′ ( ⋅ ) ( R n ) , the generalized Hölder’s inequality holds in the form

(2.5) ∫ R n ∣ f ( x ) g ( x ) ∣ d x ≤ r q ‖ f ‖ L q ( ⋅ ) ( R n ) ‖ g ‖ L q ′ ( ⋅ ) ( R n ) ,

where r q = 1 + 1 ⁄ q − − 1 ⁄ q + .

Definition 1

Let q ( ⋅ ) ∈ P ( R n ) and ω be a nonnegative measurable function on R n . Then, the weighted variable exponent Lebesgue space L q ( ⋅ ) ( ω ) is the set of all complex-valued measurable functions f such that f ω ∈ L q ( ⋅ ) . The space L q ( ⋅ ) ( ω ) is a Banach space equipped with the norm

‖ f ‖ L q ( ⋅ ) ( ω ) ≔ ‖ f ω ‖ L q ( ⋅ ) .

Lemma 2

[18, Definition 2] Let q ( ⋅ ) ∈ P ( R n ) , and a positive measurable function ω is said to be in A q ( ⋅ ) , if exists a positive constant C for all balls B in R n such that

1 B ‖ ω χ B ‖ L q ( ⋅ ) ‖ ω − 1 χ B ‖ L q ′ ( ⋅ ) ≤ C .

The variable Muckenhoupt A q ( ⋅ ) was introduced by Cruz-Uribe et al. [19]. It is easy to see that if q ( ⋅ ) ∈ P ( R n ) and ω ∈ A q ( ⋅ ) , then ω − 1 ∈ A q ′ ( ⋅ ) .

Lemma 3

[11, Lemma 6] Let q ( ⋅ ) ∈ P log ( R n ) , ω ∈ A q ( ⋅ ) , then there exist constants δ 1 , δ 2 ∈ ( 0 , 1 ) and C > 0 , such that for all balls in R n and all measurable subsets S ⊂ B ,

‖ χ S ‖ L q ( ⋅ ) ( ω ) ‖ χ B ‖ L q ( ⋅ ) ( ω ) ≤ C ∣ S ∣ ∣ B ∣ δ 1 , ‖ χ S ‖ L q ′ ( ⋅ ) ( ω − 1 ) ‖ χ B ‖ L q ′ ( ⋅ ) ( ω − 1 ) ≤ C ∣ S ∣ ∣ B ∣ δ 2 .

A locally integrable function b is called a BMO function, if it satisfies

‖ b ‖ * ≔ sup x ∈ R n , r > 0 1 ∣ B ∣ ∫ B ∣ b ( y ) − b B ∣ d y < ∞ ,

where and in the sequel B is ball-centered at x and radius of r , b B = 1 ∣ B ∣ ∫ B b ( t ) d t .

Given a positive integer m and 1 ≤ j ≤ m , we denote by C j m the family of all finite subsets σ = { σ ( 1 ) , … , σ ( j ) } of { 1 , … , m } with j different elements. For any σ ∈ C j m , the complementary sequence σ c = { 1 , … , m } \ σ . For b = ( b 1 , … , b m ) and σ = { σ ( 1 ) , … , σ ( j ) } ∈ C j m with 1 ≤ j ≤ m , we denote

b σ ≔ ( b σ ( 1 ) , … , b σ ( j ) ) , [ b ( x ) − b ( y ) ] σ ≔ [ b σ ( 1 ) ( x ) − b σ ( 1 ) ( y ) ] … [ b σ ( j ) ( x ) − b σ ( j ) ( y ) ] , [ ( b ) B − b ( y ) ] σ ≔ [ ( b σ ( 1 ) ) B − b σ ( 1 ) ( y ) ] … [ ( b σ ( j ) ) B − b σ ( j ) ( y ) ] , ‖ b σ ‖ * = ‖ b σ ( 1 ) ‖ * … ‖ b σ ( j ) ‖ * for b σ ( i ) ∈ BMO ( R n ) .

In particular,

‖ b ‖ * = ‖ b 1 ‖ * … ‖ b m ‖ * .

Lemma 4

[13, Lemma 2.4] Let ω ∈ A q ( ⋅ ) , q ( ⋅ ) ∈ ℬ ( R n ) , b i ∈ BMO ( R n ) , i = 1 , 2 , … , m , k > j ( k , j ∈ N ) . Then, we have

sup B ⊂ R n 1 ‖ χ B ‖ L q ( ⋅ ) ( ω ) ∏ i = 1 m ( b i − ( b i ) B ) χ B L q ( ⋅ ) ( ω ) ≈ ∏ i = 1 m ∥ b i ∥ *

and

∏ i = 1 m ( b i − ( b i ) B j ) χ B k L q ( ⋅ ) ( ω ) ≲ ( k − j ) m ∏ i = 1 m ∥ b i ∥ * ‖ χ B k ‖ L q ( ⋅ ) ( ω ) .

Now, we recall the definition of weighted grand of Herz-Morrey spaces with variable exponent.

Definition 2

[13, Definition 3.1] Let 0 ≤ λ < ∞ , k ∈ Z , 1 ≤ p < ∞ , q ( ⋅ ) ∈ P 0 ( R n ) , α ( ⋅ ) : R n → R , α ( ⋅ ) ∈ L ∞ ( R n ) , and θ > 0 . We define the homogeneous weighted grand Herz-Morrey space with variable exponents is denoted by

M K . p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) = { f ∈ L loc q ( ⋅ ) ( R n \ { 0 } , ω ) : ‖ f ‖ M K . p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) < ∞ } ,

where

‖ f ‖ M K . p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ≔ sup ε > 0 sup k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ‖ 2 k α ( ⋅ ) f χ k ‖ L q ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) = sup ε > 0 ε θ p ( 1 + ε ) ‖ f ‖ M K . p ( 1 + ε ) , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

The non-homogeneous weighted grand Herz-Morrey space with variable exponents is denoted by

M K p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) = { f ∈ L loc q ( ⋅ ) ( R n , ω ) : ‖ f ‖ M K p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) < ∞ } ,

where

‖ f ‖ M K p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ≔ sup ε > 0 sup k 0 ∈ N 0 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ‖ 2 k α ( ⋅ ) f χ k ‖ L q ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) = sup ε > 0 ε θ p ( 1 + ε ) ‖ f ‖ M K p ( 1 + ε ) , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Let 0 < γ < n . Then, the fractional sublinear operator T γ is defined by

T γ f ( x ) ≔ ∫ R n f ( y ) ∣ x − y ∣ n − γ d y .

Let b = ( b 1 , b 2 , … , b m ) , b j ∈ BMO ( R n ) , j ∈ { 1 , 2 , … , m } , m ∈ N , x ∉ supp f, and define the multilinear commutators of fractional sublinear operators T γ b by

(2.6) T γ b ( f ) ( x ) = ∫ R n ∏ j = 1 m ∣ b j ( x ) − b j ( y ) ∣ K ( x , y ) f ( y ) d y ,

where K ( x , y ) is the integral kernel of the operator T [1]. We are now ready to state the main results of the article, Theorems 1 and 2. Theorem 1 is a generalization of theorems [12, Theorem 3.2] and [13, Theorem 4.2]. Motivated by [12] and [13], we show that fractional sublinear operators are bounded on homogeneous weighted grand Herz-Morrey spaces.

Theorem 1

Let 1 < p < ∞ , q i ( ⋅ ) ∈ P log ( R n ) ( i = 1,2 . ) , ω ∈ A q ( ⋅ ) , α ( ⋅ ) ∈ L ∞ ( R n ) ∩ P 0 log ( R n ) ∩ P ∞ log ( R n ) ∩ P 0 ( R n ) such that − n δ 1 < α ( 0 ) , α ∞ < n δ 2 − γ , where 0 < δ 1 , δ 2 < 1 are the constants in Lemma 3. Let 1 q 2 ( ⋅ ) = 1 q 1 ( ⋅ ) − γ n and suppose that fractional sublinear operator T γ satisfies the size conditions

(2.7) ∣ T γ f ( x ) ∣ ≲ ‖ f ‖ L 1 ( R n ) ⁄ ∣ x ∣ n − γ ,

when supp f ⊆ R k and ∣ x ∣ ≥ 2 k + 1 with k ∈ Z and

(2.8) ∣ T γ f ( x ) ∣ ≲ 2 − k ( n − γ ) ‖ f ‖ L 1 ( R n ) ,

when supp f ⊆ R k and ∣ x ∣ ≤ 2 k − 2 with k ∈ Z . Then, if T γ is bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , T γ is bounded from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Corollary 1

Let p , α ( ⋅ ) and q i ( ⋅ ) , i = 1,2 as in Theorem 1. If a sublinear operator T γ satisfies the condition

(2.9) ∣ T γ f ( x ) ∣ ≲ ∫ R n ∣ f ( y ) ∣ ∣ x − y ∣ n − γ d y , x ∉ supp f ,

for any integrable function f with compact support, and T γ is bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , then T γ is bounded from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Then, we introduce multilinear commutators of T γ b that are bounded from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) , which generalizes the previous work of Zhang et al. [13, Theorem 5.1] on homogeneous weighted grand Herz-Morrey spaces.

Theorem 2

Let b = ( b 1 , b 2 , … , b m ) , b j ∈ BMO ( R n ) , j ∈ { 1 , 2 , … , m } , m ∈ N . For 1 < p < ∞ , q i ( ⋅ ) ∈ P log ( R n ) ( i = 1 , 2 … ) , ω ∈ A q ( ⋅ ) , α ( ⋅ ) ∈ L ∞ ( R n ) ∩ P 0 log ( R n ) ∩ P ∞ log ( R n ) ∩ P 0 ( R n ) , such that − n δ 1 < α ( 0 ) , α ∞ < n δ 2 − γ , where 0 < δ 1 , δ 2 < 1 are the constants in Lemma 3. Suppose that sublinear operator T γ is satisfying the size conditions (2.7) and (2.8). If T γ b is bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , then T γ b is bounded from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Corollary 2

Let p , α ( ⋅ ) and q ( ⋅ ) as in Theorem  2. If a fractional sublinear operator T γ satisfies condition (2.9), for any integrable function f with compact support, and T γ b is bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , then T γ b is bounded from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

As applications, we have established the boundedness of I λ b from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) and the boundedness of M λ b from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) . These applications are a natural extension [20].

Remark 1

For any 0 < λ < n and x ∈ R n , the Riesz potential operator I λ is defined by

(2.10) I λ f ( x ) ≔ ∫ R n f ( y ) ∣ x − y ∣ n − λ d y .

It is known that the Riesz potential operator I λ satisfies the size condition (2.9), and if I λ b is bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , then I λ b is bounded from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Remark 2

For any 0 < λ < n and x ∈ R n , the fractional maximal function M λ is defined by

(2.11) M λ f ( x ) ≔ sup r > 0 r λ − n ∫ R n ∣ f ( y ) ∣ d y .

Then, the fractional maximal function M λ also satisfies (2.9). Moreover, we have

M λ ( f ) ≲ I λ ( ∣ f ∣ ) .

If M λ b is bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , then M λ b is bounded from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

3 Proofs of main results

The section is devoted to the proofs of Theorems 1 and 2. To this end, we state several technical lemmas as follows.

Lemma 5

[11, Lemma 6] Let X be a Banach function space on R n . If f ∈ X , then we have

∣ B ∣ ‖ χ B ‖ X − 1 ‖ χ B ‖ X ′ − 1 ≤ 1 ,

where X ′ denotes the associated space of X .

Lemma 6

[13, Lemma 4.1] Let q ( ⋅ ) ∈ P 0 ( R n ) , ω be a weight, λ ∈ [ 0 , ∞ ) , α ( ⋅ ) : R n → R , α ( ⋅ ) ∈ L ∞ ( R n ) , k ∈ Z , and 1 ≤ p < ∞ . If α ( ⋅ ) is log-Hölder continuous both at the origin and at infinity, then

‖ f ‖ M K . p , q ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ≈ max sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ‖ f χ k ‖ L q ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , × sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ − 1 2 k α ( 0 ) p ( 1 + ε ) ‖ f χ k ‖ L q ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ‖ f χ k ‖ L q ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) .

Proof of Theorem 1

We show this theorem by borrowing some ideas from the proofs of [12, Theorem 3.2] and [13, Theorem 4.2]. Let f ∈ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) , and we decompose

f ( x ) = ∑ l = − ∞ ∞ f ( x ) χ l ( x ) .

From Lemma 6, we have

(3.1) ‖ T γ f ‖ M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ≈ max sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ‖ T γ ( f ) χ k ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , × sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ − 1 2 k α ( 0 ) p ( 1 + ε ) ‖ T γ ( f ) χ k ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ‖ T γ ( f ) χ k ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≕ max { E , F + G } ,

where

E = sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) χ k ∑ l = − ∞ ∞ T γ ( f χ l ) L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , F = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ − 1 2 k α ( 0 ) p ( 1 + ε ) χ k ∑ l = − ∞ ∞ T γ ( f χ l ) L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , G = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) χ k ∑ l = − ∞ ∞ T γ ( f χ l ) L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) .

The next is the same as the way in [13, Theorem 4.2] to be handled. Since to estimate F is essentially similar to estimate E , it suffices for us to show that E and G are bounded on homogeneous weighted grand Herz-Morrey space. It is easy to see that

E ≲ ∑ i = 1 3 E i , G ≲ ∑ i = 1 3 G i ,

where

E 1 = sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = − ∞ k − 2 ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , E 2 = sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = k − 1 k + 1 ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , E 3 = sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = k + 2 ∞ ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , G 1 = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = − ∞ k − 2 ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , G 2 = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = k − 1 k + 1 ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , G 3 = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = k + 2 ∞ ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) .

We first consider E 1 . For a.e. x ∈ R k with k ∈ Z and l ≤ k − 2 , from size condition of T γ , generalized Hölder’s inequality and by repeating the proof of [13, Formula (4.3)], we obtain

(3.2) ∣ T γ ( f χ l ) ( x ) ∣ ≲ 2 − k ( n − γ ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) .

By this, Lemmas 2, 3, [11, Formula (14)], and 5, we obtain

(3.3) ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) ≲ 2 − k ( n − γ ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) ‖ χ k ‖ L q 2 ( ⋅ ) ( ω ) ≲ 2 − k ( n − γ ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) ∣ B k ∣ ‖ χ k ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) − 1 ≲ 2 k γ ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) ‖ χ l ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) − 1 ‖ χ l ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) ‖ χ k ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) ≲ 2 k γ 2 ( l − k ) n δ 2 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) ‖ χ l ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) − 1 ≲ 2 k γ 2 ( l − k ) n δ 2 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 l ( n − γ ) ‖ χ l ‖ L q 2 ( ⋅ ) ( ω ) − 1 ‖ χ l ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) − 1 ≲ 2 ( l − k ) ( n δ 2 − γ ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) .

Let v ≔ n δ 2 − γ − α ( 0 ) > 0 . From (3.3), Hölder’s inequality, Fubini’s theorem for series, 2 − p ( 1 + ε ) < 2 − p , α ∞ < n δ 2 − γ , and Lemma 6, we obtain that

E 1 ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = − ∞ k − 2 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 ( l − k ) ( n δ 2 − γ ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = − ∞ k − 2 2 α ( 0 ) l ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 v ( l − k ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = − ∞ k − 2 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 2 v p ( 1 + ε ) ( l − k ) ⁄ 2 ∑ l = − ∞ k − 2 2 v ( l − k ) ( p ( 1 + ε ) ) ′ ⁄ 2 p ( 1 + ε ) ⁄ ( p ( 1 + ε ) ) ′ 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = − ∞ k − 2 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 2 v ( l − k ) p ( 1 + ε ) ⁄ 2 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) ∑ k = l + 2 k 0 2 v ( l − k ) p ( 1 + ε ) ⁄ 2 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) ∑ k = l + 2 k 0 2 v ( l − k ) p ⁄ 2 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

For the term E 2 , T γ is bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , − 1 ≤ l − k ≤ 1 , and Lemma 6, we conclude that

E 2 ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = k − 1 k + 1 ‖ T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = k − 1 k + 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 + 1 2 k α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 + 1 2 ( k − l ) α ( 0 ) p ( 1 + ε ) 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 + 1 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 1 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ max sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , sup ε > 0 sup k 0 = 1 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ l = 0 k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ max sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , sup ε > 0 sup k 0 = 1 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ l = 0 k 0 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ max sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ l = 0 k 0 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≈ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ,

the penultimate inequality holds because when l = 0 , we have 2 0 α ( 0 ) = 2 0 α ∞ ; and when l = 0 , we have 2 α ∞ = C 2 α ( 0 ) .

Now, we turn to estimate E 3 . For each k ∈ Z , l ≥ k + 2 and a.e. x ∈ R k , the size condition of T γ , and generalized Hölder’s inequality, by repeating the proof of (3.2), we obtain

(3.4) ∣ T γ ( f χ l ) ( x ) ∣ ≲ 2 − l ( n − γ ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) .

Combining this, Lemma 3 and [11, Formula (14)], by an estimate similar to that of (3.3), we have

(3.5) ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) ≲ 2 ( k − l ) n δ 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) .

Splitting E 3 by means of Minkowski’s inequality, we deduce

E 3 ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = k + 2 − 1 ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) + sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = 0 ∞ ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≕ E 31 + E 32 .

For the convenience of proof, define d ≔ n δ 1 + α ( 0 ) > 0 . Combining (3.5), Hölder’s inequality, l > k + 2 , Fubini’s theorem for series, 2 − p ( 1 + ε ) < 2 − p , − n δ 1 < α ( 0 ) , and Lemma 6, by an estimate similar to that of E 1 , we obtain that

E 31 ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = k + 2 − 1 2 k α ( 0 ) 2 ( k − l ) n δ 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = k + 2 − 1 2 l α ( 0 ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 ( n δ 1 + α ( 0 ) ) ( k − l ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = k + 2 − 1 2 l α ( 0 ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 d ( k − l ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) ∑ k = − ∞ l − 2 2 d ( k − l ) p ⁄ 2 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

To deal with E 32 , by (3.5), k 0 ≤ 0 , Hölder’s inequality, − n δ 1 < α ( 0 ) , α ∞ < n δ 2 − γ , and note that h ≔ n δ 1 + α ∞ > 0 , and Lemma 6, we obtain

E 32 ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k ( α ( 0 ) + n δ 1 ) p ( 1 + ε ) ∑ l = 0 ∞ 2 − l n δ 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = 0 ∞ 2 l α ∞ ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 − l h p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = 0 ∞ 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) ∑ l = 0 ∞ 2 − l h ( p ( 1 + ε ) ) ′ p ( 1 + ε ) ⁄ ( p ( 1 + ε ) ) ′ 1 p ( 1 + ε ) ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ,

which together with the estimate of E 31 implies that

E 3 ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Thus,

E ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω )

holds true.

We now prove that G ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

To prove G ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) , we borrow some ideas from those used in the proof of [13, Theorem 4.2].

Splitting G 1 by means of Minkowski’s inequality, we deduce

G 1 ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = − ∞ − 1 ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) + sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = 0 k − 2 ‖ χ k T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≕ G 11 + G 12 .

For the term G 11 , by (3.3) and using the fact that e ≔ n δ 2 − γ − α ∞ > 0 , Hölder’s inequality, v = n δ 2 − α ( 0 ) − γ > 0 , and Lemma 6, we obtain that

G 11 ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = − ∞ − 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 ( l − k ) ( n δ 2 − γ ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 − k e p ( 1 + ε ) ∑ l = − ∞ − 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 l ( n δ 2 − γ ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 l ( n δ 2 − γ ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 2 l α ( 0 ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 l v p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) ∑ l = − ∞ − 1 2 l v ( p ( 1 + ε ) ) ′ p ( 1 + ε ) ⁄ ( p ( 1 + ε ) ) ′ 1 p ( 1 + ε ) ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

For the term G 12 , by (3.3), Hölder’s inequality, e ≔ n δ 2 − γ − α ∞ > 0 , l ≤ k − 2 , Fubini’s theorem for series, 2 − p ( 1 + ε ) < 2 − p , and Lemma 6, by an estimate similar to that of E 1 , we have

G 12 ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

For G 2 , from T γ bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , − 1 ≤ l − k ≤ 1 , 2 − α ∞ = C 2 − α ( 0 ) , and Lemma 6, we conclude that

G 2 ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = k − 1 k + 1 ‖ T γ ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = k − 1 k + 1 ‖ ( f χ l ) ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − 1 k 0 + 1 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 1 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − 1 k 0 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 1 , k 0 ∈ Z 2 − k 0 λ ε θ 2 − 1 α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ l = 0 k 0 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 1 , k 0 ∈ Z 2 − k 0 λ ε θ 2 − 1 α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ l = 0 k 0 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ 2 − 1 α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ l = 0 k 0 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ max sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ l = 0 k 0 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≈ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

For G 3 , combining (3.5), Hölder’s inequality, l ≥ k + 2 , Fubini’s theorem for series, 2 − p ( 1 + ε ) < 2 − p , − n δ 1 < α ( 0 ) , α ∞ < n δ 2 − γ , and note that h ≔ n δ 1 + α ∞ > 0 , k > 0 , and Lemma 6, by an estimate similar to that of E 1 , we obtain that

G 3 ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Therefore, combining the estimates for E and G , we deduce that

‖ T γ f ‖ M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ≲ ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ,

which implies that T γ is bounded from M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) to M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) , which holds true and hence completes the proof of Theorem 1.□

Proof of Theorem 2

We show this theorem by borrowing some ideas from the proofs of Theorem 1 and [13, Theorem 5.1]. Let f ∈ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) , and we decompose

f ( x ) = ∑ l = − ∞ ∞ f ( x ) χ l ( x ) .

From Lemma 6, we have

‖ T γ b f ‖ M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ≈ max sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ‖ T γ b ( f ) χ k ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ − 1 2 k α ( 0 ) p ( 1 + ε ) ‖ T γ b ( f ) χ k ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) + ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ‖ T γ b ( f ) χ k ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≕ max { A , N + S } ,

where

A = sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) χ k ∑ l = − ∞ ∞ T γ b ( f χ l ) L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , N = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ − 1 2 k α ( 0 ) p ( 1 + ε ) χ k ∑ l = − ∞ ∞ T γ b ( f χ l ) L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , S = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) χ k ∑ l = − ∞ ∞ T γ b ( f χ l ) L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) .

The next is the same as the way the (3.1) is handled. Since to estimate N is essentially similar to estimate A , it suffices for us to show that A and S are bounded on homogeneous grand weighted Herz-Morrey spaces. It is easy to see that

A ≲ ∑ i = 1 3 A i , S ≲ ∑ i = 1 3 S i ,

where

A 1 = sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = − ∞ k − 2 ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , A 2 = sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = k − 1 k + 1 ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , A 3 = sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = k + 2 ∞ ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , S 1 = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = − ∞ k − 2 ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , S 2 = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = k − 1 k + 1 ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) , S 3 = sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = k + 2 ∞ ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) .

First, we consider A 1 . For a.e. x ∈ R k with k ∈ Z , and l ≤ k − 2 , from size condition of T , (2.6), and generalized Hölder’s inequality, we have

(3.6) ∣ T γ b ( f χ l ) ( x ) ∣ ≲ 2 − k ( n − γ ) ∫ R l ∏ j = 1 m ∣ b j ( x ) − b j ( y ) ∣ ∣ f ( y ) ∣ d y = 2 − k ( n − γ ) ∫ R l ∏ j = 1 m ∣ b j ( x ) − ( b j ) B l + ( b j ) B l − b j ( y ) ∣ ∣ f ( y ) ∣ d y ≲ 2 − k ( n − γ ) ∑ j = 0 m ∑ σ ∈ C j m ∣ [ b ( x ) − ( b ) B l ] σ ∣ ∫ R l ∣ [ b ( y ) − ( b ) B l ] σ c ∣ ∣ f ( y ) ∣ d y ≲ 2 − k ( n − γ ) ∑ j = 0 m ∑ σ ∈ C j m ∣ [ b ( x ) − ( b ) B l ] σ ∣ ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ [ b ( y ) − ( b ) B l ] σ c χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) .

By Lemmas 4, 2, 3, [11, Formula (14)], and 5, we obtain

(3.7) ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) ≲ ‖ b ‖ * 2 − k ( n − γ ) ( k − l ) m ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) ‖ χ k ‖ L q 2 ( ⋅ ) ( ω ) ≲ ‖ b ‖ * 2 k γ ( k − l ) m ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) ‖ χ k ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) − 1 = ‖ b ‖ * 2 k γ ( k − l ) m ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) ‖ χ l ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) − 1 ‖ χ l ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) ‖ χ k ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) ≲ ‖ b ‖ * ( k − l ) m 2 k γ 2 ( l − k ) n δ 2 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) ‖ χ l ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) − 1 ≲ ‖ b ‖ * ( k − l ) m 2 k γ 2 ( l − k ) n δ 2 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) 2 l ( n − γ ) ‖ χ l ‖ L q 2 ( ⋅ ) ( ω ) − 1 ‖ χ l ‖ L q ′ 2 ( ⋅ ) ( ω − 1 ) − 1 ≲ ‖ b ‖ * ( k − l ) m 2 ( l − k ) ( n δ 2 − γ ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) .

From (3.7), Hölder’s inequality, v ≔ n δ 2 − α ( 0 ) − γ > 0 , Fubini’s theorem for series, 2 − p ( 1 + ε ) < 2 − p , and repeating the proof of [13, Theorem 5.1] with some slight modifications, we obtain that

A 1 ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = − ∞ k − 2 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 2 v p ( 1 + ε ) ( l − k ) ⁄ 2 × ∑ l = − ∞ k − 2 ( k − l ) m ( p ( 1 + ε ) ) ′ 2 v ( l − k ) ( p ( 1 + ε ) ) ′ ⁄ 2 p ( 1 + ε ) ⁄ ( p ( 1 + ε ) ) ′ 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) ∑ k = l + 2 k 0 2 v ( l − k ) p ⁄ 2 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

For A 2 , T γ b is bounded from L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) , 2 0 α ( 0 ) = 2 0 α ∞ , 2 α ∞ = C 2 α ( 0 ) and Lemma 6; similar to the estimation for E 2 , we conclude that

A 2 ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 1 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

For A 3 , by repeating the proof of [13, Formula (5.3)], we obtain

(3.8) ∣ T γ b ( f χ l ) ( x ) ∣ ≲ 2 − l ( n − γ ) ∑ j = 0 m ∑ σ ∈ C j m ∣ [ b ( x ) − ( b ) B l ] σ ∣ ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ‖ [ b ( y ) − ( b ) B l ] σ c χ l ‖ L q ′ 1 ( ⋅ ) ( ω − 1 ) .

This together with Lemmas 4, 3 and [11, Formula (14)] further implies that

(3.9) ‖ χ k T γ b f χ l ‖ L q 2 ( ⋅ ) ( ω ) ≲ ‖ b ‖ * ( l − k ) m 2 ( k − l ) n δ 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) .

The next is the same as the way the E 3 is handled. Splitting A 3 by means of Minkowski’s inequality, we have

A 3 ≲ sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = k + 2 − 1 ‖ χ k T b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) + sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k α ( 0 ) p ( 1 + ε ) ∑ l = 0 ∞ ‖ χ k T b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≕ A 31 + A 32 .

For the term A 31 , by (3.9), Hölder’s inequality, d = n δ 1 + α ( 0 ) > 0 , Fubini’s theorem for series, 2 − p ( 1 + ε ) < 2 − p , Lemma 6, and repeating the proof of [13, Theorem 5.1] with some slight modifications, we obtain that

A 31 ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = k + 2 − 1 2 α ( 0 ) l ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ( l − k ) m 2 d ( k − l ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 ∑ l = k + 2 − 1 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 2 d p ( 1 + ε ) ( k − l ) ⁄ 2 × ∑ l = k + 1 − 1 ( l − k ) m ( p ( 1 + ε ) ) ′ 2 d ( k − l ) ( p ( 1 + ε ) ) ′ ⁄ 2 p ( 1 + ε ) ⁄ ( p ( 1 + ε ) ) ′ 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 α ( 0 ) l p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) ∑ k = − ∞ l − 2 2 d ( k − l ) p ⁄ 2 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ k 0 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

For the term A 32 , applying (3.9), Hölder’s inequality, h = n δ 1 + α ∞ > 0 , and Lemma 6, we deduce that

A 32 ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = − ∞ k 0 2 k ( α ( 0 ) + n δ 1 ) p ( 1 + ε ) ∑ l = 0 ∞ ( l − k ) m 2 − l n δ 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = 0 ∞ 2 l α ∞ ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ( l − k ) m 2 − l h p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 ≤ 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = 0 ∞ 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) × ∑ l = 0 ∞ ( l − k ) m ( p ( 1 + ε ) ) ′ 2 − l h ( p ( 1 + ε ) ) ′ p ( 1 + ε ) ⁄ ( p ( 1 + ε ) ) ′ 1 p ( 1 + ε ) ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Therefore, combining the estimates for A 31 and A 32 deduces that

A 3 ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

Thus,

A ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω )

holds true.

Next, we consider S 1 . The next is the same as the way the E 3 is handled. Splitting S 1 by means of Minkowski’s inequality, we deduce

S 1 ≲ sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = − ∞ − 1 ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) + sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k α ∞ p ( 1 + ε ) ∑ l = 0 k − 2 ‖ χ k T γ b ( f χ l ) ‖ L q 2 ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≕ S 11 + S 12 .

To show S 11 using (3.7) and e = n δ 2 − α ∞ − γ > 0 , Hölder’s inequality, and using the fact that v = n δ 2 − α ( 0 ) − γ > 0 , Lemma 6 and by repeating the proof of [13, Theorem 5.1] with regular modifications, we obtain that

S 11 ≲ ‖ b ‖ * sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 k ( α ∞ − n δ 2 − γ ) p ( 1 + ε ) × ∑ l = − ∞ − 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ( k − l ) m 2 l ( n δ 2 − γ ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ k = 0 k 0 2 − k e p ( 1 + ε ) × ∑ l = − ∞ − 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ( k − l ) m 2 l ( n δ 2 − γ ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ( k − l ) m 2 l ( n δ 2 − γ ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 2 l α ( 0 ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) ( k − l ) m 2 l v p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * sup ε > 0 sup k 0 > 0 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − ∞ − 1 2 l α ( 0 ) p ( 1 + ε ) ‖ f χ l ‖ L q 1 ( ⋅ ) ( ω ) p ( 1 + ε ) ∑ l = − ∞ − 1 ( k − l ) m ( p ( 1 + ε ) ) ′ 2 l v ( p ( 1 + ε ) ) ′ p ( 1 + ε ) ⁄ ( p ( 1 + ε ) ) ′ 1 p ( 1 + ε ) ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

For the term S 12 , by Hölder’s inequality, e = n δ 2 − α ∞ − γ > 0 , k > 0 , Fubini’s theorem for series, 2 − p ( 1 + ε ) < 2 − p , and Lemma 6, by an estimate similar to that of A 1 , we have

S 12 ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

To deal with S 2 , in the view of the L q 1 ( ⋅ ) ( ω ) to L q 2 ( ⋅ ) ( ω ) boundedness of T γ b , − 1 ≤ l − k ≤ 1 , 2 − α ∞ = C 2 − α ( 0 ) , and Lemma 6, similar to the estimation for G 2 , we obtain

S 2 ≲ ‖ b ‖ * sup ε > 0 sup k 0 > 1 , k 0 ∈ Z 2 − k 0 λ ε θ ∑ l = − 1 k 0 2 l α ∞ p ( 1 + ε ) ‖ f χ l ‖ L q ( ⋅ ) ( ω ) p ( 1 + ε ) 1 p ( 1 + ε ) ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) .

The estimate for S 3 can be obtained by similar way to A 31 and using the fact that h = n δ 1 + α ∞ > 0 , k > 0 .

From the estimates of A and S , we see that

‖ T γ b f ‖ M K . p , q 2 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω ) ≲ ‖ b ‖ * ‖ f ‖ M K . p , q 1 ( ⋅ ) α ( ⋅ ) , λ , θ ( ω )

and hence completes the proof of Theorem 2.□

  1. Funding information: This study was supported by National Natural Science Foundation of China (Grant No. 12361019), University-Level Research Project of YiLi Normal University (Grant No. 2023YSYB003), and the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant No. 2021D01C463).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. WZ and JZ collected relevant literature. JZ provided the relevant proof ideas and, alongside ZY and WZ, proved the relevant conclusions. ZY and WZ completed the first draft. ZY and JZ revised and improved the first draft. ZY prepared the manuscript with contributions from all co-authors.

  3. Conflict of interest: The authors declare that there is no financial or nonfinancial conflicts of interest regarding the publication of this article.

  4. Data availability statement: The authors declare that this article has no associated data.

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Received: 2024-07-25
Revised: 2025-02-16
Accepted: 2025-03-29
Published Online: 2025-06-03

© 2025 the author(s), published by De Gruyter

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

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  61. Tilings, sub-tilings, and spectral sets on p-adic space
  62. The higher mapping cone axiom
  63. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  64. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  65. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  66. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  67. On a nonlinear boundary value problems with impulse action
  68. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  69. Decompositions of the extended Selberg class functions
  70. Subharmonic functions and associated measures in ℝn
  71. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  72. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  73. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  74. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  75. Green's graphs of a semigroup
  76. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  77. Infinitely many solutions for a class of Kirchhoff-type equations
  78. On an uncertainty principle for small index subgroups of finite fields
https://doi.org/10.1515/math-2025-0143
Schlagwörter für diesen Artikel
fractional sublinear operator; multilinear commutators; grand Herz-Morrey space; weight
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  65. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
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  78. On an uncertainty principle for small index subgroups of finite fields
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