Home Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
Article Open Access

Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents

  • Kieu Huu Dung EMAIL logo and Pham Thi Kim Thuy
Published/Copyright: July 3, 2023
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we obtain some sufficient conditions for the boundedness of commutators of p -adic Hardy-Littlewood operators with symbols in central bounded mean oscillation space and Lipschitz space on the p -adic function spaces with variable exponents such as the p -adic local central Morrey, p -adic Morrey-Herz, and p -adic local block spaces with variable exponents.

Keywords: Hardy-Littlewood operator; commutator; local central Morrey space; Morrey-Herz space; local block space; central BMO space; Lipschitz space; variable exponent; p-adic analysis
MSC 2010: 42B35; 47B38; 46E30

1 Introduction

The theory of functions from Q p n into C plays a significant role in p -adic probability, p -adic quantum mechanics, p -adic harmonic analysis, and p -adic partial differential equations [1–9]. Furthermore, there has been a lot of interest in the theory of p -adic operators. For instance, the p -adic Hardy-Littlewood operator studied by Rim and Lee [10]

ℋ φ p ( f ) ( x ) = ∫ Z p ∗ φ ( t ) f ( t x ) d t , x ∈ Q p n ,

where Z p ∗ = { t ∈ Q p \ { 0 } : ∣ t ∣ p < 1 } and φ ∈ L loc 1 ( Z p ∗ ) .

We now discuss the commutator of p -adic Hardy-Littlewood operator as follows:

ℋ φ p , b ( f ) ( x ) = ∫ Z p ∗ φ ( t ) ( b ( x ) − b ( t x ) ) f ( t x ) d t , x ∈ Q p n ,

where b ∈ ℳ ( Q p n , C ) . For example, if n = 1 and φ = 1 , then the operator ℋ φ p , b will become the commutator of Hardy operators H p , b given by

H p , b ( f ) ( z ) = 1 ∣ z ∣ p ∫ ∣ t ∣ p ≤ ∣ z ∣ p ( b ( z ) − b ( t ) ) f ( t ) d t , z ∈ Q p \ { 0 } .

Lately, the commutators of the p -adic Hardy type operators have been studied in the works [11–15]. In the case of n = 1 , we denote U φ p ≔ ℋ φ p and U φ p , b ≔ ℋ φ p , b . As an application, we present the Cauchy problem of p -adic pseudo-differential equation:

(1) D α u = b ( ∣ x ∣ p ) ⋅ f ( ∣ x ∣ p ) , x ∈ Q p , α > 0 , u ( 0 ) = 0 .

Here, b and f are continuous functions, u is a solution, and the Vladimirov operator D α [1,3,5,6] is defined by

D α h ( z ) = 1 − p α 1 − p − α − 1 ∫ Q p h ( z − t ) − h ( z ) ∣ t ∣ p α + 1 d t .

In case of b = 1 , the readers can find equation (1) in [16]. Moreover, a generalization of equation (1) is the fractional p -adic Brownian motion equation [6, Section 5.3]. It is well known that the equation (1) has a radial solution u as follows:

u ( x ) = 1 − p − α 1 − p α − 1 ∫ ∣ t ∣ p ≤ ∣ x ∣ p ( ∣ x − t ∣ p α − 1 − ∣ t ∣ p α − 1 ) b ( t ) ⋅ f ( t ) d t = ( b ( x ) ⋅ U ψ p f ( x ) − U ψ p , b f ( x ) ) ∣ x ∣ p α ,

where

ψ ( t ) = 1 − p − α 1 − p α − 1 { ∣ 1 − t ∣ p α − 1 − ∣ t ∣ p α − 1 } .

Thereby, the regularity of the solution of equation (1) depends on the boundedness of U ψ p and U ψ p , b .

Variable exponent spaces have some essential applications in image processing, harmonic analysis, and differential equations [17–23]. In 2009, Izuki [24] introduced the variable exponent Morrey-Herz spaces. Next, the nonhomogeneous central Morrey spaces of variable exponent are defined in [25]. Besides, Yee et al. [26] stated the local block space with variable exponent LB u , p ( ⋅ ) ( R n ) as follows:

LB u , p ( ⋅ ) ( R n ) = ∑ k = 1 ∞ λ k b k : ∑ k = 1 ∞ ∣ λ k ∣ < ∞ and b k is a local ( u , L p ( ⋅ ) ) -block .

Here,

‖ f ‖ LB u , p ( ⋅ ) ( R n ) = inf ∑ k = 1 ∞ ∣ λ k ∣ : f = ∑ k = 1 ∞ λ k b k a.e. .

Especially, the dual space of the space LB u , p ( ⋅ ) ( R n ) is the local variable exponent Morrey space L M u p ′ ( ⋅ ) ( R n ) . On p -adic fields, Chacón-Cortés and Rafeiro [27] launched the initial research on the p -adic Lebesgue spaces with a variable exponent.

Motivated by the aforementioned results, we obtain the Lipschitz estimates for the operator ℋ φ p , b on the p -adic local central Morrey space B ω , loc q ( ⋅ ) , λ ( Q p n ) , the p -adic Morrey-Herz space M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) , and the p -adic local block space LB u , q ( ⋅ ) ( Q p n ) with variable exponents. Furthermore, the boundedness of ℋ φ p , b defined by the central variable exponent BMO space on the p -adic above spaces is also discussed.

Section 2 introduces the p -adic Lebesgue, p -adic local central Morrey, p -adic Morrey-Herz, p -adic local block, p -adic central bounded mean oscillation spaces with variable exponents, and the p -adic Lipschitz spaces. In Section 3, our main results are presented.

2 Preliminaries

Let us fix a prime number p . On the field of rational numbers Q , we set the p -adic norm ∣ ⋅ ∣ p as follows: if x = 0 , ∣ 0 ∣ p = 0 ; if x = p α m n , where m , n / ⋮ p , α = α ( x ) ∈ Z , then ∣ x ∣ p = p − α . The completion of the field Q with the p -adic norm ∣ ⋅ ∣ p leads to the field Q p of p -adic numbers. Then

  1. ∣ x ∣ p ≥ 0 , ∀ x ∈ Q p , ∣ x ∣ p = 0 ⇔ x = 0 ;

  2. ∣ x y ∣ p = ∣ x ∣ p ∣ y ∣ p , ∀ x , y ∈ Q p ;

  3. ∣ x + y ∣ p ≤ max ( ∣ x ∣ p , ∣ y ∣ p ) , ∀ x , y ∈ Q p , and ∣ x + y ∣ p = max ( ∣ x ∣ p , ∣ y ∣ p ) with ∣ x ∣ p ≠ ∣ y ∣ p .

The space Q p n = Q p × ⋯ × Q p consists of all points x = ( x 1 , … , x n ) , where x i ∈ Q p , i = 1 , … , n , n ≥ 1 . The p -adic norm of Q p n is defined by

(2) ∣ x ∣ p = max 1 ≤ j ≤ n ∣ x j ∣ p .

Let ℳ ( U , F ) be the set of all measurable functions f ( ⋅ ) : U → F , W ( Q p n ) be the set of all nonnegative weighted functions on Q p n , and ‖ ℋ ‖ X → Y be the norm of ℋ between two normed vector spaces X and Y .

Let B k ( a ) and S k ( a ) define, respectively, the ball and sphere of Q p n with a center at a ∈ Q p n and radius p k :

(3) B k ( a ) = { x ∈ Q p n : ∣ x − a ∣ p ≤ p k } and S k ( a ) = { x ∈ Q p n : ∣ x − a ∣ p = p k } .

We set B k = B k ( 0 ) and S k = S k ( 0 ) . Besides, χ k is the characteristic function of the sphere S k .

As is known, there exists a Haar measure d x on Q p n , which is unique up to positive constant multiple and is translation invariant. We normalize the measure d x ,

∫ B 0 ( 0 ) d x = ∣ B 0 ( 0 ) ∣ = 1 ,

where ∣ B ∣ denotes the Haar measure of a measurable subset B of Q p n . Then ∣ B k ( a ) ∣ = p n k and ∣ S k ( a ) ∣ = p n k ( 1 − p − n ) , for any a ∈ Q p n .

The Lebesgue space L q ( Q p n ) ≔ { f ∈ ℳ ( Q p n , C ) : ‖ f ‖ L q ( Q p n ) < ∞ } , where ‖ f ‖ L q ( Q p n ) = ∫ Q p n ∣ f ( x ) ∣ q d x 1 / q and q ∈ ( 0 , ∞ ) .

The space L loc q ( V ) ≔ { f ∈ ℳ ( V , C ) : ∫ K ∣ f ( x ) ∣ q d x < ∞ , for any compact subset K of  V } . For f ∈ L loc 1 ( Q p n ) ,

∫ Q p n f ( x ) d x = lim α → ∞ ∫ B α f ( x ) d x = lim α → ∞ ∑ − ∞ < γ ≤ α ∫ S γ f ( x ) d x .

The set P ( Q p n ) ≔ { q ∈ ℳ ( Q p n , ( 1 , ∞ ) ) : 1 < q − ≤ q ( x ) ≤ q + < ∞ , for all x ∈ Q p n } , where q − = ess inf x ∈ Q p n q ( x ) and q + = ess sup x ∈ Q p n q ( x ) .

For more details, the readers can find the book [8].

Definition 2.1

For q ( ⋅ ) ∈ P ( Q p n ) , the p -adic variable exponent Lebesgue space L q ( ⋅ ) ( Q p n ) is defined by

L q ( ⋅ ) ( Q p n ) = { f ∈ ℳ ( Q p n , C ) : F q ( ⋅ ) ( f / η ) < ∞ for some η > 0 } ,

where

F q ( ⋅ ) ( f / η ) = ∫ Q p n ∣ f ( x ) ∣ η q ( x ) d x .

The norm is given by

∥ f ∥ L q ( ⋅ ) ( Q p n ) = inf η > 0 : F q f η ≤ 1 .

For q ∈ P ( Q p n ) , for all f ∈ L q ( ⋅ ) ( Q p n ) , we have

(4) min { A 1 / q − , A 1 / q + } ≤ ‖ f ‖ L q ( ⋅ ) ( Q p n ) ≤ max { A 1 / q − , A 1 / q + } whenever F q ( f ) ≤ A .

For q ∈ P ( Q p n ) , we define q ′ ( ⋅ ) as follows:

1 q ( x ) + 1 q ′ ( x ) = 1 , for all x ∈ Q p n .

Let q ∈ P ( Q p n ) and ω ∈ W ( Q p n ) . The space L ω q ( ⋅ ) ( Q p n ) is given by

L ω q ( ⋅ ) ( Q p n ) = { f ∈ ℳ ( Q p n , C ) : ‖ f ‖ L ω q ( ⋅ ) ( Q p n ) < ∞ } ,

where ‖ f ‖ L ω q ( ⋅ ) ( Q p n ) = ∥ f ω ∥ L q ( ⋅ ) ( Q p n ) .

The space L ω , loc q ( ⋅ ) ( Q p n \ { 0 } ) is defined by

L ω , loc q ( ⋅ ) ( Q p n \ { 0 } ) = { f ∈ ℳ ( Q p n \ { 0 } , C ) : f . χ K ∈ L ω q ( ⋅ ) ( Q p n ) , for any compact subset K of  Q p n \ { 0 } } .

Denote by C 0 log ( Q p n ) the set of all log-Hölder continuous functions α ( ⋅ ) : Q p n → R satisfying at the origin,

∣ α ( x ) − α ( 0 ) ∣ ≤ C 0 α log ( e + ∣ x ∣ p − 1 ) , for all x ∈ Q p n .

Denote by C ∞ log ( Q p n ) the set of all log-Hölder continuous functions α ( ⋅ ) : Q p n → R satisfying at infinity,

∣ α ( x ) − α ∞ ∣ ≤ C ∞ α log ( e + ∣ x ∣ p ) , for all x ∈ Q p n ,

where lim ∣ x ∣ p → ∞ a ( x ) = α ∞ ∈ R .

Next, we would like to introduce the p -adic local central Morrey and p -adic Morrey-Herz spaces with variable exponents (see [21,22] for the real field).

Definition 2.2

Let λ ∈ R , q ( ⋅ ) ∈ P ( Q p n ) , and ω ∈ W ( Q p n ) . The p -adic variable exponent local central Morrey space B ω , loc q ( ⋅ ) , λ ( Q p n ) is defined by

B ω , loc q ( ⋅ ) , λ ( Q p n ) = { f ∈ L ω , loc q ( ⋅ ) ( Q p n ) : ‖ f ‖ B ω , loc q ( ⋅ ) , λ ( Q p n ) < ∞ } ,

where ‖ f ‖ B ω , loc q ( ⋅ ) , λ ( Q p n ) = sup k ≤ 0 , k ∈ Z ‖ f ‖ L ω q ( ⋅ ) ( B k ) ∣ B k ∣ λ ‖ 1 ‖ L q ( ⋅ ) ( B k ) . In case ω = 1 , we denote B loc q ( ⋅ ) , λ ( Q p n ) ≔ B ω , loc q ( ⋅ ) , λ ( Q p n ) .

Definition 2.3

Let λ ∈ [ 0 , ∞ ) , ℓ ∈ ( 0 , ∞ ) , q ( ⋅ ) ∈ P ( Q p n ) , ω ∈ W ( Q p n ) , and α ( ⋅ ) : Q p n → R with α ( ⋅ ) ∈ L ∞ ( Q p n ) . The p -adic variable exponent Morrey-Herz space M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) is defined by

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

where ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) = sup k 0 ∈ Z p − k 0 λ ∑ k = − ∞ k 0 ‖ p k α ( ⋅ ) f χ k ‖ L ω q ( ⋅ ) ( Q p n ) ℓ 1 / ℓ .

If ω = 1 , then M K ˙ ℓ , q ( ⋅ ) α ( ⋅ ) , λ ( Q p n ) ≔ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) .

Theorem 2.4

[21, Proposition 2.5] If λ ∈ [ 0 , ∞ ) , ℓ ∈ ( 0 , ∞ ) , α ∈ L ∞ ( Q p n ) ∩ C 0 log ( Q p n ) ∩ C ∞ log ( Q p n ) , ω ∈ W ( Q p n ) , and q ( ⋅ ) ∈ P ( Q p n ) , then

‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) ≈ max { sup k 0 ∈ Z − ∪ { 0 } ℳ 1 , k 0 , sup k 0 ∈ Z + ( ℳ 2 , k 0 + ℳ 3 , k 0 ) } .

Here,

ℳ 1 , k 0 = p − k 0 λ ∑ k = − ∞ k 0 p k α ( 0 ) ℓ ‖ f χ k ‖ L ω q ( ⋅ ) ( Q p n ) ℓ 1 / ℓ , ℳ 2 , k 0 = p − k 0 λ ∑ k = − ∞ − 1 p k α ( 0 ) ℓ ‖ f χ k ‖ L ω q ( ⋅ ) ( Q p n ) ℓ 1 / ℓ ,

ℳ 3 , k 0 = p − k 0 λ ∑ k = 0 k 0 p k α ∞ ℓ ‖ f χ k ‖ L ω q ( ⋅ ) ( Q p n ) ℓ 1 / ℓ .

By Theorem 2.4, we obtain the following lemma.

Lemma 2.5

If λ ∈ [ 0 , ∞ ) , ℓ ∈ ( 0 , ∞ ) , α ∈ L ∞ ( Q p n ) ∩ C 0 log ( Q p n ) ∩ C ∞ log ( Q p n ) , ω ∈ W ( Q p n ) , and q ( ⋅ ) ∈ P ( Q p n ) , then

∥ f χ j ∥ L ω q ( ⋅ ) ( Q p n ) ≲ p j ( λ − α ( 0 ) ) ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) , for all j ∈ Z − ∪ { 0 } , ∥ f χ j ∥ L ω q ( ⋅ ) ( Q p n ) ≲ p j ( λ − α ∞ ) ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) , for a l l j ∈ Z + .

Following [26], we state the p -adic variable exponent local block space.

Definition 2.6

Let q ∈ ℳ ( Q p n , ( 0 , ∞ ) ) and u ∈ ℳ ( ( 0 , ∞ ) , ( 0 , ∞ ) ) . A function b ∈ ℳ ( Q p n , C ) is a local ( u , L q ( ⋅ ) ) -block if it is supported in B k , k ∈ N and

‖ b ‖ L q ( ⋅ ) ( Q p n ) ≤ 1 u ( p k ) .

The p -adic variable exponent local block space LB u , q ( ⋅ ) ( Q p n ) is defined as follows.

LB u , q ( ⋅ ) ( Q p n ) = ∑ k = 0 ∞ λ k b k : ∑ k = 0 ∞ ∣ λ k ∣ < ∞ and b k is a local ( u , L q ( ⋅ ) ) -block .

Here,

‖ f ‖ LB u , q ( ⋅ ) ( Q p n ) = inf ∑ k = 0 ∞ ∣ λ k ∣ : f = ∑ k = 0 ∞ λ k b k a.e. .

Proposition 2.7

Let q ∈ ℳ ( Q p n , ( 0 , ∞ ) ) , u ∈ ℳ ( ( 0 , ∞ ) , ( 0 , ∞ ) ) , and f ∈ LB u , q ( ⋅ ) ( Q p n ) . If g ∈ ℳ ( Q p n , C ) and ∣ g ∣ ≤ ∣ f ∣ , then g ∈ LB u , q ( ⋅ ) ( Q p n ) .

Definition 2.8

Let q ∈ ℳ ( Q p n , ( 0 , ∞ ) ) and u ∈ ℳ ( ( 0 , ∞ ) , ( 0 , ∞ ) ) . We say that u ∈ L W q ( ⋅ ) ( Q p n ) if there exists a positive constant C such that

(5) C ≤ u ( p k ) , for all k ∈ N ,

(6) ‖ χ B k ‖ L q ( ⋅ ) ( Q p n ) ≤ C u ( p k ) , for all k ∈ Z \ N ,

(7) ∑ j = 0 ∞ ‖ χ B k ‖ L q ( ⋅ ) ( Q p n ) ‖ χ B k + j + 1 ‖ L q ( ⋅ ) ( Q p n ) u ( p k + j + 1 ) ≤ C u ( p k ) , for all k ∈ Z .

Because of the proof of [26, Theorem 4], we have the following result.

Theorem 2.9

If q ∈ ℳ ( Q p n , ( 0 , ∞ ) ) and u ∈ L W q ′ ( ⋅ ) ( Q p n ) , then LB u , q ( ⋅ ) ( Q p n ) ⊂ L loc 1 ( Q p n ) and LB u , q ( ⋅ ) ( Q p n ) is a Banach space.

Next, the p -adic variable exponent central BMO space CMO . q ( ⋅ ) ( Q p n ) (see [28] for the real field) and the p -adic Lipschitz space Lip β ( Q p n ) are introduced as follows.

Definition 2.10

Let q ( ⋅ ) ∈ P ( Q p n ) . The space CMO . q ( ⋅ ) ( Q p n ) ≔ { f ∈ L loc q ( ⋅ ) ( Q p n ) : ‖ f ‖ CMO . q ( ⋅ ) ( Q p n ) < ∞ } , where

‖ f ‖ CMO . q ( ⋅ ) ( Q p n ) = sup γ ∈ Z ‖ f − f B γ ‖ L q ( ⋅ ) ( B γ ) ‖ 1 ‖ L q ( ⋅ ) ( B γ ) and f B γ = 1 ∣ B γ ∣ ∫ B γ f ( x ) d x .

Definition 2.11

Let β ∈ ( 0 , 1 ] . The space Lip β ( Q p n ) ≔ { f : Q n → C satisfying ‖ f ‖ Lip β ( Q p n ) < ∞ } , where

‖ f ‖ Lip β ( Q p n ) = sup u , v ∈ Q p n , u ≠ v ∣ f ( u ) − f ( v ) ∣ ∣ u − v ∣ p β .

Definition 2.12

The set B ( Q p n ) ≔ { q ∈ P ( Q p n ) : ‖ M ‖ L q ( ⋅ ) ( Q p n ) → L q ( ⋅ ) ( Q p n ) < ∞ } , where M is the Hardy-Littlewood maximal operator on Q p n defined by

M ( f ) ( x ) = sup γ ∈ Z 1 p γ n ∫ B γ ( x ) ∣ f ( y ) ∣ d y .

By Lemmas 1 and 2 in [29], we obtain the following lemma.

Lemma 2.13

Let q ( ⋅ ) ∈ B ( Q p n ) .

  1. Then

    ‖ χ B ‖ L q ( ⋅ ) ( Q p n ) ‖ χ S ‖ L q ( ⋅ ) ( Q p n ) ≲ ∣ B ∣ ∣ S ∣ and ‖ χ B ‖ L q ′ ( ⋅ ) ( Q p n ) ‖ χ S ‖ L q ′ ( ⋅ ) ( Q p n ) ≲ ∣ B ∣ ∣ S ∣ ,

    for all balls B in Q p n and all measurable subsets S ⊂ B .

  2. Then

    ‖ χ B k ‖ L q ( ⋅ ) ( Q p n ) ‖ χ B k ‖ L q ′ ( ⋅ ) ( Q p n ) ≃ p k n , for a l l k ∈ Z .

3 Main results and their proofs

For simplicity, we denote

P φ ( Q p n ) = { v ∈ P ( Q p n ) : v ( t − 1 ⋅ ) = v ( ⋅ ) , for almost everywhere t ∈ supp ( φ ) } .

First, let us present the boundedness of ℋ φ p , b on the p -adic local central Morrey space with variable exponent.

Theorem 3.1

Let q , q 1 ∈ P φ ( Q p n ) , r 1 ∈ P φ ( Q p n ) ∩ B ( Q p n ) , λ , λ 1 ∈ R such that

(8) 1 q ( ⋅ ) = 1 q 1 ( ⋅ ) + 1 r 1 ( ⋅ ) ,

(9) λ 1 = λ + 1 q − − 1 r 1 + − 1 q 1 + .

If b ∈ CMO . r 1 ( ⋅ ) ( Q p n ) and

C 1 = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p n ξ d t < ∞ ,

where ξ = λ 1 + 1 / q 1 + − 1 / q 1 − − 1 / r 1 − , then ℋ φ p , b is bounded from B ˙ loc q 1 ( ⋅ ) , λ 1 ( Q p n ) to B ˙ loc q ( ⋅ ) , λ ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ B ˙ loc q 1 ( ⋅ ) , λ 1 ( Q p n ) → B ˙ loc q ( ⋅ ) , λ ( Q p n ) ≲ C 1 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) .

Proof

For any k ∈ Z , by the Minkowski inequality and the Hölder inequality,

(10) ∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( B k ) ≲ ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ‖ b ( ⋅ ) − b ( t ⋅ ) ‖ L r 1 ( ⋅ ) ( B k ) ‖ f ( t ⋅ ) ‖ L q 1 ( ⋅ ) ( B k ) d t .

Next, by putting log p ∣ t ∣ p = m ,

(11) ‖ b ( ⋅ ) − b ( t ⋅ ) ‖ L r 1 ( ⋅ ) ( B k ) ≤ ‖ b ( ⋅ ) − b B k ‖ L r 1 ( ⋅ ) ( B k ) + ‖ b B k − b B k + m ‖ L r 1 ( ⋅ ) ( B k ) + ‖ b ( t ⋅ ) − b B k + m ‖ L r 1 ( ⋅ ) ( B k ) ≔ I 1 + I 2 + I 3 .

Moreover, we immediately have

(12) I 1 ≤ ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ∥ b ∥ CMO . r 1 ( ⋅ ) ( Q p n ) .

By r 1 ∈ B ( Q p n ) and Lemma 2.13.(ii), for any k ∈ Z ,

∣ b B k − b B k + 1 ∣ ≤ 1 ∣ B k ∣ ∫ B k ∣ b ( x ) − b B k + 1 ∣ d x ≲ ‖ 1 ‖ L r 1 ′ ( ⋅ ) ( B k + 1 ) ∣ B k + 1 ∣ ‖ b − b B k + 1 ‖ L r 1 ( ⋅ ) ( B k + 1 ) ≤ ‖ 1 ‖ L r 1 ′ ( ⋅ ) ( B k + 1 ) ‖ 1 ‖ L r 1 ( ⋅ ) ( B k + 1 ) ∣ B k + 1 ∣ ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ≲ ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) .

Thus, by m ≤ 0 and m ∈ Z ,

∣ b B k + m − b B k ∣ ≤ ∣ b B k + m − b B k + m + 1 ∣ + ⋯ + ∣ b B k − 1 − b B k ∣ ≲ − m ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) = log p 1 t p ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) .

Then

(13) I 2 ≤ ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ∣ b B k − b B k + m ∣ ≲ ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) log p 1 t p ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) .

By the formula for change of variables and ∣ t ∣ p ∈ ( 0 , 1 ] , we have

F r 1 ( ( b ( t ⋅ ) − b B k + m ) χ B k / η ) = ∫ t B k ∣ b ( z ) − b B k + m ∣ η r 1 ( z ) ∣ t ∣ p − n d z ≤ ∫ B k + m ∣ t ∣ p − n / r 1 − ∣ b ( z ) − b B k + m ∣ η r 1 ( z ) d z .

Thus, by using ‖ 1 ‖ L r 1 ( ⋅ ) ( B k + m ) ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ≤ 1 ,

(14) I 3 ≤ ∣t∣ p − n / r 1 − ‖ b ( ⋅ ) − b B k + m ‖ L r 1 ( ⋅ ) ( B k + m ) ≤ ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ∣ t ∣ p − n / r 1 − ‖ 1 ‖ L r 1 ( ⋅ ) ( B k + m ) ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ≤ ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ∣ t ∣ p − n / r 1 − ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) .

Next, by considering ∣ t ∣ p ∈ ( 0 , 1 ] , we also have

F q 1 ( f ( t ⋅ ) χ B k / η ) = ∫ t B k ∣ f ( z ) ∣ η q 1 ( z ) ∣ t ∣ p − n d z ≤ ∫ B k + m ∣ t ∣ p − n / q 1 − ∣ f ( z ) ∣ η q 1 ( z ) d z .

This gives ‖ f ( t ⋅ ) ‖ L q 1 ( ⋅ ) ( B k ) ≤ ∣ t ∣ p − n / q 1 − ‖ f ‖ L q 1 ( ⋅ ) ( B k + m ) . Consequently, by the inequalities (10)–(14), for any k ∈ Z , m = log p ∣ t ∣ p , we obtain

(15) ‖ ℋ φ p , b ( f ) ‖ L q ( ⋅ ) ( B k ) ≲ ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − log p 1 t p + ∣ t ∣ p − n / r 1 − ‖ f ‖ L q 1 ( ⋅ ) ( B k + m ) d t ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ≲ ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − ‖ f ‖ L q 1 ( ⋅ ) ( B k + m ) d t ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) .

Hence, for any k ≤ 0 and k ∈ Z ,

‖ ℋ φ p , b ( f ) ‖ L q ( ⋅ ) ( B k ) ∣ B k ∣ λ ‖ 1 ‖ L q ( ⋅ ) ( B k ) ≲ ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − K 1 d t ‖ f ‖ B ˙ loc q 1 ( ⋅ ) , λ 1 ( Q p n ) ,

where K 1 = ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ∣ B k + m ∣ λ 1 ‖ 1 ‖ L q 1 ( ⋅ ) ( B k + m ) ∣ B k ∣ λ ‖ 1 ‖ L q ( ⋅ ) ( B k ) .

By the relations (4) and (9) with m = log p ∣ t ∣ p ≤ 0 , we deduce

K 1 ≲ p max { k n / r 1 + , k n / r 1 − } p ( k + m ) n λ 1 p max { ( k + m ) n / q 1 + , ( k + m ) n / q 1 − } p k n λ p min { k n / q + , k n / q − } ≤ p k ( n / r 1 + + n λ 1 + n / q 1 + − n λ − max { n / q + , n / q − } ) p m n λ 1 p max { m n / q 1 + , m n / q 1 − } ≲ ∣ t ∣ p n λ 1 p m min { n / q 1 + , n / q 1 − } = ∣ t ∣ p n λ 1 + n / q 1 + .

Then

‖ ℋ φ p , b ( f ) ‖ B ˙ loc q ( ⋅ ) , λ ( Q p n ) ≲ C 1 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ B ˙ loc q 1 ( ⋅ ) , λ 1 ( Q p n ) .

This completes our proof.□

Theorem 3.2

Let q ∈ P φ ( Q p n ) , λ ∈ R , β ∈ ( 0 , 1 ] , ω ( x ) = ∣ x ∣ p β . If b ∈ Lip β ( Q p n ) and

C 2 = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p n λ − n / q − − β d t < ∞ ,

then ℋ φ p , b is bounded from B ˙ ω , loc q ( ⋅ ) , λ ( Q p n ) to B ˙ loc q ( ⋅ ) , λ ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ B ˙ ω , loc q ( ⋅ ) , λ ( Q p n ) → B ˙ loc q ( ⋅ ) , λ ( Q p n ) ≲ C 2 ‖ b ‖ Lip β ( Q p n ) .

Proof

For any k ∈ Z , by the Minkowski inequality,

(16) ∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( B k ) ≲ ‖ b ‖ Lip β ( Q p n ) ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∥ f ( t ⋅ ) ∣ ⋅ ∣ p β ∥ L q ( ⋅ ) ( B k ) d t .

For η ∈ ( 0 , ∞ ) and ∣ t ∣ p ∈ ( 0 , 1 ] , we have

∫ Q p n ∣ f ( t x ) ∣ χ B k ( x ) ∣ x ∣ p β η q ( x ) d x = ∫ t B k ∣ f ( z ) ∣ ∣ t − 1 z ∣ p β η q ( t − 1 z ) ∣ t ∣ p − n d z ≤ ∫ B k + m ∣ t ∣ p − n / q − − β ∣ f ( z ) ∣ ω ( z ) η q ( z ) d z .

where m = log p ∣ t ∣ p . Thus, ∥ f ( t ⋅ ) ∥ L ω q ( ⋅ ) ( B k ) ≤ ∣ t ∣ p − n / q − − β ‖ f ‖ L ω q ( ⋅ ) ( B k + m ) . Combining this with (16),

(17) ∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( B k ) ≲ ‖ b ‖ Lip β ( Q p n ) ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p − n / q − − β ‖ f ‖ L ω q ( ⋅ ) ( B k + m ) d t .

From this, for any k ≤ 0 and k ∈ Z ,

‖ ℋ φ p , b ( f ) ‖ L q ( ⋅ ) ( B k ) ∣ B k ∣ λ ‖ 1 ‖ L q ( ⋅ ) ( B k ) ≲ ‖ b ‖ Lip β ( Q p n ) ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p − n / q − − β K 2 d t ‖ f ‖ B ˙ ω , loc q ( ⋅ ) , λ ( Q p n ) ,

where K 2 = ∣ B k + m ∣ λ ‖ 1 ‖ L q ( ⋅ ) ( B k + m ) ∣ B k ∣ λ ‖ 1 ‖ L q ( ⋅ ) ( B k ) ≤ ∣ B k + m ∣ λ ∣ B k ∣ λ = p m n λ = ∣ t ∣ p n λ . Hence,

‖ ℋ φ p , b ( f ) ‖ B ˙ loc q ( ⋅ ) , λ ( Q p n ) ≲ C 2 ‖ b ‖ Lip β ( Q p n ) ‖ f ‖ B ˙ ω , loc q ( ⋅ ) , λ ( Q p n ) .

This finishes our proof.□

We discuss the boundedness of ℋ φ p , b on the p -adic Morrey-Herz space with variable exponents.

Theorem 3.3

Let ℓ , λ ∈ ( 0 , ∞ ) , q , q 1 ∈ P φ ( Q p n ) , r 1 ∈ P φ ( Q p n ) ∩ B ( Q p n ) , α , α 1 ∈ L ∞ ( Q p n ) ∩ C 0 log ( Q p n ) ∩ C ∞ log ( Q p n ) such that

(18) α ( 0 ) ≥ max − n r 1 + + α 1 ( 0 ) , − n r 1 + + α 1 ∞ ,

(19) and α ∞ ≤ min − n r 1 − + α 1 ( 0 ) , − n r 1 − + α 1 ∞ .

Assume that b ∈ CMO . r 1 ( ⋅ ) ( Q p n ) and the condition (8) in Theorem 3.1 holds. If

C 3 = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − max { ∣ t ∣ p λ − α 1 ( 0 ) , ∣ t ∣ p λ − α 1 ∞ } d t < ∞ ,

then ℋ φ p , b is bounded from M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) to M K ˙ ℓ , q ( ⋅ ) α ( ⋅ ) , λ ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) → M K ˙ ℓ , q ( ⋅ ) α ( ⋅ ) , λ ( Q p n ) ≲ C 3 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) .

Proof

By (15) and (4), for any k ∈ Z ,

(20) ‖ ℋ φ p , b ( f ) ‖ L q ( ⋅ ) ( S k ) ≲ ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ 1 ‖ L r 1 ( ⋅ ) ( B k ) ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − ‖ f ‖ L q 1 ( ⋅ ) ( S k + m ) d t ≲ p ζ 1 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − ‖ f ‖ L q 1 ( ⋅ ) ( S k + m ) d t ,

where m = log p ∣ t ∣ p and ζ 1 = max { k n / r 1 + , k n / r 1 − } . Besides, by Lemma 2.5,

‖ f ‖ L q 1 ( ⋅ ) ( S k + m ) ≲ p max { ( k + m ) ( λ − α 1 ( 0 ) ) , ( k + m ) ( λ − α 1 ∞ ) } ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) = p ζ 2 max { p m ( λ − α 1 ( 0 ) ) , p m ( λ − α 1 ∞ ) } ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) ≲ p ζ 2 max { ∣ t ∣ p λ − α 1 ( 0 ) , ∣ t ∣ p λ − α 1 ∞ } ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) ,

with ζ 2 = max { k ( λ − α 1 ( 0 ) ) , k ( λ − α 1 ∞ ) } . From this, by (20),

(21) ∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( S k ) ≲ p ζ 1 + ζ 2 C 3 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) .

Here,

(22) ζ 1 + ζ 2 = k n r 1 + + min { λ − α 1 ( 0 ) , λ − α 1 ∞ } , if k < 0 k n r 1 − + max { λ − α 1 ( 0 ) , λ − α 1 ∞ } , otherwise .

By Theorem 2.4, we have

(23) ∥ ℋ φ p , b ( f ) ∥ M K ˙ ℓ , q ( ⋅ ) α ( ⋅ ) , λ ( Q p n ) ≲ max { sup k 0 ∈ Z − ∪ { 0 } ℳ 1 , sup k 0 ∈ Z + ( ℳ 2 + ℳ 3 ) } ,

where

ℳ 1 = p − k 0 λ ∑ k = − ∞ k 0 p k α ( 0 ) ℓ ∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( S k ) ℓ 1 / ℓ , ℳ 2 = p − k 0 λ ∑ k = − ∞ − 1 p k α ( 0 ) ℓ ∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( S k ) ℓ 1 / ℓ ,

ℳ 3 = p − k 0 λ ∑ k = 0 k 0 p k α ∞ ℓ ∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( S k ) ℓ 1 / ℓ .

By (21) and (22), we obtain

ℳ 1 ≤ C 3 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) p − k 0 λ ∑ k = − ∞ k 0 p k α ( 0 ) ℓ + ( ζ 1 + ζ 2 ) ℓ 1 / ℓ ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) = C 3 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) p − k 0 λ ∑ k = − ∞ k 0 p k ℓ ζ 3 1 / ℓ ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) .

Here, ζ 3 = min { n r 1 + + λ − α 1 ( 0 ) + α ( 0 ) , n r 1 + + λ − α 1 ∞ + α ( 0 ) } . By having ζ 3 > 0 ,

(24) ℳ 1 ≲ C 3 p k 0 ( ζ 3 − λ ) ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) .

Similarly,

(25) ℳ 2 ≲ C 3 p − k 0 λ ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) .

Next, by putting ζ 4 = max { n r 1 − + λ − α 1 ( 0 ) + α ∞ , n r 1 − + λ − α 1 ∞ + α ∞ } and using (21) and (22), ℳ 3 is controlled as follows:

ℳ 3 ≲ C 3 p − k 0 λ ∑ k = 0 k 0 p k α ∞ ℓ + ( ζ 1 + ζ 2 ) ℓ 1 / ℓ ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) = C 3 p − k 0 λ ∑ k = 0 k 0 p k ℓ ζ 4 1 / ℓ ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) ≲ C 3 ( p k 0 ( ζ 4 − λ ) + p − k 0 λ ) ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) .

In view of the aforementioned argument, by the relations (23)–(25), λ > 0 , (18), and (19),

‖ ℋ φ p , b ( f ) ‖ M K ˙ ℓ , q ( ⋅ ) α ( ⋅ ) , λ ( Q p n ) ≲ C 3 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ M K ˙ ℓ , q 1 ( ⋅ ) α 1 ( ⋅ ) , λ ( Q p n ) .

This completes our proof.□

Theorem 3.4

Let q ∈ P φ ( Q p n ) , ℓ , λ ∈ ( 0 , ∞ ) , β ∈ ( 0 , 1 ] , ω ( x ) = ∣ x ∣ p β , and α ∈ L ∞ ( Q p n ) ∩ C 0 log ( Q p n ) ∩ C ∞ log ( Q p n ) with α ( 0 ) − α ∞ ≥ 0 . If b ∈ Lip β ( Q p n ) and

C 4 = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p n λ − n / q − − β max { ∣ t ∣ p λ − α ( 0 ) , ∣ t ∣ p λ − α ∞ } d t < ∞ ,

then ℋ φ p , b is bounded from M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) to M K ˙ ℓ , q ( ⋅ ) ( Q p n ) α ( ⋅ ) , λ . Moreover,

‖ ℋ φ p , b ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) → M K ˙ ℓ , q ( ⋅ ) α ( ⋅ ) , λ ( Q p n ) ≲ C 4 ‖ b ‖ Lip β ( Q p n ) .

Proof

By (17), for any k ∈ Z ,

∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( S k ) ≲ ‖ b ‖ Lip β ( Q p n ) ∫ Z p ∗ ∣ φ ( t ) ∣ × ∣ 1 − t ∣ p β ∣ t ∣ p − n / q − − β ‖ f ‖ L ω q ( ⋅ ) ( S k + m ) d t ,

where m = log p ∣ t ∣ p . Next, by Lemma 2.5, we obtain

‖ f ‖ L ω q ( ⋅ ) ( S k + m ) ≲ p max { ( k + m ) ( λ − α ( 0 ) ) , ( k + m ) ( λ − α ∞ ) } ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) = max { p m ( λ − α ( 0 ) ) , p m ( λ − α ∞ ) } p max { k ( λ − α ( 0 ) ) , k ( λ − α ∞ ) } ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) = max { ∣ t ∣ p λ − α ( 0 ) , ∣ t ∣ p λ − α ∞ } p max { k ( λ − α ( 0 ) ) , k ( λ − α ∞ ) } ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) .

As a consequence,

(26) ∥ ℋ φ p , b ( f ) ∥ L q ( ⋅ ) ( S k ) ≲ C 4 p max { k ( λ − α ( 0 ) ) , k ( λ − α ∞ ) } ‖ b ‖ Lip β ( Q p n ) ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) .

By Theorem 2.4, one has

(27) ∥ ℋ φ p , b ( f ) ∥ M K ˙ ℓ , q ( ⋅ ) α ( ⋅ ) , λ ( Q p n ) ≲ max { sup k 0 ∈ Z − { 0 } ℳ 1 , sup k 0 ∈ Z + ( ℳ 2 + ℳ 3 ) } .

Here, ℳ 1 , ℳ 2 , and ℳ 3 are given in the proof of Theorem 3.3.

Now, by the inequality (26) with k 0 ∈ Z − ∪ { 0 } ,

ℳ 1 ≲ C 4 ‖ b ‖ Lip β ( Q p n ) p − k 0 λ ∑ k = − ∞ k 0 p k α ( 0 ) ℓ + max { k ( λ − α ( 0 ) ) ℓ , k ( λ − α ∞ ) ℓ } 1 / ℓ ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) = C 4 ‖ b ‖ Lip β ( Q p n ) p − k 0 λ ∑ k = − ∞ k 0 p k ℓ min { λ , λ − α ∞ + α ( 0 ) } 1 / ℓ ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) .

Thus, by min { λ , λ − α ∞ + α ( 0 ) } > 0 , α ( 0 ) − α ∞ ≥ 0 , k 0 ∈ Z − ∪ { 0 } ,

(28) ℳ 1 ≲ C 4 ‖ b ‖ Lip β ( Q p n ) p k 0 ( min { λ , λ − α ∞ + α ( 0 ) } − λ ) ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) = C 4 ‖ b ‖ Lip β ( Q p n ) p k 0 . min { 0 , α ( 0 ) − α ∞ } ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) = C 4 ‖ b ‖ Lip β ( Q p n ) ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) .

By the aforementioned argument, we obtain

(29) ℳ 2 ≲ C 4 ‖ b ‖ Lip β ( Q p n ) p − k 0 λ ∥ f ∥ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) .

Next, ℳ 3 is estimated as follows:

ℳ 3 ≲ C 4 ‖ b ‖ Lip β ( Q p n ) p − k 0 λ ∑ k = 0 k 0 p k α ∞ ℓ + max { k ( λ − α ( 0 ) ) ℓ , k ( λ − α ∞ ) ℓ } 1 / ℓ ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) = C 4 ‖ b ‖ Lip β ( Q p n ) p − k 0 λ ∑ k = 0 k 0 p k ℓ max { λ − α ( 0 ) + α ∞ , λ } 1 / ℓ ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) = C 4 ‖ b ‖ Lip β ( Q p n ) p − k 0 λ ∑ k = 0 k 0 p k ℓ λ 1 / ℓ ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) ≲ C 4 ‖ b ‖ Lip β ( Q p n ) p − k 0 λ ( p k 0 λ + 1 ) ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) .

Hence, by (27)–(29),

‖ ℋ φ p , b ( f ) ‖ M K ˙ ℓ , q ( ⋅ ) α ( ⋅ ) , λ ( Q p n ) ≲ C 4 ‖ b ‖ Lip β ( Q p n ) ‖ f ‖ M K ˙ ℓ , q ( ⋅ ) , ω α ( ⋅ ) , λ ( Q p n ) .

This finishes our proof.□

Finally, we establish the boundedness of ℋ φ p , b on the p -adic local block space with variable exponent.

Theorem 3.5

Let q ∈ P φ ( Q p n ) ∩ B ( Q p n ) , β ∈ ( 0 , 1 ] , and u ∈ L W q ′ ( ⋅ ) ( Q p n ) . If b ∈ Lip β ( Q p n ) and

C 5 = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p − n / q − − n d t < ∞ ,

then ℋ φ p , b is bounded from LB u , q ( ⋅ ) ( Q p n ) to LB u , q ( ⋅ ) ( Q p n ) . Moreover,

‖ ℋ φ p , b ∣ ⋅ ∣ p − β ‖ LB u , q ( ⋅ ) ( Q p n ) → LB u , q ( ⋅ ) ( Q p n ) ≲ C 5 ‖ b ‖ Lip β ( Q p n ) .

Proof

Let f ∈ LB u , q ( ⋅ ) ( Q p n ) . Then,

f = ∑ k = 0 ∞ λ k b k ,

with ∑ k = 0 ∞ ∣ λ k ∣ ≲ ‖ f ‖ LB u , q ( ⋅ ) ( Q p n ) , b k a local ( u , L q ( ⋅ ) ) -block such that supp ( b k ) ⊂ B k for k ∈ N and

‖ b k ‖ L q ( ⋅ ) ( Q p n ) ≤ 1 u ( p k ) .

Now, we estimate

∣ ℋ φ p , b ( f ) ( ⋅ ) ∣ ≤ ∑ k = 0 ∞ ∣ λ k ∣ ℋ ˜ φ p , b ( b k ) ( ⋅ ) .

Here,

(30) ℋ ˜ φ p , b ( b k ) ( x ) = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ b ( x ) − b ( t x ) ∣ ∣ b k ( t x ) ∣ d t = ∑ j = 0 ∞ h φ , j p , b ( b k ) ( x ) ,

with h φ , j p , b ( b k ) ( x ) = ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ b ( x ) − b ( t x ) ‖ b k ( t x ) ∣ d t . This lead to that supp ( h φ , j p , b ) ⊂ B k + j . Moreover,

‖ h φ , j p , b ( b k ) ∣ ⋅ ∣ p − β ‖ L q ( ⋅ ) ( Q p n ) ≤ ‖ b ‖ Lip β ( Q p n ) ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ‖ b k ( t ⋅ ) ‖ L q ( ⋅ ) ( Q p n ) d t ≤ ‖ b ‖ Lip β ( Q p n ) ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p − n / q − ‖ b k ‖ L q ( ⋅ ) ( Q p n ) d t ≲ ‖ b ‖ Lip β ( Q p n ) ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p − n / q − d t 1 u ( p k + j ) . u ( p k + j ) u ( p k ) .

By u ∈ L W q ′ ( ⋅ ) ( Q p n ) , q ∈ B ( Q p n ) , and Lemma 2.13.(i),

u ( p k + j ) u ( p k ) ≲ ‖ χ B k + j ‖ L q ′ ( ⋅ ) ( Q p n ) ‖ χ B k ‖ L q ′ ( ⋅ ) ( Q p n ) ≲ ∣ B k + j ∣ ∣ B k ∣ ≲ p j n .

Based on the aforementioned arguments,

‖ h φ , j p , b ( b k ) ∣ ⋅ ∣ p − β ‖ L q ( ⋅ ) ( Q p n ) ≲ h j ∗ ‖ b ‖ Lip β ( Q p n ) . 1 u ( p k + j ) ,

where h j ∗ = ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p − n / q − − n d t .

By setting

h ˜ φ , j p , b ( b k ) = h φ , j p , b ( b k ) h j ∗ ‖ b ‖ Lip β ( Q p n ) if h j ∗ ‖ b ‖ Lip β ( Q p n ) ≠ 0 0 , otherwise .

Thus, by (30),

ℋ ˜ φ p , b ( b k ) ( x ) ∣ x ∣ p − β = ∑ j = 0 ∞ h j ∗ ‖ b ‖ Lip β ( Q p n ) . h ˜ φ , j p , b ( b k ) ( x ) ∣ x ∣ p − β , for a.e. x ∈ Q p n .

Note that, for any j ∈ N , h ˜ φ , j p , b ( b k ) ∣ ⋅ ∣ p − β is local ( u , L q ( ⋅ ) ) -block. Then,

‖ ℋ ˜ φ p , b ( b k ) ∣ ⋅ ∣ p − β ‖ LB u , q ( ⋅ ) ( Q p n ) ≲ ‖ b ‖ Lip β ( Q p n ) ∑ j = 0 ∞ h j ∗ .

Namely,

‖ ℋ ˜ φ p , b ( b k ) ∣ ⋅ ∣ p − β ‖ LB u , q ( ⋅ ) ( Q p n ) ≲ C 5 ‖ b ‖ Lip β ( Q p n ) ,

for all k ∈ N . Since LB u , q ( ⋅ ) ( Q p n ) is Banach space, we have

‖ ℋ φ p , b ( f ) ∣ ⋅ ∣ p − β ‖ LB u , q ( ⋅ ) ( Q p n ) ≤ ∑ k = 0 ∞ ∣ λ k ∣ ℋ ˜ φ p , b ( b k ) ∣ ⋅ ∣ p − β LB u , q ( ⋅ ) ( Q p n ) ≤ ∑ k = 0 ∞ ∣ λ k ∣ ⋅ ‖ ℋ ˜ φ p , b ( b k ) ∣ ⋅ ∣ p − β ‖ LB u , q ( ⋅ ) ( Q p n ) ≲ C 5 ‖ b ‖ Lip β ( Q p n ) ∑ k = 0 ∞ ∣ λ k ∣ ≤ C 5 ‖ b ‖ Lip β ( Q p n ) ‖ f ∣ ‖ LB u , q ( ⋅ ) ( Q p n ) .

Then, the proof of the theorem is achieved.□

Theorem 3.6

Let q ∈ P φ ( Q p n ) , q 1 , r 1 ∈ P φ ( Q p n ) ∩ B ( Q p n ) , u ∈ L W q 1 ′ ( ⋅ ) ( Q p n ) , and v ∈ L W q ′ ( ⋅ ) ( Q p n ) with v ( z ) = z − n / r 1 − u ( z ) , for all z > 0 . Assume that b ∈ CMO . r 1 ( ⋅ ) ( Q p n ) and the condition (8) in Theorem 3.1 holds.

If

C 6 = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − − n d t < ∞ ,

then ℋ φ p , b is bounded from LB u , q 1 ( ⋅ ) ( Q p n ) to LB v , q ( ⋅ ) ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ LB u , q 1 ( ⋅ ) ( Q p n ) → LB v , q ( ⋅ ) ( Q p n ) ≲ C 6 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) .

Proof

For any f ∈ LB u , q 1 ( ⋅ ) ( Q p n ) , we have

f = ∑ k = 0 ∞ λ k b k ,

with ∑ k = 0 ∞ ∣ λ k ∣ ≲ ‖ f ‖ LB u , q 1 ( ⋅ ) ( Q p n ) , b k a local ( u , L q 1 ( ⋅ ) ) -block such that supp ( b k ) ⊂ B k for k ∈ N and

‖ b k ‖ L q 1 ( ⋅ ) ( Q p n ) ≤ 1 u ( p k ) .

We recall ℋ ˜ φ p , b ( f ) ( ⋅ ) = ∑ j = 0 ∞ h φ , j p , b ( b k ) ( ⋅ ) with h φ , j p , b ( b k ) ( ⋅ ) = ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ b ( ⋅ ) − b ( t ⋅ ) ‖ b k ( t ⋅ ) ∣ d t .

By (15) and (4), for any j ∈ N ,

‖ h φ , j p , b ( b k ) ‖ L q ( ⋅ ) ( B k + j ) ≤ ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − ‖ b k ‖ L q 1 ( ⋅ ) ( Q p n ) d t ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ 1 ‖ L r 1 ( ⋅ ) ( B k + j ) ≤ p max { ( k + j ) n / r 1 + , ( k + j ) n / r 1 − } ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − v ( p k + j ) u ( p k ) d t . 1 v ( p k + j ) .

By v , u ∈ L W q 1 ′ ( ⋅ ) ( Q p n ) , q 1 ∈ B ( Q p n ) , and Lemma 2.13.(i),

p max { ( k + j ) n / r 1 + , ( k + j ) n / r 1 − } v ( p k + j ) u ( p k ) = u ( p k + j ) u ( p k ) ≲ ‖ χ B k + j ‖ L q 1 ′ ( ⋅ ) ( Q p n ) ‖ χ B k ‖ L q 1 ′ ( ⋅ ) ( Q p n ) ≲ ∣ B k + j ∣ ∣ B k ∣ ≲ p j n .

Thus,

(31) ‖ h φ , j p , b ( b k ) ‖ L q ( ⋅ ) ( B k + j ) ≲ h ˜ j ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) . 1 v ( p k + j ) ,

where

h ˜ j = ∫ S − j ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q 1 − − n / r 1 − − n d t .

Let us denote

m φ , j p , b ( b k ) = h φ , j p , b ( b k ) h ˜ j ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) if h ˜ j ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ≠ 0 0 , otherwise .

Hence,

ℋ ˜ φ p , b ( b k ) ( ⋅ ) = ∑ j = 0 ∞ h ˜ j ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) m φ , j p , b ( b k ) ( ⋅ ) .

Note that, for j ∈ N , by supp ( h φ , j p , b ( b k ) ) ⊂ B k + j and (31), the function m φ , j p , b ( b k ) is a local ( v , L q ( ⋅ ) ) -block. As a consequence,

‖ ℋ ˜ φ p , b ( b k ) ‖ LB v , q ( ⋅ ) ( Q p n ) ≲ ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ∑ j = 0 ∞ ∣ h ˜ j ∣ = C 6 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ,

for all k ∈ N .

Since LB v , q ( ⋅ ) ( Q p n ) is Banach space, we obtain

‖ ℋ φ p , b ( f ) ‖ LB v , q ( ⋅ ) ( Q p n ) ≤ ∑ k = 0 ∞ ∣ λ k ∣ ℋ ˜ φ p , b ( b k ) LB v , q ( ⋅ ) ( Q p n ) ≤ ∑ k = 0 ∞ ∣ λ k ∣ ⋅ ‖ ℋ ˜ φ p , b ( b k ) ‖ LB v , q ( ⋅ ) ( Q p n ) ≲ C 6 ‖ b ‖ CMO . r 1 ( ⋅ ) ( Q p n ) ‖ f ‖ LB u , q 1 ( ⋅ ) ( Q p n ) .

This completes our proof.□

If q ( ⋅ ) , q 1 ( ⋅ ) , r 1 ( ⋅ ) , α ( ⋅ ) , and α 1 ( ⋅ ) are constants, we obtain the following useful result.

Corollary 3.7

Let q , q 1 , r 1 ∈ ( 1 , ∞ ) , λ ∈ R such that

(32) 1 q = 1 q 1 + 1 r 1 .

If b ∈ CMO . r 1 ( Q p n ) and

C 1 ∗ = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p n ( λ − 1 / r 1 ) d t < ∞ ,

then ℋ φ p , b is bounded from B ˙ loc q 1 , λ ( Q p n ) to B ˙ loc q , λ ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ B ˙ loc q 1 , λ ( Q p n ) → B ˙ loc q , λ ( Q p n ) ≲ C 1 ∗ ‖ b ‖ CMO . r 1 ( Q p n ) .

Corollary 3.8

Let q ∈ ( 1 , ∞ ) , λ ∈ R , β ∈ ( 0 , 1 ] , ω ( x ) = ∣ x ∣ p β . If b ∈ Lip β ( Q p n ) and

C 2 ∗ = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p n λ − n / q − β d t < ∞ ,

then ℋ φ p , b is bounded from B ˙ ω , loc q , λ ( Q p n ) to B ˙ loc q , λ ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ B ˙ ω , loc q , λ ( Q p n ) → B ˙ loc q , λ ( Q p n ) ≲ C 2 ∗ ‖ b ‖ Lip β ( Q p n ) .

Corollary 3.9

Let ℓ , λ ∈ ( 0 , ∞ ) , q , q 1 , r 1 ∈ ( 1 , ∞ ) , and α , α 1 ∈ R such that α = − n / r 1 + α 1 . Assume that b ∈ CMO . r 1 ( Q p n ) and the condition (32) in Corollary 3.7 holds. If

C 3 ∗ = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q + λ − α 1 1 d t < ∞ ,

then ℋ φ p , b is bounded from M K ˙ ℓ , q 1 α 1 , λ ( Q p n ) to M K ˙ ℓ , q α , λ ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ M K ˙ ℓ , q 1 α 1 , λ ( Q p n ) → M K ˙ ℓ , q α , λ ( Q p n ) ≲ C 3 ∗ ‖ b ‖ CMO . r 1 ( Q p n ) .

Corollary 3.10

Let q ∈ ( 1 , ∞ ) , ℓ , λ ∈ ( 0 , ∞ ) , β ∈ ( 0 , 1 ] , ω ( x ) = ∣ x ∣ p β , and α ∈ R . If b ∈ Lip β ( Q p n ) and

C 4 ∗ = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p n λ − n / q − β + λ − α d t < ∞ ,

then ℋ φ p , b is bounded from M K ˙ ℓ , q , ω α , λ ( Q p n ) to M K ˙ ℓ , q α , λ ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ M K ˙ ℓ , q , ω α , λ ( Q p n ) → M K ˙ ℓ , q α , λ ( Q p n ) ≲ C 4 ∗ ‖ b ‖ Lip β ( Q p n ) .

Corollary 3.11

Let q ∈ ( 1 , ∞ ) , β ∈ ( 0 , 1 ] , and u ∈ L W q ′ ( Q p n ) . If b ∈ Lip β ( Q p n ) and

C 5 ∗ = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ 1 − t ∣ p β ∣ t ∣ p − n / q − n d t < ∞ ,

then ℋ φ p , b is bounded from LB u , q ( Q p n ) to LB u , q ( Q p n ) . Moreover,

‖ ℋ φ p , b ∣ ⋅ ∣ p − β ‖ LB u , q ( Q p n ) → LB u , q ( Q p n ) ≲ C 5 ∗ ‖ b ‖ Lip β ( Q p n ) .

Corollary 3.12

Let q , q 1 , r 1 ∈ ( 1 , ∞ ) , u ∈ L W q 1 ′ ( Q p n ) , and v ∈ L W q ′ ( Q p n ) with v ( z ) = z − n / r 1 u ( z ) , for all z > 0 . Assume that b ∈ CMO . r 1 ( Q p n ) and the condition (32) in Corollary 3.7 holds.

If

C 6 ∗ = ∫ Z p ∗ ∣ φ ( t ) ∣ ⋅ ∣ t ∣ p − n / q − n d t < ∞ ,

then ℋ φ p , b is bounded from LB u , q 1 ( Q p n ) to LB v , q ( Q p n ) . Moreover,

‖ ℋ φ p , b ‖ LB u , q 1 ( Q p n ) → LB v , q ( Q p n ) ≲ C 6 ∗ ‖ b ‖ CMO . r 1 ( Q p n ) .

4 Comments

In 2021, Dung et al. [13] investigated the commutator of the p -adic Hardy-Cesàro operator, which is generalized of the operator ℋ φ p , b . Some sufficient criteria for the boundedness of commutators of Hardy-Cesàro operators with symbols in central BMO spaces on the p -adic Herz spaces, p -adic Morrey spaces, and p -adic Morrey-Herz spaces are discussed in [13]. In addition, Wang et al. [28] introduced the central BMO spaces with variable exponent and Yee et al. [26] presented the local block spaces with variable exponent. As a natural development, Theorems 3.1–3.6 solved the boundedness of ℋ φ p , b with symbols in central BMO spaces with variable exponent and Lipschitz spaces on the p -adic function spaces.

The study of the necessary and sufficient conditions for the boundedness of ℋ φ p , b with symbols in central BMO spaces with the variable exponent on some function spaces is an interesting one. We will continue investigating this problem in the hopes of obtaining some conclusions.

Acknowledgments

The authors are grateful to the anonymous referee for the valuable suggestions and comments which lead to the improvement of the article.

  1. Funding information: Kieu Huu Dung would like to thank Van Lang University, Vietnam for funding this work.

  2. Conflict of interest: The authors declare no conflict of interest.

  3. Data availability statement: No data were used to support this study.

References

[1] A. V. Antoniouk, A. Y. Khrennikov, and A. N. Kochubei, Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables, J. Pseudo-Differ. Oper. Appl. 11 (2020), no. 1, 311–343, DOI: https://doi.org/10.1007/s11868-019-00320-3. 10.1007/s11868-019-00320-3Search in Google Scholar

[2] S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, Harmonic analysis in the p-adic Lizorkin spaces: fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems, J. Fourier Anal. Appl. 12 (2006), no. 4, 393–425, DOI: https://doi.org/10.1007/s00041-006-6014-0. 10.1007/s00041-006-6014-0Search in Google Scholar

[3] B. Dragovich, A. Y. Khrennikov, S. V. Kozyrev, I. V. Volovich, and E. I. Zelenov, p-Adic mathematical physics: the first 30 years, p-Adic Numbers Ultrametric Anal. Appl. 9 (2017), no. 2, 87–121, DOI: https://doi.org/10.1134/S2070046617020017. 10.1134/S2070046617020017Search in Google Scholar

[4] K. H. Dung, D. V. Duong, and N. D. Duyet, Weighted Triebel-Lizorkin and Herz spaces estimates for p-adic Hausdorff type operator and its applications, Anal. Math. 48 (2022), no. 3, 717–740, DOI: https://doi.org/10.1007/s10476-022-0129-7. 10.1007/s10476-022-0129-7Search in Google Scholar

[5] A. Y. Khrennikov and A. N. Kochubei, On the p-adic Navier-Stokes equation, Appl. Anal. 99 (2020), no. 8, 1425–1435, DOI: https://doi.org/10.1080/00036811.2018.1533120. 10.1080/00036811.2018.1533120Search in Google Scholar

[6] A. Y. Khrennikov, S. V. Kozyrev, and W. A. Zúnnnniga-Galindo, Ultrametric Pseudodifferential Equations and Applications, Cambridge University Press, Cambridge, 2018. 10.1017/9781316986707Search in Google Scholar

[7] V. S. Vladimirov and I. V. Volovich, p-adic quantum mechanics, Comm. Math. Phys. 123 (1989), no. 4, 659–676, DOI: https://doi.org/10.1007/BF01218590. 10.1007/BF01218590Search in Google Scholar

[8] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994. 10.1142/1581Search in Google Scholar

[9] S. S. Volosivets, Hausdorff operators on p-adic linear spaces and their properties in Hardy, BMO, and Hölder spaces, Math. Notes 93 (2013), no. 3, 382–391, DOI: https://doi.org/10.1134/S0001434613030048. 10.1134/S0001434613030048Search in Google Scholar

[10] K. S. Rim and J. Lee, Estimates of weighted Hardy-Littlewood averages on the p-adic vector space, J. Math. Anal. Appl. 324 (2006), no. 2, 1470–1477, DOI: https://doi.org/10.1016/j.jmaa.2006.01.038. 10.1016/j.jmaa.2006.01.038Search in Google Scholar

[11] N. M. Chuong, D. V. Duong, and K. H. Dung, Some estimates for p-adic rough multilinear Hausdorff operators and commutators on weighted Morrey-Herz type spaces, Russ. J. Math. Phys. 26 (2019), no. 1, 9–31, DOI: https://doi.org/10.1134/S1061920819010023. 10.1134/S1061920819010023Search in Google Scholar

[12] N. M. Chuong, H. D. Hung, and N. T. Hong, Bounds of p-adic weighted Hardy-Cesàro operators and their commutators on p-adic weighted spaces of Morrey types, p-Adic Numbers Ultrametric Anal. Appl. 8 (2016), no. 1, 31–44, DOI: https://doi.org/10.1134/S2070046616010039. 10.1134/S2070046616010039Search in Google Scholar

[13] K. H. Dung, D. V. Duong, and T. N. Luan, Weighted central BMO type space estimates for commutators of p-adic Hardy-Cesàro operators, p-Adic Numbers Ultrametric Anal. Appl. 13 (2021), no. 4, 266–279, DOI: https://doi.org/10.1134/S2070046621040026. 10.1134/S2070046621040026Search in Google Scholar

[14] G. Gao and Y. Zhong, Some estimates of Hardy operators and their commutators on Morrey-Herz spaces, J. Math. Inequal. 11 (2017), 49–58, DOI: https://dx.doi.org/10.7153/jmi-11-05. 10.7153/jmi-11-05Search in Google Scholar

[15] R. Liu and J. Zhou, Weighted multilinear p-adic Hardy operators and commutators, Open Math. 15 (2017), no. 1, 1623–1634, DOI: https://doi.org/10.1515/math-2017-0139. 10.1515/math-2017-0139Search in Google Scholar

[16] A. Kochubei, Radial solutions of non-Archimedean pseudodifferential equations, Pacific J. Math. 269 (2014), no. 2, 355–369, DOI: https://doi.org/10.2140/pjm.2014.269.355. 10.2140/pjm.2014.269.355Search in Google Scholar

[17] A. Almeida and D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl. 394 (2012), no. 2, 781–795, DOI: https://doi.org/10.1016/j.jmaa.2012.04.043. 10.1016/j.jmaa.2012.04.043Search in Google Scholar

[18] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer, Basel, 2013. 10.1007/978-3-0348-0548-3Search in Google Scholar

[19] K. H. Dung, D. L. C. Minh, and P. T. K. Thuy, Commutators of Hardy-Cesàro operators on Morrey-Herz spaces with variable exponents, AIMS Math. 7 (2022), no. 10, 19147–19166, DOI: https://doi.org/10.3934/math.20221051. 10.3934/math.20221051Search in Google Scholar

[20] L. Diening and M. Ružička, Calderón-Zygmund operators on generalized Lebesgue spaces Lp(⋅) and problems related to fluid dynamics, J. Reine. Angew. Math. 563 (2003), 197–220, DOI: https://doi.org/10.1515/crll.2003.081. 10.1515/crll.2003.081Search in Google Scholar

[21] Y. Lu and Y. P. Zhu, Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents, Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 7, 1180–1194, DOI: https://doi.org/10.1007/s10114-014-3410-2. 10.1007/s10114-014-3410-2Search in Google Scholar

[22] J. L. Wu and W. J. Zhao, Boundedness for fractional Hardy-type operator on variable-exponent Herz-Morrey spaces, Kyoto J. Math. 56 (2016), no. 4, 831–845, DOI: https://doi.org/10.1215/21562261-3664932. 10.1215/21562261-3664932Search in Google Scholar

[23] S. Wang and J. Xu, Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents, Open Math. 19 (2021), no. 1, 412–426, DOI: https://doi.org/10.1515/math-2021-0024. 10.1515/math-2021-0024Search in Google Scholar

[24] M. Izuki, Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponent, J. Math. Sci. Res. 13 (2009), no. 10, 243–253. 10.11650/twjm/1500405453Search in Google Scholar

[25] Y. Mizuta, T. Ohno, and T. Shimomura, Boundedness of maximal operators and Sobolevas theorem for non-homogeneous central Morrey spaces of variable exponent, Hokkaido Math. J. 44 (2015), no. 2, 185–201, DOI: https://doi.org/10.14492/hokmj/1470053290. 10.14492/hokmj/1470053290Search in Google Scholar

[26] T. L. Yee, K. L. Cheung, K. P. Ho, and C. K. Suen, Local sharp maximal functions, geometrical maximal functions and rough maximal functions on local Morrey spaces with variable exponents, Math. Inequal. Appl. 23 (2020), no. 4, 1509–1528, DOI: https://doi.org/10.7153/mia-2020-23-10810.7153/mia-2020-23-108Search in Google Scholar

[27] L. F. Chacón-Cortés and H. Rafeiro, Variable exponent Lebesgue spaces and Hardy-Littlewood maximal function on p-adic numbers, p-Adic Numbers Ultrametric Anal. Appl. 12 (2020), no. 2, 90–111, DOI: https://doi.org/10.1134/S2070046620020028. 10.1134/S2070046620020028Search in Google Scholar

[28] D. H. Wang, Z. G. Liu, J. Zhou, and Z. D. Teng, Central BMO spaces with variable exponent, arXiv:1708.00285, 2017, DOI: https://doi.org/10.48550/arXiv.1708.00285. Search in Google Scholar

[29] M. Izuki, Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J. 40 (2010), no. 3, 343–355, DOI: https://doi.org/10.32917/hmj/1291818849. 10.32917/hmj/1291818849Search in Google Scholar
Received: 2022-07-18
Revised: 2023-02-03
Accepted: 2023-03-28
Published Online: 2023-07-03

© 2023 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2022-0579
Keywords for this article
Hardy-Littlewood operator; commutator; local central Morrey space; Morrey-Herz space; local block space; central BMO space; Lipschitz space; variable exponent; p-adic analysis
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2022-0579/html
Scroll to top button