Home Mathematics Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
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Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents

  • Shengrong Wang and Jingshi Xu EMAIL logo
Published/Copyright: May 27, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

If vector-valued sublinear operators satisfy the size condition and the vector-valued inequality on weighted Lebesgue spaces with variable exponent, then we obtain their boundedness on weighted Herz-Morrey spaces with variable exponents.

Keywords: subilinear operator; vector-valued inequality; Muckenhoupt weight; variable exponent; Herz-Morrey space
MSC 2010: 42B25; 42B35

1 Introduction

Since the fundamental paper [1] by Kováčik and Rákosník appeared in 1991, the Lebesgue spaces with variable exponent have been extensively studied by many authors; see [2,3,4]. Motivated by applications to fluid dynamics, image restoration and partial differential equations with non-standard growth conditions, many variable spaces were introduced, such as Besov and Triebel-Lizorkin spaces with variable exponents [5,6,7, 8,9,10, 11,12], Besov-type and Triebel-Lizorkin-type spaces with variable exponents [13,14,15, 16,17,18, 19,20,21], Hardy spaces with variable exponent [22], Bessel potential spaces with a variable exponent [23,24] and Morrey spaces with variable exponents [25]. The list is not exhausted.

Herz spaces were introduced in [26]. After that the theory of these spaces had a remarkable development in part due to its usefulness in applications. For instance, they appear in the characterization of multipliers on Hardy spaces [27], in the summability of Fourier transforms [28] and in regularity theory for elliptic equations in divergence form [29]. For more details of the theory and applications of Herz spaces, we refer the reader to the monograph [30]. Herz spaces with variable exponents were studied in [31,32, 33,34]. As a generalization, Herz-Morrey spaces with variable exponents were introduced in [35]. Indeed, Izuki [35] obtained the boundedness of vector-valued sublinear operators satisfying a size condition on Herz-Morrey spaces with variable exponent M K ˙ q , p ( ⋅ ) α , λ ( R n ) . Furthermore, Dong and Xu of the paper generalized Izuki’s result for the M K ˙ q , p ( ⋅ ) α ( ⋅ ) , λ ( R n ) in [36]. Wang and Shu [37] obtained the boundedness of some sublinear operators on weighted variable Herz-Morrey spaces M K ˙ q , p ( ⋅ ) α , λ ( R n , w ) .

Motivated by the mentioned work, in this paper, we will prove the boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) . As a result, we obtain the boundedness of vector-valued Hardy-Littlewood maximal operator on weighted Herz-Morrey spaces with variable exponents M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) . It is well known that the boundedness of vector-valued Hardy-Littlewood maximal operator on non-weighted and weighted Lebesgue spaces play a key role in the theory of function spaces. The paper is organized as follows. In Section 2, we collect some notations and state the main result. The proof of the main result is given in Section 3.

2 Notations and main result

In this section, we first recall some definitions and notations, then we state our result. Let p ( ⋅ ) be a measurable function on R n taking values in [ 1 , ∞ ) , then the Lebesgue space with variable exponent L p ( ⋅ ) ( R n ) is defined by

L p ( ⋅ ) ( R n ) ≔ f is measurable: ∫ R n ∣ f ( x ) ∣ λ p ( x ) d x < ∞ for some λ > 0 .

The Lebesgue space L p ( ⋅ ) ( R n ) becomes a Banach function space equipped with the norm

∥ f ∥ L p ( ⋅ ) ≔ inf λ > 0 : ∫ R n ∣ f ( x ) ∣ λ p ( x ) d x ≤ 1 .

The space L loc p ( ⋅ ) ( R n ) is defined by L loc p ( ⋅ ) ( R n ) ≔ { f : f χ K ∈ L p ( ⋅ ) ( R n ) for all compact subsets K ⊂ R n } , where and what follows, χ S denotes the characteristic function of a measurable set S ⊂ R n . Let p ( ⋅ ) : R n → ( 0 , ∞ ) , we denote p − ≔ ess inf x ∈ R n p ( x ) , p + ≔ ess sup x ∈ R n p ( x ) . The set P ( R n ) consists of all measurable function p ( ⋅ ) satisfying p − > 1 and p + < ∞ ; P 0 ( R n ) consists of all measurable function p ( ⋅ ) satisfying p − > 0 and p + < ∞ . L p ( ⋅ ) can be similarly defined as above for p ( ⋅ ) ∈ P 0 ( R n ) . p ′ ( ⋅ ) is the conjugate exponent of p ( ⋅ ) ∈ P ( R n ) , which means 1 / p ( ⋅ ) + 1 / p ′ ( ⋅ ) = 1 .

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

∥ f ∥ L p ( ⋅ ) ( w ) ≔ ∥ f w ∥ L p ( ⋅ ) .

Definition 1

Let α ( ⋅ ) be a real-valued measurable function on R n .

  1. The function α ( ⋅ ) is locally log-Hölder continuous if there exists a constant C 1 such that

    ∣ α ( x ) − α ( y ) ∣ ≤ C 1 log ( e + 1 / ∣ x − y ∣ ) , x , y ∈ R n , ∣ x − y ∣ < 1 2 .

  2. The function α ( ⋅ ) is log-Hölder continuous at the origin if there exists a constant C 2 such that

    ∣ α ( x ) − α ( 0 ) ∣ ≤ C 2 log ( e + 1 / ∣ x ∣ ) , ∀ x ∈ R n .

    Denote by P 0 log ( R n ) the set of all log-Hölder continuous functions at the origin.

  3. The function α ( ⋅ ) is log-Hölder continuous at infinity if there exists α ∞ ∈ R and a constant C 3 such that

    ∣ α ( x ) − α ∞ ∣ ≤ C 3 log ( e + ∣ x ∣ ) , ∀ x ∈ R n .

    Denote by P ∞ log ( R n ) the set of all log-Hölder continuous functions at infinity.

  4. The function α ( ⋅ ) is global log-Hölder continuous if α ( ⋅ ) are both locally log-Hölder continuous and log-Hölder continuous at infinity. Denote by P log ( R n ) the set of all global log-Hölder continuous functions.

Definition 2

Let p ( ⋅ ) ∈ P ( R n ) , a positive measurable function w is said to be in A p ( ⋅ ) , if exists a positive constant C for all balls B in R n such that

1 ∣ B ∣ ∥ w χ B ∥ L p ( ⋅ ) ∥ w − 1 χ B ∥ L p ′ ( ⋅ ) ≤ C .

Remark 1

The variable Muckenhoupt A p ( ⋅ ) was introduced by Cruz-Uribe et al. in [38]. For more details, see [38, 39,40, 41,42]. It is easy to see that if p ( ⋅ ) ∈ P ( R n ) and w ∈ A p ( ⋅ ) , then w − 1 ∈ A p ′ ( ⋅ ) .

Let f ∈ L loc 1 ( R n ) . Then the standard Hardy-Littlewood maximal function of f is defined by

M f ( x ) ≔ sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ f ( y ) ∣ d y , ∀ x ∈ R n ,

where the supremum is taken over all balls Q containing x in R n .

In general, the Hardy-Littlewood maximal operator is not bounded on weighted variable Lebesgue spaces. But one has the following lemma [38, Theorem 1.5, p. 746].

Lemma 1

If p ( ⋅ ) ∈ P log ( R n ) ∩ P ( R n ) and w ∈ A p ( ⋅ ) , then there is a positive constant C such that for each f ∈ L p ( ⋅ ) ( w ) ,

∥ ( M f ) w ∥ L p ( ⋅ ) ≤ C ∥ f w ∥ L p ( ⋅ ) .

To give the definitions of the weighted Herz-Morrey space with variable exponents, we use the following notations. For each k ∈ Z we define B k ≔ { x ∈ R n : ∣ x ∣ ≤ 2 k } , D k ≔ B k \ B k − 1 , χ k ≔ χ D k , χ ˜ m = χ m , m ≥ 1 , χ ˜ 0 = χ B 0 . We also need the notation of the variable mixed sequence space ℓ q ( ⋅ ) ( L p ( ⋅ ) ) , which is first defined by Almeida and Hästö in [5]. Let w be a nonnegative measurable function. Given a sequence of functions { f j } j ∈ Z , define the modular

ρ ℓ q ( ⋅ ) ( L p ( ⋅ ) ( w ) ) ( { f j } j ) ≔ ∑ j ∈ Z inf λ j : ∫ R n ∣ f j ( x ) w ( x ) ∣ λ j 1 q ( x ) p ( x ) d x ≤ 1 ,

where λ 1 / ∞ = 1 . If q + < ∞ or q ( ⋅ ) ≤ p ( ⋅ ) , the above can be written as

ρ ℓ q ( ⋅ ) ( L p ( ⋅ ) ( w ) ) ( { f j } j ) = ∑ j ∈ Z ∥ ∣ f j w ∣ q ( ⋅ ) ∥ L p ( ⋅ ) q ( ⋅ ) .

The norm is

∥ { f j } j ∥ ℓ q ( ⋅ ) ( L p ( ⋅ ) ( w ) ) ≔ inf { μ > 0 : ρ ℓ q ( ⋅ ) ( L p ( ⋅ ) ( w ) ) ( { f j / μ } j ) ≤ 1 } .

Definition 3

Let p ( ⋅ ) , q ( ⋅ ) ∈ P 0 ( R n ) , λ ∈ [ 0 , ∞ ) . Let α ( ⋅ ) be a bounded real-valued measurable function on R n . The homogeneous weighted Herz-Morrey space M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) and non-homogeneous weighted Herz-Morrey space M K q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) are defined, respectively, by

M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ≔ { f ∈ L loc p ( ⋅ ) ( R n ⧹ { 0 } , w ) : ∥ f ∥ M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) < ∞ }

and

M K q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ≔ { f ∈ L loc p ( ⋅ ) ( R n , w ) : ∥ f ∥ M K q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) < ∞ } ,

where

∥ f ∥ M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ≔ sup L ∈ Z 2 − L λ ∥ ( 2 α ( ⋅ ) k f χ k ) k ≤ L ∥ ℓ q ( ⋅ ) ( L p ( ⋅ ) ( w ) )

and

∥ f ∥ M K q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ≔ sup L ∈ N 0 2 − L λ ∥ ( 2 α ( ⋅ ) k f χ ˜ k ) k = 0 L ∥ ℓ q ( ⋅ ) ( L p ( ⋅ ) ( w ) ) .

For any quantities A and B , if there exists a constant C > 0 such that A ≤ C B , we write A ≲ B . If A ≲ B and B ≲ A , we write A ≈ B . The following Proposition 1 is from [43, Proposition 1, pp. 5–6].

Proposition 1

Let p ( ⋅ ) , q ( ⋅ ) ∈ P 0 ( R n ) , w be a weight, λ ∈ [ 0 , ∞ ) , and α ( ⋅ ) ∈ L ∞ ( R n ) .

  1. If α ( ⋅ ) , q ( ⋅ ) ∈ P 0 log ( R n ) ∩ P ∞ log ( R n ) , then for any f ∈ L loc p ( ⋅ ) ( R n \ { 0 } , w ) ,

    ∥ f ∥ M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ≈ max { sup L ⩽ 0 , L ∈ Z 2 − L λ ∥ ( 2 k α ( 0 ) f χ k ) k ≤ L ∥ ℓ q 0 ( L p ( ⋅ ) ( w ) ) , sup L > 0 , L ∈ Z [ 2 − L λ ∥ ( 2 k α ( 0 ) f χ k ) k < 0 ∥ ℓ q 0 ( L p ( ⋅ ) ( w ) ) + 2 − L λ ∥ ( 2 k α ∞ f χ k ) k = 0 L ∥ ℓ q ∞ ( L p ( ⋅ ) ( w ) ) ] } ,

    where and hereafter, q 0 ≔ q ( 0 ) .

  2. If α ( ⋅ ) , q ( ⋅ ) ∈ P ∞ log ( R n ) , then

    M K q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) = M K q ∞ , p ( ⋅ ) α ∞ , λ ( w ) .

Lemma 2 has been proved by Izuki and Noi [44, pp. 9–10].

Lemma 2

If p ( ⋅ ) ∈ P log ( R n ) ∩ P ( R n ) and w ∈ A p ( ⋅ ) , then there exist constants δ 1 , δ 2 ∈ ( 0 , 1 ) and C > 0 such that for all balls B in R n and all measurable subsets S ⊂ B ,

(1) ∥ χ S ∥ L p ( ⋅ ) ( w ) ∥ χ B ∥ L p ( ⋅ ) ( w ) ≤ C ∣ S ∣ ∣ B ∣ δ 1 ,

(2) ∥ χ S ∥ L p ′ ( ⋅ ) ( w − 1 ) ∥ χ B ∥ L p ′ ( ⋅ ) ( w − 1 ) ≤ C ∣ S ∣ ∣ B ∣ δ 2 .

Our main result is as follows.

Theorem 1

Let r ∈ ( 1 , ∞ ) , p ( ⋅ ) ∈ P log ( R n ) ∩ P ( R n ) , α ( ⋅ ) , q ( ⋅ ) ∈ L ∞ ( R n ) ∩ P 0 log ( R n ) ∩ P ∞ log ( R n ) ∩ P 0 ( R n ) , w ∈ A p ( ⋅ ) , λ − n δ 1 < α ( 0 ) , α ∞ < n δ 2 , where δ 1 , δ 2 ∈ ( 0 , 1 ) are the constants in Lemma 2 for the exponent p ( ⋅ ) and the weight w . Suppose that T is a sublinear operator satisfies the size condition,

(3) ∣ T f ( x ) ∣ ≤ C ∫ R n ∣ x − y ∣ − n ∣ f ( y ) ∣ d y

for all f ∈ L loc 1 ( R n ) and a.e. x ∉ supp f . If the sublinear operator T satisfies vector-valued inequality on L p ( ⋅ ) ( w ) ,

(4) ∑ j = 1 ∞ ∣ T f j ∣ r 1 r L p ( ⋅ ) ( w ) ≤ C ∑ j = 1 ∞ ∣ f j ∣ r 1 r L p ( ⋅ ) ( w )

for all sequences { f j } j = 1 ∞ of locally integrable functions on R n , then

∑ j = 1 ∞ ∣ T f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ≤ C ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ,

where C is independent of { f j } j = 1 ∞ .

The following Lemma 3 is Corollary 3.2 in [42, p. 11].

Lemma 3

Let p ( ⋅ ) ∈ P ( R n ) and w be a weight. If the maximal operator M is bounded on L p ( ⋅ ) ( w ) and L p ′ ( ⋅ ) ( w − 1 ) and r ∈ ( 1 , ∞ ) , then there is a positive constant C such that

∑ j = 1 ∞ ( M f j ) r 1 r L p ( ⋅ ) ( w ) ≤ C ∑ j = 1 ∞ ∣ f j ∣ r 1 r L p ( ⋅ ) ( w ) .

From Theorem 1 and Lemma 3, we obtain the following corollary.

Corollary 1

Let r ∈ ( 1 , ∞ ) , p ( ⋅ ) ∈ P log ( R n ) ∩ P ( R n ) , α ( ⋅ ) , q ( ⋅ ) ∈ L ∞ ( R n ) ∩ P 0 log ( R n ) ∩ P ∞ log ( R n ) ∩ P 0 ( R n ) , w ∈ A p ( ⋅ ) , λ − n δ 1 < α ( 0 ) , α ∞ < n δ 2 , where δ 1 , δ 2 ∈ ( 0 , 1 ) are the constants in Lemma 2 for the exponent p ( ⋅ ) and the weight w , then

∑ j = 1 ∞ ∣ M f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ≤ C ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ,

where C is independent of { f j } j = 1 ∞ of locally integrable functions on R n .

3 Proof of Theorem 1

To prove Theorem 1, we need the following lemma, which is well known. For example, see [45, Proposition 1.2, p. 6].

Lemma 4

Let 0 < p ≤ ∞ , ε > 0 . Then there is a positive constant C such that

(5) ∑ j = − ∞ ∞ ∑ k = − ∞ ∞ 2 − ∣ k − j ∣ ε a k p 1 / p ≤ C ∑ j = − ∞ ∞ a j p 1 / p

for non-negative sequences { a j } j = − ∞ ∞ . Here, when p = ∞ , it is understood that (5) stands for

sup j ∈ Z ∑ k = − ∞ ∞ 2 − ∣ k − j ∣ ε a k ≤ C sup j ∈ Z a j .

Proof of Theorem 1

Since the set of all bounded compact supported functions is dense in weighted variable Lebesgue spaces (see [42, Lemma 3.1, p. 10]), we only consider bounded compact supported functions. Let { f j } be a sequence of bounded compact supported functions, we decompose

f j ( x ) = ∑ l = − ∞ ∞ f j l χ l ≕ ∑ l = − ∞ ∞ f j l , j ∈ N .

By Proposition 1, we have

∑ j = 1 ∞ ∣ T f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ≈ max sup L ≤ 0 , L ∈ Z 2 − L λ 2 k α ( 0 ) ∑ j = 1 ∞ ∣ T f j ∣ r 1 r χ k k ≤ L l q 0 ( L p ( ⋅ ) ( w ) ) , sup L > 0 , L ∈ Z 2 − L λ 2 k α ( 0 ) ∑ j = 1 ∞ ∣ T f j ∣ r 1 r χ k k < 0 l q 0 ( L p ( ⋅ ) ( w ) ) + 2 − L λ 2 k α ∞ ∑ j = 1 ∞ ∣ T f j ∣ r 1 r χ k k = 0 L l q ∞ ( L p ( ⋅ ) ( w ) ) ≔ max { E , H } ,

where

E ≔ sup L ≤ 0 , L ∈ Z 2 − L λ 2 k α ( 0 ) ∑ j = 1 ∞ ∣ T f j ∣ r 1 r χ k k ≤ L l q 0 ( L p ( ⋅ ) ( w ) ) , H ≔ sup L > 0 , L ∈ Z { F + G } , F ≔ 2 − L λ 2 k α ( 0 ) ∑ j = 1 ∞ ∣ T f j ∣ r 1 r χ k k < 0 l q 0 ( L p ( ⋅ ) ( w ) ) , L > 0 , G ≔ 2 − L λ 2 k α ∞ ∑ j = 1 ∞ ∣ T f j ∣ r 1 r χ k k = 0 L l q ∞ ( L p ( ⋅ ) ( w ) ) , L > 0 .

Since to estimate F is essentially similar to estimate E , so we suffice to show that

E , G ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

To do so, we have

E ≤ C ∑ i = i 3 E i , G ≤ C ∑ i = i 3 G i ,

where

E 1 ≔ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ k = − ∞ L 2 k α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∑ l = − ∞ k − 2 T f j l r 1 r χ k L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) , E 2 ≔ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ k = − ∞ L 2 k α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∑ l = k − 1 k + 1 T f j l r 1 r χ k L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ,

E 3 ≔ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ k = − ∞ L 2 k α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∑ l = k + 2 ∞ T f j l r 1 r χ k L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) , G 1 ≔ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ j = 1 ∞ ∑ l = − ∞ k − 2 T f j l r 1 r χ k L p ( ⋅ ) ( w ) q ∞ 1 q ∞ , G 2 ≔ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ j = 1 ∞ ∑ l = k − 1 k + 1 T f j l r 1 r χ k L p ( ⋅ ) ( w ) q ∞ 1 q ∞ , G 3 ≔ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ j = 1 ∞ ∑ l = k + 2 ∞ T f j l r 1 r χ k L p ( ⋅ ) ( w ) q ∞ 1 q ∞ .

We shall use the following estimates. If l < k − 1 , then by Lemma 2 and Definition 2, we have

(6) 2 − k n ∫ R n ∑ j = 1 ∞ ∣ f j l ( y ) ∣ r 1 r d y χ k L p ( ⋅ ) ( w ) ≤ C 2 − k n ∥ χ B k ∥ L p ( ⋅ ) ( w ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r w χ l L p ( ⋅ ) ∥ χ l w − 1 ∥ L p ′ ( ⋅ ) ≤ C 2 − k n ∣ B k ∣ ∥ χ B k ∥ L p ′ ( ⋅ ) ( w − 1 ) − 1 ∥ χ B l ∥ L p ′ ( ⋅ ) ( w − 1 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) ≤ C 2 ( l − k ) n δ 2 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) .

If l ≥ k + 1 , then

(7) 2 − k n ∫ R n ∑ j = 1 ∞ ∣ f j l ( y ) ∣ r 1 r d y χ k L p ( ⋅ ) ( w ) ≤ C 2 − k n ∥ χ B k ∥ L p ( ⋅ ) ( w ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r w χ l L p ( ⋅ ) ∥ χ l w − 1 ∥ L p ′ ( ⋅ ) ≤ C 2 − k n ∥ χ B k ∥ L p ( ⋅ ) ( w ) ∥ χ B l ∥ L p ( ⋅ ) ( w ) ∥ χ B l ∥ L p ( ⋅ ) ( w ) − 1 ∥ χ B l ∥ L p ′ ( ⋅ ) ( w − 1 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) ≤ C 2 ( l − k ) n ( 1 − δ 1 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) .

To estimate E 1 , since l ≤ k − 2 , we deduce that

∣ x − y ∣ ≥ ∣ x ∣ − ∣ y ∣ > 2 k − 1 − 2 l ≥ 2 k − 2 , x ∈ D k , y ∈ D l .

Thus, by (3) for ∀ x ∈ D k , we have

∣ T f j l ∣ ≲ 2 − k n ∫ R n ∣ f j l ( y ) ∣ d y .

Therefore, by the Minkowski inequality, we obtain

(8) ∑ j = 1 ∞ ∑ l = − ∞ k − 2 T f j l r 1 r χ k L p ( ⋅ ) ( w ) ≲ ∑ j = 1 ∞ ∑ l = − ∞ k − 2 2 − k n ∫ R n ∣ f j l ( y ) ∣ d y r 1 r χ k L p ( ⋅ ) ( w ) ≲ ∑ l = − ∞ k − 2 2 − k n ∫ R n ∑ j = 1 ∞ ∣ f j l ( y ) ∣ r 1 r d y χ k L p ( ⋅ ) ( w ) .

By (6) and Lemma 4, we obtain

∑ k = − ∞ L 2 k α ( 0 ) q ( 0 ) ∑ l = − ∞ k − 2 2 − k n ∫ R n ∑ j = 1 ∞ ∣ f j l ( y ) ∣ r 1 r d y χ k L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ ∑ k = − ∞ L 2 k α ( 0 ) q ( 0 ) ∑ l = − ∞ k − 2 2 ( l − k ) n δ 2 ∑ j = 1 ∞ ∣ f j ∣ r 1 r L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) = ∑ k = − ∞ L ∑ l = − ∞ k − 2 2 l α ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( l − k ) ( n δ 2 − α ( 0 ) ) q ( 0 ) 1 q ( 0 ) ≲ ∑ l = − ∞ L − 2 2 l α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ,

where 2 − ∣ k − l ∣ ( n δ 2 − α ( 0 ) ) = 2 − ∣ k − l ∣ ε for ε = n δ 2 − α ( 0 ) > 0 . Hence,

E 1 ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

To estimate E 2 . For k − 1 ≤ l ≤ k + 1 , ∀ x ∈ D k , since T satisfies (4), then by the Minkowski inequality, we obtain

(9) ∑ j = 1 ∞ ∑ l = k − 1 k + 1 T f j l r 1 r χ k L p ( ⋅ ) ( w ) ≲ ∑ l = k − 1 k + 1 ∑ j = 1 ∞ ∣ T f j l ∣ r 1 r χ k L p ( ⋅ ) ( w ) ≲ ∑ l = k − 1 k + 1 ∑ j = 1 ∞ ∣ T f j l ∣ r 1 r χ k L p ( ⋅ ) ( w ) ≲ ∑ l = k − 1 k + 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) .

Thus, we have

E 2 ≲ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ l = k − 1 k + 1 ∑ k = − ∞ L 2 k α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ l = − ∞ L + 1 2 l α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

To estimate E 3 , since l ≥ k + 2 , we have

∣ x − y ∣ ≥ ∣ y ∣ − ∣ x ∣ > 2 l − 2 , x ∈ D k , y ∈ D l .

For ∀ x ∈ D k , since the sublinear operator T satisfies (3), we have

∣ T f j l ∣ ≲ 2 − l n ∫ R n ∣ f j l ( y ) ∣ d y .

Therefore, by the Minkowski inequality, we have

(10) ∑ j = 1 ∞ ∑ l = k + 2 ∞ T f j l r 1 r χ k L p ( ⋅ ) ( w ) ≲ ∑ j = 1 ∞ ∑ l = k + 2 ∞ 2 − l n ∫ R n ∣ f j l ( y ) ∣ d y r 1 r χ k L p ( ⋅ ) ( w ) ≲ ∑ l = k + 2 ∞ 2 − l n ∫ R n ∑ j = 1 ∞ ∣ f j l ( y ) ∣ r 1 r d y χ k L p ( ⋅ ) ( w ) .

By (7), we obtain

∑ k = − ∞ L 2 k α ( 0 ) q ( 0 ) ∑ l = k + 2 ∞ 2 − l n ∫ R n ∑ j = 1 ∞ ∣ f j l ( y ) ∣ r 1 r d y χ k L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ ∑ k = − ∞ L 2 k α ( 0 ) q ( 0 ) ∑ l = k + 2 ∞ 2 ( k − l ) n δ 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ ∑ k = − ∞ L ∑ l = k + 2 L 2 l α ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( k − l ) ( n δ 1 + α ( 0 ) ) q ( 0 ) 1 q ( 0 ) + ∑ k = − ∞ L 2 k α ( 0 ) ∑ l = L + 1 0 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( k − l ) n δ 1 q ( 0 ) 1 q ( 0 ) + ∑ k = − ∞ L 2 k α ( 0 ) ∑ l = 1 ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( k − l ) n δ 1 q ( 0 ) 1 q ( 0 ) ≔ I 3 , 1 + I 3 , 2 + I 3 , 3 .

Therefore,

E 3 ≲ sup L ≤ 0 , L ∈ Z 2 − L λ I 3 , 1 + sup L ≤ 0 , L ∈ Z 2 − L λ I 3 , 2 + I sup L ≤ 0 , L ∈ Z 2 3 , 3 − L λ ≔ E 3 , 1 + E 3 , 2 + E 3 , 3 .

We consider E 3 , 1 . By Lemma 4, we have

E 3 , 1 ≲ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ k = − ∞ L ∑ l = k + 2 L 2 l α ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( k − l ) ( n δ 1 + α ( 0 ) ) q ( 0 ) 1 q ( 0 ) ≲ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ l = − ∞ L + 2 2 l α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ,

where 2 − ∣ k − l ∣ ( n δ 1 + α ( 0 ) ) = 2 − ∣ k − l ∣ η for η = n δ 1 + α ( 0 ) > 0 .

We consider E 3 , 2 . Since n δ 1 + α ( 0 ) − λ > 0 , we obtain

E 3 , 2 ≲ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ k = − ∞ L 2 k ( n δ 1 + α ( 0 ) ) ∑ l = L + 1 0 2 l α ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 − l ( n δ 1 + α ( 0 ) ) q ( 0 ) 1 q ( 0 ) ≲ sup L ≤ 0 , L ∈ Z sup l ≤ 0 2 − l λ 2 l α ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 − L λ ∑ k = − ∞ L 2 k ( n δ 1 + α ( 0 ) ) ∑ l = L + 1 0 2 − l ( n δ 1 + α ( 0 ) − λ ) q ( 0 ) 1 q ( 0 ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) sup L ≤ 0 , L ∈ Z 2 L ( − n δ 1 − α ( 0 ) ) ∑ k = − ∞ L 2 k ( n δ 1 + α ( 0 ) ) q ( 0 ) 1 q ( 0 ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

We consider E 3 , 3 . Since n δ 1 + α ( 0 ) − λ > 0 , we obtain

E 3 , 3 ≲ sup L ≤ 0 , L ∈ Z 2 − L λ ∑ k = − ∞ L 2 k ( n δ 1 + α ( 0 ) ) ∑ l = 1 ∞ 2 l α ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 − l ( n δ 1 + α ∞ ) q ( 0 ) 1 q ( 0 ) ≲ sup L ≤ 0 , L ∈ Z sup l ≥ 1 2 − l λ 2 l α ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 − L λ ∑ k = − ∞ L 2 k ( n δ 1 + α ( 0 ) ) ∑ l = 1 ∞ 2 − l ( n δ 1 + α ∞ − λ ) q ( 0 ) 1 q ( 0 ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) sup L ≤ 0 , L ∈ Z 2 − L λ ∑ k = − ∞ L 2 k ( n δ 1 + α ( 0 ) ) q ( 0 ) 1 q ( 0 ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) sup L ≤ 0 , L ∈ Z 2 L ( − λ + n δ 1 + α ( 0 ) ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

To go on, we need further preparation.

If l < 0 , by Proposition 1, we have

(11) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) = 2 − l α ( 0 ) 2 l α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ 2 − l α ( 0 ) ∑ t = − ∞ l 2 t α ( 0 ) q ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ t L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ 2 l ( λ − α ( 0 ) ) 2 − l λ ∑ t = − ∞ l 2 t α ( 0 ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ t L p ( ⋅ ) ( w ) q ( 0 ) 1 q ( 0 ) ≲ 2 l ( λ − α ( 0 ) ) ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

Next, we estimate G . To estimate G 1 , by (8) and (6), we have

G 1 ≲ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = − ∞ k − 2 2 ( l − k ) n δ 2 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 1 q ∞ ≲ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = − ∞ − 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( l − k ) n δ 2 + ∑ l = 0 k ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( l − k ) n δ 2 q ∞ 1 q ∞ ≲ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = − ∞ − 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( l − k ) n δ 2 q ∞ 1 q ∞ + 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = 0 k ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( l − k ) n δ 2 q ∞ 1 q ∞ ≕ G 1 , 1 + G 1 , 2 .

If q ∞ ≥ 1 , since n δ 2 − α ∞ > 0 and n δ 2 − α ( 0 ) > 0 , then by the Minkowski inequality and (11), we obtain

G 1 , 1 = 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = − ∞ − 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( l − k ) n δ 2 q ∞ 1 q ∞ ≲ 2 − L λ ∑ l = − ∞ − 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) ∑ k = 0 L ( 2 k α ∞ 2 ( l − k ) n δ 2 ) q ∞ 1 q ∞ ≲ 2 − L λ ∑ l = − ∞ − 1 2 l n δ 2 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) ∑ k = 0 L 2 − k ( n δ 2 − α ∞ ) q ∞ 1 q ∞

≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) 2 − L λ ∑ l = − ∞ − 1 2 l ( n δ 2 + λ − α ( 0 ) ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

If q ∞ < 1 , since n δ 2 − α ∞ > 0 and n δ 2 − α ( 0 ) > 0 , then by (11), we have

G 1 , 1 ≲ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = − ∞ − 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 2 ( l − k ) n δ 2 q ∞ 1 q ∞ = 2 − L λ ∑ l = − ∞ − 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 2 l n δ 2 q ∞ ∑ k = 0 L 2 k α ∞ q ∞ 2 − k n δ 2 q ∞ 1 q ∞ = 2 − L λ ∑ l = − ∞ − 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 2 l n δ 2 q ∞ ∑ k = 0 L 2 − k ( n δ 2 − α ∞ ) q ∞ 1 q ∞ ≲ 2 − L λ ∑ l = − ∞ − 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 2 l n δ 2 q ∞ 1 q ∞ ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) 2 − L λ ∑ l = − ∞ − 1 2 l ( n δ 2 + λ − α ( 0 ) ) q ∞ 1 q ∞ ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

We consider G 1 , 2 . Since n δ 2 − α ∞ > 0 , by Lemma 4, we have

G 1 , 2 = 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = 0 k ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( l − k ) n δ 2 q ∞ 1 q ∞ = 2 − L λ ∑ k = 0 L ∑ l = 0 k 2 l α ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( l − k ) ( n δ 2 − α ∞ ) q ∞ 1 q ∞ ≲ 2 − L λ ∑ l = 0 k 2 l α ∞ q ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 1 q ∞ ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ,

where 2 − ∣ k − l ∣ ( n δ 2 − α ∞ ) ≲ 2 − ∣ k − l ∣ η for η = n δ 2 − α ∞ .

To estimate G 2 , by (9), we have

G 2 ≲ 2 − L λ ∑ l = k − 1 k + 1 ∑ k = 0 L 2 k α ∞ q ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 1 q ∞ ≲ 2 − L λ ∑ l = − 1 L + 1 2 l α ∞ q ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 1 q ∞ ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

To estimate G 3 , by (10), we have

G 3 ≤ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = k + 2 ∞ 2 − l n ∫ R n ∑ j = 1 ∞ ∣ f j l ( y ) ∣ r 1 r d y χ k L p ( ⋅ ) ( w ) q ∞ 1 q ∞ .

By (7), we obtain

G 3 ≲ 2 − L λ ∑ k = 0 L 2 k α ∞ q ∞ ∑ l = k + 2 ∞ 2 ( k − l ) n δ 1 ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 1 q ∞ ≲ 2 − L λ ∑ k = 0 L ∑ l = k + 2 L + 2 2 l α ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( k − l ) ( n δ 1 + α ∞ ) q ∞ 1 q ∞ + 2 − L λ ∑ k = 0 L 2 k α ∞ ∑ l = L + 3 ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( k − l ) n δ 1 q ∞ 1 q ∞ ≔ G 3 , 1 + G 3 , 2 .

Now we estimate G 3 , 1 . By Lemma 4, we obtain

G 3 , 1 ≲ 2 − L λ ∑ k = 0 L ∑ l = k + 2 L + 2 2 l α ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 ( k − l ) ( n δ 1 + α ∞ ) q ∞ 1 q ∞ ≲ 2 − L λ ∑ l = 0 L + 2 2 l α ∞ q ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) q ∞ 1 q ∞ ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ,

where 2 − ∣ k − l ∣ ( n δ 1 + α ∞ ) = 2 − ∣ k − l ∣ ζ for ζ = n δ 1 + α ∞ > 0 .

Then we estimate G 3 , 2 . Since n δ 1 + α ∞ − λ > 0 ,

G 3 , 2 ≲ 2 − L λ ∑ k = 0 L 2 k ( n δ 1 + α ∞ ) ∑ l = L + 3 ∞ 2 l α ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 − l ( n δ 1 + α ∞ ) q ∞ 1 q ∞ ≲ sup l ≥ 1 2 − l λ 2 l α ∞ ∑ j = 1 ∞ ∣ f j ∣ r 1 r χ l L p ( ⋅ ) ( w ) 2 − L λ ∑ k = 0 L 2 k ( n δ 1 + α ∞ ) ∑ l = L + 3 ∞ 2 − l ( n δ 1 + α ∞ − λ ) q ∞ 1 q ∞ ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) 2 − L λ ∑ k = 0 L 2 k ( n δ 1 + α ∞ ) q ∞ 2 − L ( n δ 1 + α ∞ − λ ) 1 q ∞ ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) 2 − L λ 2 ( n δ 1 + α ∞ ) L 2 − L ( n δ 1 + α ∞ − λ ) ≲ ∑ j = 1 ∞ ∣ f j ∣ r 1 r M K ˙ q ( ⋅ ) , p ( ⋅ ) α ( ⋅ ) , λ ( w ) .

This completes the proof.□

Acknowledgments

The authors would like to thank the referees for valuable suggestions.

  1. Funding information: Jingshi Xu was partially supported by the National Natural Science Foundation of China (Grant No. 11761026, 11761027) and the Natural Science Foundation of Guangxi Province (Grant No. 2020GXNSFAA159085).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-03-12
Revised: 2021-01-19
Accepted: 2021-01-29
Published Online: 2021-05-27

© 2021 Shengrong Wang and Jingshi Xu, published by De Gruyter

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

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  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0024
Keywords for this article
subilinear operator; vector-valued inequality; Muckenhoupt weight; variable exponent; Herz-Morrey space
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  38. Inviscid, zero Froude number limit of the viscous shallow water system
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  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
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  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
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  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
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  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
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  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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