Home B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
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B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces

  • Javanshir J. Hasanov , Rabil Ayazoglu and Simten Bayrakci EMAIL logo
Published/Copyright: July 10, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this article, we consider the Laplace-Bessel differential operator

Δ B k , n = ∑ i = 1 k ∂ 2 ∂ x i 2 + γ i x i ∂ ∂ x i + ∑ i = k + 1 n ∂ 2 ∂ x i 2 , γ 1 > 0 , … , γ k > 0 .

Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator M b , γ and the commutator [ b , A γ ] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator [ b , I α , γ ] of the B-Riesz potential on B-Morrey spaces L p , λ , γ , when b ∈ BMO γ .

Keywords: commutator; generalized shift operator; B-Riesz potential; B-maximal function; B-Morrey space; bounded mean oscillation (BMO) space
MSC 2010: 42B20; 42B25; 42B35

1 Introduction

The Laplace-Bessel differential operator, defined by

Δ B k , n = ∑ i = 1 k ∂ 2 ∂ x i 2 + γ i x i ∂ ∂ x i + ∑ i = k + 1 n ∂ 2 ∂ x i 2 , γ 1 > 0 , … , γ k > 0 ,

is an important technical tool in the Fourier-Bessel harmonic analysis and applications. It has been investigated by many researchers, such as Muckenhoupt, Stein, Kipriyanov, Trimeche, Lyakhov, Stempak, Gadjiev, Aliev, Guliyev, Bayrakci, Hasanov, Serbetci, Keskin and Ekincioglu [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].

Given a linear operator T acting on functions and given a function b, the commutator [ T , b ] is formally defined by

[ T , b ] f = T ( b f ) − b T ( f ) .

The first result on commutators was obtained by Coifman, Rochberg and Weiss [18]. They showed that if T is a classical Calderon-Zygmund singular integral operator and b ∈ BMO , then the commutator [ T , b ] is bounded on L p ( ℝ n ) , 1 < p < ∞ . Later, Chanillo [19] proved that commutators characterize Riesz potentials on the bounded mean oscillation (BMO) space.

Classical Morrey spaces L p , λ ( ℝ n ) , 0 < λ < n , 1 ≤ p < ∞ were introduced in 1938 by Morrey [20] to study the local behavior of solutions to second-order elliptic partial differential equations. Morrey spaces defined as the set of all functions f ∈ L loc p ( ℝ n ) such that the following norm is finite

∥ f ∥ L p , λ ≡ f L p , λ ( ℝ n ) = sup x ∈ ℝ n , r > 0 1 r λ ∫ B ( x , r ) | f ( y ) | p d y 1 / p ,

where B ( x , r ) is the ball in ℝ n centered at x and with radius r > 0 . Moreover, for the readers, it is convenient to declare in what sense L p , λ ( ℝ n ) is trivial when λ < 0 , λ > n and L p , 0 ( ℝ n ) ≅ L p ( ℝ n ) , L p , n ( ℝ n ) ≅ L ∞ ( ℝ n ) .

Classical Morrey spaces are investigated by Adams [21] in a study related to Riesz potentials and recently also by Chiarenza and Frasca [22] in studying the boundedness of the Hardy-Littlewood maximal operator, Riesz potentials and singular integral operators. Moreover, Mixed Morrey Spaces, a generalization of Morrey spaces, are defined by Scapellato [23,24] and their applications to partial differential equations of parabolic type are examined. Furthermore, Morrey spaces generated by generalized translation operator and their applications have been research areas for many mathematicians such as Guliyev, Hasanov, Zeren, Ekincioglu and Uygur [5,9,10,25,26].

In this article, we consider Morrey spaces L p , λ , γ called B-Morrey space, B-maximal commutators, commutators of singular integral operators and Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we prove the boundedness of the B-maximal commutator M b , γ and the commutator [ b , A γ ] of the B-singular integral operator on the B-Morrey space, 1 < p < ∞ when b ∈ BMO γ . We also obtain the Hardy-Littlewood-Sobolev-type theorem for the commutator [ b , I α , γ ] of the B-Riesz potential when b ∈ BMO γ .

The article is organized as follows. Section 2 contains some basic definitions and results on the Lebesgue and Morrey spaces. Main results and their proofs are given in Section 3.

2 Preliminaries

Let ℝ k , + n = { x = ( x 1 , … , x n ) ∈ ℝ n : x 1 > 0 , … , x k > 0 , 1 ≤ k ≤ n } , E ( x , r ) = { y ∈ ℝ k , + n ; | x − y | < r } , γ = ( γ 1 , … , γ k ) , γ 1 > 0 , … , γ k > 0 , | γ | = γ 1 + … + γ k , x ′ = ( x 1 , … , x k ) ∈ ℝ k and ( x ′ ) γ d x = x 1 γ 1 ⋯ x k γ k d x 1 … d x k . For a measurable set E ⊂ ℝ k , + n , let | E | γ = ∫ E ( x ′ ) γ d x and | E ( 0 , r ) | γ = ω ( n , k , γ ) r Q , Q = n + | γ | , where

ω ( n , k , γ ) = ∫ E ( 0 , 1 ) ( x ′ ) γ d x = π n − k / 2 2 k ∏ i = 1 k Γ γ i + 1 2 Γ γ i 2 .

Let L p , γ ≡ L p , γ ( ℝ k , + n ) , 1 ≤ p < ∞ be the space of all measurable functions on ℝ k , + n with the norm

∥ f ∥ L p , γ = ∫ ℝ k , + n | f ( x ) | p ( x ′ ) γ d x 1 p < ∞ .

In the case p = ∞ , the space L ∞ , γ ( ℝ k , + n ) is defined by means of the usual modification

∥ f ∥ L ∞ = ess sup | f ( x ) | , x ∈ ℝ k , + n .

The weak- L p , γ space W L p , γ ≡ W L p , γ ( ℝ k , + n ) is defined by

∥ f ∥ W L p , γ = sup r > 0 r | { x ∈ ℝ k , + n   : | f ( x ) | > r } | γ 1 / p , 1 ≤ p < ∞ .

The weighted space L p , ω , γ ≡ L p , ω , γ ( ℝ k , + n ) is the space of measurable functions on ℝ k , + n defined by

∥ f ∥ L p , ω , γ = ∫ ℝ k , + n | f ( x ) | p ω ( x ) ( x ′ ) γ d x 1/ p , 1 ≤ p < ∞

and L ∞ , ω , γ ( ℝ k , + n ) = L ∞ ( ℝ k , + n ) .

Definition 2.1

For 1 < p < ∞ and a nonnegative locally integrable function ω on ℝ k , + n , ω is in the Muckenhoupt class A p , γ if it satisfies the condition

sup x ∈ ℝ k , + n , r > 0 1 | E ( x , r ) | γ ∫ E ( x , r ) ω ( y ) ( y ′ ) γ d y 1 | E ( x , r ) | γ ∫ E ( x , r ) ω − 1 p − 1 ( y ) ( y ′ ) γ d y p − 1 < ∞

and a weight function ω belongs to the class A 1, γ if there exists a positive constant C such that for any x ∈ ℝ k , + n and r > 0 ,

| E ( x , r ) | γ − 1 ∫ E ( x , r ) ω ( y ) ( y ′ ) γ d y ≤ C ess sup y ∈ E ( x , r ) 1 ω ( y ) .

Definition 2.2

The generalized shift operator T y is defined by

T y f ( x ) = C γ , k ∫ 0 π … ∫ 0 π f ( ( x ′ , y ′ ) β , x ″ − y ″ ) d ν ( β ) ,

where

C γ , k = π − k 2 ∏ i = 1 k Γ γ i + 1 2 Γ γ i 2 = 2 k π k ω ( 2 k , k , γ )

x ′ ∈ ℝ k , x ″ ∈ ℝ n − k , 1 ≤ k ≤ n , ( x ′ , y ′ ) β = ( ( x 1 , y 1 ) β 1 , … , ( x k , y k ) β k ) , ( x i , y i ) β i = ( x i 2 − 2 x i y i cos β i + y i 2 ) 1 / 2 , 1 ≤ i ≤ k and d ν ( β ) = ∏ i = 1 k sin γ i − 1 β i d β 1 … d β k .

It is well known that T y is closely related to the Laplace-Bessel differential operator Δ B k , n , see [13,27] for details.

Lemma 2.3

For all x ∈ ℝ k , + n

∫ E ( x , t ) g ( y ) ( y ′ ) γ d y = C γ , k − 1 ∫ B ( ( x , 0 ) , t ) g z 1 2 + z ¯ 1 2 , … , z k 2 + z ¯ k 2 , z ″ d ν ( z , z ′ ¯ ) ,

where B ( ( x , 0 ) , t ) = { ( z , z ′ ¯ ) ∈ ℝ n × ( 0 , ∞ ) k : | ( x 1 − z 1 2 + z ¯ 1 2 , … , x k − z k 2 + z ¯ k 2 , x ″ − z ″ ) | < t } , d ν ( z , z ′ ¯ ) = ( z ′ ¯ ) γ − 1 d z d z ′ ¯ , d z ′ ¯ = d z ¯ 1 ′ ⋯ d z ¯ k ′ and ( z ′ ¯ ) γ − 1 = ( z ¯ 1 ′ ) γ 1 − 1 ⋯ ( z ¯ k ′ ) γ k − 1 .

Lemma 2.4

For all x ∈ ℝ k , + n

∫ E (0, r ) T y g ( x ) ( y ′ ) γ d y = ∫ E ( ( x ,0), r ) g z 1 2 + z ¯ 1 2 , … , z k 2 + z ¯ k 2 , z ″ d ν ( z , z ¯ ′ ) ,

where E ( ( x , 0 ) , r ) = { ( z , z ′ ¯ ) ∈ ℝ n × ( 0 , ∞ ) k : | ( x − z , z ′ ¯ ) | < r } .

Lemma 2.5

For all x ∈ ℝ k , + n

∫ ℝ k , + n T y g ( x ) φ ( y ) ( y ′ ) γ d y = ∫ ℝ n × ( 0 , ∞ ) k g z 1 2 + z ¯ 1 2 , … , z k 2 + z ¯ k 2 , z ″ φ ( z , z ′ ¯ ) d ν ( z , z ′ ¯ ) .

The proofs of Lemmas 2.3–2.5 are immediately obtained by changing the variables as follows: z ″ = y ″ , z i = y i cos α i , z ¯ i = y i sin α i , 0 ≤ α i < π , i = 1 , … , k , y ∈ ℝ k , + n z ′ ¯ = ( z ¯ 1 , … , z ¯ k ) and ( z , z ¯ ′ ) ∈ ℝ n × ( 0 , ∞ ) k , 1 ≤ k ≤ n .

Lemma 2.6

[28] Let 0 < α < Q . Then, | T y | x | α − Q − | y | α − Q | ≤ 2 Q − α + 1 | y | α − Q − 1 | x | , 2 | x | ≤ | y | .

Definition 2.7

[7] Let 1 ≤ p < ∞ , 0 ≤ λ ≤ Q . B-Morrey space L p , λ , γ = L p , λ , γ ( ℝ k , + n ) , associated with the Laplace-Bessel differential operator, is defined as the space of locally integrable functions f with the finite norm

∥ f ∥ L p , λ , γ = sup t > 0 , x ∈ ℝ k , + n t − λ ∫ E ( 0 , t ) T y | f | p ( x ) ( y ′ ) γ d y 1 / p .

Note that L p , 0 , γ = L p , γ , L p , Q , γ = L ∞ . If λ < 0 or λ > Q , then L p , λ , γ = Θ .

Definition 2.8

[9] Let 1 ≤ p < ∞ , 0 ≤ λ ≤ Q . The weak B-Morrey space W L p , λ , γ = W L p , λ , γ ( ℝ k , + n ) is defined by

∥ f ∥ W L p , λ , γ = sup r > 0 r sup t > 0 , x ∈ ℝ k , + n t − λ ∫ { y ∈ E ( 0 , t ) : T y [ | f | ] ( x ) > r } ( y ′ ) γ d y 1 / p .

Note that L p , λ , γ ⊂ W L p , λ , γ and ∥ f ∥ W L p , λ , γ ≤ ∥ f ∥ L p , λ , γ .

Definition 2.9

The B -BMO space generated by the generalized shift operator and denoted by BMO γ = BMO γ ( ℝ k , + n ) is defined as the space of locally integrable functions f with the finite norm

∥ f ∥ BMO γ = sup t > 0 , x ∈ ℝ k , + n | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) T y f ( x ) − f E ( 0 , t ) ( x ) | ( y ′ ) γ d y |

or

∥ f ∥ BMO γ = inf C sup t > 0 , x ∈ ℝ k , + n | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) | T y f ( x ) − C | ( y ′ ) γ d y ,

where f E ( 0 , t ) ( x ) = | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) T y f ( x ) ( y ′ ) γ d y . Moreover, the BMO ( X , ν ) space is defined as the space of all functions f such that the following norm is finite

∥ f ∥ BMO ( X , ν ) = sup t > 0 , x ∈ X ν ( B ( x , t ) ) − 1 ∫ B ( x , t ) | f ( y ) − f B ( x , t ) | d ν ( y ) < ∞ ,

where f B ( x , t ) = ν ( B ( x , t ) ) − 1 ∫ B ( x , t ) f ( y ) d ν ( y ) .

We also define the sharp maximal operator M γ # generated by the generalized shift operator. Besides its close relation to B -BMO , it is also a very convenient tool for a pointwise control of many operators arising in the Fourier-Bessel harmonic analysis.

The sharp maximal operator M γ # is defined by

M γ # f ( x ) = sup t > 0 | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) | T y f ( x ) − f E ( 0 , t ) ( x ) | ( y ′ ) γ d y .

If ν is a doubling measure (i.e., there exists a constant C such that ν ( 2 Q ) ≤ C ν ( Q ) for all cubes Q ), then it satisfies the following John-Nirenberg inequality.

Theorem 2.10

[29] Let f ∈ BMO ( X , ν ) and ν be a doubling measure. Then, for every r > 0 there exists two constants c 1 and c 2 , independent of f , r such that

ν ( { y ∈ B ( x , t )   : | f ( x ) − f B ( x , t ) | > r } ) ≤ c 1 ν ( B ( x , t ) ) e − c 2 r / ∥ f ∥ BMO ( X , ν ) .

It is clear that BMO ( X , ν ) = BMO p ( X , ν ) provided that the John-Nirenberg inequality holds.

Theorem 2.11

[30]

  1. Let f ∈ L 1 , γ loc ( ℝ k , + n ) . If

    ∥ f ∥ BMO p , γ = sup t > 0 , x ∈ ℝ k , + n | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) | T y f ( x ) − f E ( 0 , t ) ( x ) | p ( y ′ ) γ d y 1 / p < ∞ ,

    then for any 1 < p < ∞

    ∥ f ∥ BMO γ ≤ ∥ f ∥ BMO p , γ ≤ A p ∥ f ∥ BMO γ ,

    where the constant A p depends on p.

  2. Let f ∈ BMO γ . Then, there exists a constant c > 0 such that

| f E ( 0 , r ) − f E ( 0 , t ) | ≤ c ∥ f ∥ BMO γ ln t r , 0 < 2 r < t ,

where c is independent of f , x , r and t.

Now let us define the B-maximal operator M γ and the B-Riesz potential I α , γ generated by the generalized shift operator as follows:

M γ f ( x ) = sup r > 0 1 | E ( 0 , r ) | γ ∫ E ( 0 , r ) T y ( | f | ) ( x ) ( y ′ ) γ d y , I α , γ f ( x ) = ∫ ℝ k , + n f ( y ) T y | x | α − Q ( y ′ ) γ d y , 0 < α < Q .

The following theorem deals with the boundedness of the B-maximal operator in the weighted L p , ω , γ space.

Theorem 2.12

[31]

  1. If f ∈ L 1 , ω , γ , ω ∈ A 1 , γ , then M γ f ∈ W L 1 , ω , γ and

    ∥ M γ f ∥ W L 1 , ω , γ ≤ c 1 ∥ f ∥ L 1 , ω , γ ,

    where c 1 depends on ω , γ and n.

  2. If f ∈ L p , ω , γ , ω ∈ A p , γ , 1 < p < ∞ , then M γ f ∈ L p , ω , γ and

∥ M γ f ∥ L p , ω , γ ≤ c 2 ∥ f ∥ L p , ω , γ ,

where c 2 depends on p , ω , γ and n.

The following theorems deal with the boundedness of B-Riesz potentials I α , γ and the B-maximal operator M γ in B-Morrey spaces and the first one known as Hardy-Littlewood-Sobolev-type inequality for B-Riesz potentials.

Theorem 2.13

[25] Let 0 < α < Q , 0 ≤ λ < Q − α and 1 ≤ p ≤ Q − λ α .

  1. If 1 < p < Q − λ α , then the condition 1 p − 1 q = α Q − λ is necessary and sufficient condition for the boundedness of I α , γ from L p , λ , γ to L q , λ , γ .

  2. If p = 1 , then the condition 1 − 1 q = α Q − λ is necessary and sufficient condition for the boundedness of I α , γ from L 1 , λ , γ to W L q , λ , γ .

Theorem 2.14

[9,25]

  1. If f ∈ L 1 , λ , γ , 0 ≤ λ < Q , then M γ f ∈ W L 1 , λ , γ and

∥ M γ f ∥ W L 1 , λ , γ ≤ c 1 ∥ f ∥ L 1 , λ , γ ,

where c 1 depends only on λ , γ , k and n.

  1. If f ∈ L p , λ , γ , 1 < p < ∞ , 0 ≤ λ < Q , then M γ f ∈ L p , λ , γ and

∥ M γ f ∥ L p , λ , γ ≤ c 2 ∥ f ∥ L p , λ , γ ,

where c 2 depends on p , λ , γ , k and n.

3 Main results and proofs

3.1 Maximal commutators in L p , λ , γ

Given a measurable function b, the commutator of the B-maximal operator M γ is formally defined by

[ M γ , b ] f = M γ ( b f ) − b M γ ( f )

and the B-maximal commutator is defined by

M b , γ ( f ) ( x ) = sup r > 0 | E ( 0 , r ) | γ − 1 ∫ E ( 0 , r ) T y | ( b ( x ) − b ( y ) ) f ( x ) | ( y ′ ) γ d y , ∀ x ∈ ℝ k , + n .

Lemma 3.1.1

Let 1 < s < ∞ , b ∈ BMO γ . Then, there exists c 1 > 0 such that

M γ # ( M b , γ f ) ( x ) ≤ c 1 ∥ b ∥ BMO γ ( ( M γ ( M γ f ) s ) 1 / s ( x ) + M γ ( M γ | f | s ) 1 / s ( x ) ) , x ∈ ℝ k , + n .

Proof

By the boundedness of M γ and the pointwise inequality, we have

M γ # ( M b , γ f ) ( x ) ≤ 2 M γ ( M b , γ f ) ( x ) , x ∈ ℝ k , + n .

Using Hölder’s inequality, we have

| E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) T z ( | b ( x ) − b ( z ) | | f ( x ) | ) ( z ′ ) γ d z ≤ | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) T z ( | b ( x ) − b E ( 0 , t ) | | f ( x ) | ) ( z ′ ) γ d z + | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) | b ( z ) − b E ( 0 , t ) | T z ( | f | ) ( x ) ( z ′ ) γ d z ≤ | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) T z | b ( x ) − b E ( 0 , t ) | s ′ ( z ′ ) γ d z 1 / s ′ ∫ E ( 0 , t ) T z ( | f | ) s ( x ) ( z ′ ) γ d z 1 / s + | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) | b ( z ) − b E ( 0 , t ) | T z ( | f | ) ( x ) ( z ′ ) γ d z ≤ C ∥ b ∥ BMO γ M γ | f | s 1 / s ( x ) + | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) | b ( z ) − b E ( 0 , t ) | T z ( | f | ) ( x ) ( z ′ ) γ d z

and

| E ( 0 , r ) | γ − 1 ∫ E ( 0 , r ) T y | E ( 0 , t ) | γ − 1 ∫ E ( 0 , t ) | b ( z ) − b E ( 0 , t ) | T z ( | f | ) ( x ) ( z ′ ) γ d z ( y ′ ) γ d y ≤ | E ( 0 , r ) | γ − 1 ∫ E ( 0 , r ) T y [ | b ( z ) − b E ( 0 , t ) | M γ f ( x ) ] ( y ′ ) γ d y ≤ | E ( 0 , r ) | γ − 1 ∫ E ( 0 , r ) T y | b ( z ) − b E ( 0 , r ) | s ′ ( y ′ ) γ d y 1 / s ′ ∫ E ( 0 , r ) T y ( M γ f ) s ( x ) ( y ′ ) γ d y 1 / s + | E ( 0 , r ) | γ − 1 ∫ E ( 0 , r ) T y [ | b E ( 0 , t ) − b E ( 0 , r ) | M γ f ( x ) ] ( y ′ ) γ d y ≤ C ∥ b ∥ BMO γ ( M γ ( M γ f ) s ) 1 / s ( x ) .

Therefore,

(3.1) M γ ( M b , γ f ) ( x ) = sup r > 0 | E ( 0 , r ) | γ − 1 ∫ E ( 0 , r ) T y [ M b , γ f ] ( x ) ( y ′ ) γ d y ≤ C ∥ b ∥ BMO γ ( M γ ( M γ f ) s ) 1 / s ( x ) + sup r > 0 | E ( 0 , r ) | γ − 1 ∫ E ( 0 , r ) T y ( M γ | f | s ) 1 / s ( x ) ( y ′ ) γ d y ≤ C ∥ b ∥ BMO γ ( ( M γ ( M γ f ) s ) 1 / s ( x ) + M γ ( M γ | f | s ) 1 / s ( x ) ) .

□

Proposition 3.1.2

[32] Let ( X , d , ν ) be a space of homogeneous type in the sense of d be a pseudometric and ν be a positive Borel regular measure satisfying the following doubling condition: there exists a constant c such that

ν ( B ( x , 2 r ) ) ≤ c ν ( B ( x , r ) ) < ∞ for all x ∈ X and r > 0 .

There exists a positive constant C such that for any weight ω and any nonnegative function f with ν ( { x ∈ X : f ( x ) > β } ) < ∞ for any β > 0 ,

  1. if ν ( X ) = ∞ , then ∫ X f ( x ) ω ( x ) d ν ( x ) ≤ C ∫ X M # f ( x ) M ω ( x ) d ν ( x ) ,

  2. if ν ( X ) < ∞ , then ∫ X f ( x ) ω ( x ) d ν ( x ) ≤ C ∫ X M # f ( x ) M ω ( x ) d ν ( x ) + C ω ( X ) ν X ( f ) ,

where ω ( X ) = ∫ X ω ( x ) d ν ( x ) , ν X ( f ) = 1 ν ( X ) ∫ X f ( x ) d ν ( x ) . The Fefferman-Stein sharp maximal operator is defined as M # f ( x ) = sup x ∈ Q 1 ν ( Q ) ∫ Q | f ( y ) − f Q | d ν ( y ) , where f Q = 1 ν ( Q ) ∫ Q f and the supremum is taken over all cubes Q containing the point x.

Lemma 3.1.3

Let f ∈ L p , γ , ω ∈ A p , γ , 1 < p < ∞ . Then, ∥ f ω 1 / p ∥ L p , γ ≤ C ∥ ω 1 / p M γ # f ∥ L p , γ , where a constant c > 0 is independent of f.

Proof

Let f ∈ L p , γ ( ℝ k , + n ) . We need to introduce another maximal operator defined on a space of homogeneous type ( X , d , ν ) . Here, X = ℝ n × ( 0 , ∞ ) k = { ( x , x ′ ¯ ) : x ∈ ℝ n , x ′ ¯ ∈ ( 0 , ∞ ) k } equipped with the continuous pseudometric d and the positive measure ν satisfying the doubling condition:

ν ( E ( ( x , x ′ ¯ ) , 2 r ) ) ≤ c ν ( E ( ( x , x ′ ¯ ) , r ) ) ,

where a constant c is independent of ( x , x ′ ¯ ) , r > 0 and E ( ( x , x ′ ¯ ) , r ) = { ( y , y ′ ¯ ) ∈ X : d ( ( x , x ′ ¯ ) , ( y , y ′ ¯ ) ) < r } , d ν ( x , x ′ ¯ ) = ( x ′ ¯ ) γ − 1 d x d x ′ ¯ , ( x ′ ¯ ) γ − 1 = ( x ¯ 1 ) γ 1 − 1 … ( x k ¯ ) γ k − 1 and d ( ( x , x ′ ¯ ) , ( y , y ′ ¯ ) ) = | ( x , x ′ ¯ ) − ( y , y ′ ¯ ) | ≡ ( | x − y | 2 + ( x ′ ¯ − y ′ ¯ ) 2 ) 1 / 2 . We define the maximal operator M ν by

M ν f ¯ ( x , x ′ ¯ ) = sup r > 0 1 ν ( E ( ( x , x ′ ¯ ) , r ) ) ∫ E ( ( x , x ′ ¯ ) , r ) | f ¯ ( y , y ′ ¯ ) | d ν ( y , y ′ ¯ ) ,

where f ¯ ( x , x ′ ¯ ) = f x 1 2 + x ¯ 1 2 , … , x k 2 + x ¯ k 2 , x ″ , x ″ ∈ ℝ n − k . Coifman and Weiss [33] showed that the maximal operator M ν is of weak type ( 1 , 1 ) and is bounded on L p = L p ( X , d , ν ) for 1 < p < ∞ . The sharp maximal operator is defined by

M ν # f ¯ ( x , x ′ ¯ ) = sup r > 0 1 ν ( E ( ( x , 0 ) , r ) ) ∫ E ( ( x , 0 ) , r ) ) | T y f ¯ ( x , x ′ ¯ ) − f ¯ E ( ( x , 0 ) , r ) ) ( x , x ′ ¯ ) | d ν ( y , y ′ ¯ ) ,

where f ¯ E ( ( x , 0 ) , r ) ) ( x , x ′ ¯ ) = 1 ν ( E ( ( x , 0 ) , r ) ) ∫ E ( ( x , 0 ) , r ) ) T y f ¯ ( x , x ′ ¯ ) d ν ( y , y ′ ¯ ) . Moreover, Guliyev and Hasanov [9,25] proved that

M γ f x 1 2 + x ¯ 1 2 , … , x k 2 + x ¯ k 2 , x ″ = M ν f ¯ x 1 2 + x ¯ 1 2 , … , x k 2 + x ¯ k 2 , x ″ , 0

and

M γ f ( x ) = M ν f ¯ ( x , 0 ) .

So according to Proposition 3.1.2, we have

∥ f ω 1 p ∥ L p , γ = ∥ f ¯ ω 1 p ∥ L p = sup ∥ g ¯ ∥ L p ′ ≤ 1 ∫ X f ¯ ( y , y ′ ¯ ) g ¯ ( y , y ′ ¯ ) ω 1 p ( y , y ′ ¯ ) d ν ( y , y ′ ¯ ) , 1 / p + 1 / p ′ = 1 ≤ C sup ∥ g ¯ ∥ L p ′ ≤ 1 ∫ X M ν # f ¯ ( y , y ′ ¯ ) M ν ( g ¯ ω 1 p ) ( y , y ′ ¯ ) d ν ( y , y ′ ¯ ) = C sup ∥ g ∥ L p ′ , γ ≤ 1 ∫ ℝ k , + n M γ # f ( y ) M γ ( g ω 1 p ) ( y ) ( y ′ ) γ d y .

Finally, using Hölder’s inequality with the conjugate exponents p, p ′ and Theorem 2.12, we get

∥ f ω 1 p ∥ L p , γ ≤ C sup ∥ g ∥ L p ′ , γ ≤ 1 ∥ ω 1 p M γ # f ∥ L p , γ ∥ ω 1 p M γ ( g ω 1 p ) ∥ L p ′ , γ ≤ C sup ∥ g ∥ L p ′ , γ ≤ 1 ∥ ω 1 p M γ # f ∥ L p , γ ∥ g ∥ L p ′ , γ ≤ C ∥ ω 1 p M γ # f ∥ L p , γ . □

Theorem 3.1.4

Let b ∈ BMO γ , 1 < p < ∞ and ω ∈ A p , γ . Then, the B-maximal commutator M b , γ is bounded on the weighted space L p , ω , γ .

Proof

By using Lemmas 3.1.1, 3.1.3 and Theorem 2.12, we have M b , γ is bounded on the space L p , ω , γ .□

The commutator M b , γ of the B-maximal operator M γ and the B-maximal commutator [ M γ , b ] are essentially different from each other. Indeed, the commutator M b , γ is a positive and sublinear operator, but [ M γ , b ] is neither positive nor sublinear. However, if b satisfies some conditions, then the operator M b , γ controls [ M γ , b ] .

A necessary and sufficient condition for the boundedness of the commutator M b , γ on the B-Morrey space L p , λ , γ is given by the following theorem.

Theorem 3.1.5

Let 1 < p < ∞ , 0 ≤ λ < Q . Then, the commutator M b , γ is bounded on the B-Morrey space L p , λ , γ if and only if b ∈ BMO γ .

Proof

Sufficiency: Let the commutator M b , γ be bounded on L p , λ , γ , 1 < p < ∞ . Since

∥ f ∥ L p , λ , γ = sup t > 0 , x ∈ ℝ k , + n t − λ ∫ E ( 0 , t ) T y [ | f | ] p ( x ) ( y ′ ) γ d y 1 / p ,

let us consider f = χ E ( 0 , r ) . We have

∥ χ E ( 0 , r ) ∥ L p , λ , γ = sup t > 0 , x ∈ ℝ k , + n t − λ ∫ E ( 0 , t ) T y [ χ E ( 0 , r ) ] p ( x ) ( y ′ ) γ d y 1 / p = sup t > 0 , x ∈ ℝ k , + n t − λ ∫ E ( x , t ) χ E ( 0 , r ) ( y ) ( y ′ ) γ d y 1 / p = sup E ( x , t ) ⊂ E ( 0 , r ) ( t − λ | E ( x , t ) ∩ E ( 0 , r ) | γ ) 1 / p ≤ r Q − λ p .

Then,

1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) | T z b ( x ) − b E ( 0 , t ) | ( z ′ ) γ d z = 1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) T z b ( x ) − 1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) T z b ( y ) ( y ′ ) γ d y ( z ′ ) γ d z ≤ 1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) 1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) | T z b ( x ) − T z b ( y ) | ( y ′ ) γ d y ( z ′ ) γ d z ≤ 1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) M b , γ χ E ( 0 , t ) ( z ) ( z ′ ) γ d z ≤ C t − Q + λ ∥ M b , γ χ E ( 0 , t ) ∥ L p , λ , γ ∥ χ E ( 0 , t ) ∥ L p ′ , λ , γ ≤ C t Q − λ p ′ + Q − λ p − Q + λ ≤ C .

This shows that b ∈ BMO γ .

Necessity: Let b ∈ BMO γ and f ∈ L p , λ , γ , 1 < p < ∞ , 0 ≤ λ < Q . Taking into account the properties of A p , γ we easily obtain ( M γ χ E ( 0 , t ) ) δ ∈ A p , γ , for any 0 < δ < 1 . Then, by using Lemma 3.1.3 and Theorem 3.1.4, we obtain

∫ E ( 0 , t ) T y ( | M b , γ f | ) p ( x ) ( y ′ ) γ d y = ∫ ℝ k , + n T y ( | M b , γ f | ) p ( x ) χ E ( 0 , t ) ( y ) ( y ′ ) γ d y ≤ ∫ ℝ k , + n T y ( | M b , γ f | ) p ( x ) ( M γ χ E ( 0 , t ) ( y ) ) δ ( y ′ ) γ d y ≤ C   ∥ b ∥ BMO γ p ∫ ℝ k , + n T y ( | f | ) p ( x ) ( M γ χ E ( 0 , t ) ( y ) ) δ ( y ′ ) γ d y ≤ C ∥ b ∥ BMO γ p ∫ E ( 0 , t ) T y ( | f | ) p ( x ) ( y ′ ) γ d y + ∑ j = 1 ∞ ∫ E ( 0 , 2 j + 1 t ) \ E ( 0 , 2 j t ) T y ( | f | ) p ( x ) ( M γ χ E ( 0 , t ) ( y ) ) δ ( y ′ ) γ d y ≤ C   ∥ b ∥ BMO γ p ∫ E ( 0 , t ) T y ( | f | ) p ( x ) ( y ′ ) γ d y + ∑ j = 1 ∞ ∫ E ( 0 , 2 j + 1 t ) \ E ( 0 , 2 j t ) T y ( | f | ) p ( x ) t Q δ ( | y | + t ) Q δ ( y ′ ) γ d y ≤ C   ∥ b ∥ BMO γ p ∥ f ∥ L p , λ , γ p t λ + ∑ j = 1 ∞ 1 ( 2 j + 1 ) Q δ ( 2 j + 1 t ) λ ≤ C   t λ b BMO γ p ∥ f ∥ L p , λ , γ p .

Finally, we have

∥ M b , γ f ∥ p , λ , γ p = sup t > 0 , x ∈ ℝ k , + n t − λ ∫ E ( 0 , t ) T y ( | M b , γ f | ) p ( x ) ( y ′ ) γ d y ≤ ∥ b ∥ BMO γ p ∥ f ∥ L p , λ , γ p . □

Theorem 3.1.6

Let 0 ≤ λ < Q , b ∈ BMO γ . Then, the commutator M b , γ is bounded from L 1 , λ , γ to W L 1 , λ , γ .

Proof

Let 0 ≤ λ < Q , f ∈ L 1 , λ , γ . The result can be obtained from f ( x ) ≤ M γ f ( x ) . Hence, by using (3.1) and Theorem 2.14, we have

∥ M b , γ f ∥ W L 1 , λ , γ ≤ ∥ M γ ( M b , γ f ) ∥ W L 1 , λ , γ ≤ ∥ b ∥ BMO γ ∥ ( M γ ( M γ f ) s ) 1 / s + M γ ( M γ | f | s ) 1 / s ∥ W L 1 , λ , γ ≤ ∥ b ∥ BMO γ ∥ f ∥ L 1 , λ , γ . □

3.2 Commutators of the B-Riesz potential in B-Morrey spaces

In this section, we consider the commutator of the B-Riesz potential defined by

[ b , I α , γ ] f ( x ) = ∫ ℝ k , + n ( b ( x ) − b ( y ) ) | y | α − Q T y f ( x ) ( y ′ ) γ d y , 0 < α < Q

and the operator | b , I α , γ | given by

| b , I α , γ | f ( x ) = ∫ ℝ k , + n | b ( x ) − b ( y ) | | y | α − Q T y [ | f | ] ( x ) ( y ′ ) γ d y , 0 < α < Q .

Theorem 3.2.1

Let 0 < α < Q , 0 ≤ λ < Q − α , 1 p − 1 q = α Q − λ and 1 < p < Q − λ α . Then, the commutator | b , I α , γ | is bounded from L p , λ , γ to L q , λ , γ if and only if b ∈ BMO γ .

Proof

Sufficiency: Let | b , I α , γ | be bounded from L p , λ , γ to L q , λ , γ , 1 < p < Q − λ α . We have

1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) | T z b ( x ) − f E ( 0 , t ) | ( z ′ ) γ d z = 1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) T z b ( x ) − 1 | E ( 0 , t ) | γ ∫ E ( 0 , t ) T z b ( y ) ( y ′ ) γ d y ( z ′ ) γ d z ≤ 1 | E ( 0 , t ) | γ 1 + α Q ∫ E ( 0 , t ) 1 | E ( 0 , t ) | γ 1 − α Q ∫ E ( 0 , t ) | T z b ( x ) − T z b ( y ) | ( y ′ ) γ d y ( z ′ ) γ d z ≤ 1 | E ( 0 , t ) | γ 1 + α Q ∫ E ( 0 , t ) ∫ E ( 0 , t ) | T z ( b ( x ) − b ( y ) ) | | y | Q − α ( y ′ ) γ d y ( z ′ ) γ d z ≤ 1 | E ( 0 , t ) | γ 1 + α Q ∫ E ( 0 , t ) | b , I α , γ | χ E ( 0 , t ) ( z ) ( z ′ ) γ d z ≤ C t − Q − α + λ ∥ | b , I α , γ | χ E ( 0 , t ) ∥ L q , λ , γ ∥ χ E ( 0 , t ) ∥ L q ′ , λ , γ ≤ C t Q − λ q ′ + Q − λ p − Q − α + λ ≤ C .

This shows that b ∈ BMO γ .

Necessity: Let f ∈ L p , λ , γ . Then,

(3.2) | b , I α , γ | f ( x ) = ∫ E ( 0 , t ) + ∫ ℝ k , + n \ E ( 0 , t ) T y | [ b − b ( x ) ] f ( x ) | | y | α − Q ( y ′ ) γ d y ≡ F 1 ( x , t ) + F 2 ( x , t ) .

First, let us estimate F 1 ( x , t ) . By using Hölder’s inequality, we have

F 1 ( x , t ) = ∫ E ( 0 , t ) T y | [ b − b ( x ) ] f ( x ) | | y | α − Q ( y ′ ) γ d y ≤ ∑ j = − ∞ − 1 ( 2 j t ) α − Q ∫ E ( 0 , 2 j + 1 t ) \ E ( 0 , 2 j t ) T y | [ b − b ( x ) ] f ( x ) | ( y ′ ) γ d y ≤ ∑ j = − ∞ − 1 ( 2 j t ) α − Q ∫ E ( 0 , 2 j + 1 t ) \ E ( 0 , 2 j t ) | T y b ( x ) − b | r ′ ( y ′ ) γ d y 1 / r ′ , 1 / r + 1 / r ′ = 1 × ∫ E ( 0 , 2 j + 1 t ) \ E ( 0 , 2 j t ) T y | f | r ( x ) ( y ′ ) γ d y 1 / r ≤ ∥ b ∥ BMO γ ( M γ ( | f | r ) ( x ) ) 1 / r ∑ j = − ∞ − 1 ( 2 j t ) α .

Hence,

(3.3) F 1 ( x , t ) ≤ C t α ∥ b ∥ BMO γ ( M γ ( | f | r ) ( x ) ) 1 / r .

Now by using Hölder’s inequality, let us estimate F 2 ( x , t ) . We get

F 2 ( x , t ) ≤ ∫ ℝ k , + n \ E ( 0 , t ) T y ( | b − b ( x ) | | f ( x ) | ) | y | α − Q ( y ′ ) γ d y ≤ ∑ j = 0 ∞ ( 2 j t ) α − Q ∫ E ( 0 , 2 j + 1 t ) \ E ( 0 , 2 j t ) T y ( | b − b ( x ) | | f ( x ) | ) ( y ′ ) γ d y ≤ ∑ j = 0 ∞ ( 2 j t ) α − Q ∫ E ( 0 , 2 j + 1 t ) \ E ( 0 , 2 j t ) | T y b ( x ) − b | p ′ ( y ′ ) γ d y 1 / p ′ × ∫ E ( 0 , 2 j + 1 t ) \ E ( 0 , 2 j t ) T y [ | f | ] p ( x ) ( y ′ ) γ d y 1 / p , 1 / p + 1 / p ′ = 1 ≤ 2 Q − λ p ′ t α − Q − λ p ∥ b ∥ BMO γ ∥ f ∥ L p , λ , γ ∑ j = 0 ∞ 2 j α − Q − λ p .

Thus,

(3.4) F 2 ( x , t ) ≤ C t α − Q − λ p ∥ b ∥ BMO γ ∥ f ∥ L p , λ , γ .

So, from (3.3) and (3.4) we have

| | b , I α , γ | f ( x ) | ≤ C t α ∥ b ∥ BMO γ ( M γ ( | f | r ) ( x ) ) 1 / r + C t α − Q − λ p ∥ b ∥ BMO γ ∥ f ∥ L p , λ , γ .

Minimizing with respect to t = [ ( M γ ( | f | r ) ( x ) ) − 1 / r ∥ f ∥ L p , λ , γ ] p / ( Q − λ ) we obtain

| | b , I α , γ | f ( x ) | ≤ C ∥ b ∥ BMO γ ( ( M γ ( | f | r ) ( x ) ) 1 / r ) p / q ∥ f ∥ L p , λ , γ 1 − p / q .

Hence, by Theorem 2.14, we get

∫ E ( 0 , t ) T y [ | | b , I α , γ | f | ] q ( x ) ( y ′ ) γ d y ≤ C ∥ b ∥ BMO γ q ∥ f ∥ L p , λ , γ q − p ∫ E ( 0 , t ) T y ( M γ ( | f | r ) ( x ) ) p / r ( y ′ ) γ d y ≤ C t λ ∥ b ∥ BMO γ q ∥ f ∥ L p , λ , γ q − p ∥ f ∥ L p , λ , γ p = C t λ ∥ b ∥ BMO γ q ∥ f ∥ L p , λ , γ q .

So, ∥ | b , I α , γ | f ∥ L p , λ , γ ≤ C ∥ b ∥ BMO γ ∥ f ∥ L p , λ , γ . □

Theorem 3.2.2

Let 0 < α < Q , 0 ≤ λ < Q − α , 1 − 1 q = α Q − λ and b ∈ BMO γ . Then, the commutator | b , I α , γ | is bounded from L 1 , λ , γ to W L q , λ , γ .

Proof

Let f ∈ L 1 , λ , γ . We have

| { x ∈ E ( 0 , t )   : | | b , I α , γ | f ( x ) | > 2 β } | γ ≤ | { x ∈ E ( 0 , t )   :   F 1 ( x , t ) > β } | γ + | { x ∈ E ( 0 , t )   :   F 2 ( x , t ) > β } | γ ,

where F 1 ( x , t ) and F 2 ( x , t ) are defined in (3.2). Now taking into account (3.3) and Theorem 2.14, we have

| { x ∈ E ( 0 , t )   :   | F 1 ( x , t ) | > β } | γ ≤ x ∈ E ( 0 , t )   :   ( M γ ( | f | r ) ( x ) ) 1 / r > β C t α ∥ b ∥ BMO γ γ ≤ C t α β ∥ b ∥ BMO γ ∥ f ∥ L 1 , λ , γ .

Also taking into account (3.4), for β = C t α − Q − λ p ∥ b ∥ BMO γ ∥ f ∥ L 1 , λ , γ we have

F 2 ( x , t ) ≤ β , that is , | { x ∈ E ( 0 , t )   :   F 2 ( x , t ) > β } | γ = 0 .

Finally,

| { x ∈ E ( 0 , t )   :   | | b , I α , γ | f ( x ) | > 2 β } | γ ≤ C 1 β q ∥ b ∥ BMO γ q 1 β q ∥ f ∥ L 1 , λ , γ q . □

The following theorem is known as the Hardy-Littlewood-Sobolev-type theorem for the commutator of the B-Riesz potentials and it gives a necessary and sufficient condition for the boundedness of the operator | b , I α , γ | from L p , λ , γ to L q , λ , γ

Theorem 3.2.3

Let 0 < α < Q , 0 ≤ λ < Q − α , b ∈ BMO γ and 1 ≤ p < Q − λ α .

  1. If 1 < p < Q − λ α , then | b , I α , γ | is bounded from L p , λ , γ to L q , λ , γ if and only if 1 p − 1 q = α Q − λ .

  2. If p = 1 , then | b , I α , γ | is bounded from L 1 , λ , γ to W L q , λ , γ if and only if 1 − 1 q = α Q − λ .

Proof

  1. Sufficiency: Let 1 < p < Q − λ α . Let us define f t ( x ) ≕ f ( t x ) , t > 0 . Then,

∥ f t ∥ L p , λ , γ = t − Q p sup r > 0 , x ∈ ℝ k , + n r − λ ∫ E ( 0 , t r ) T y [ | f | ] p ( t x ) ( y ′ ) γ d y 1 / p = t − Q − λ p ∥ f ∥ L p , λ , γ

and we have

∥ | b , I α , γ | f t ∥ L q , λ , γ = t − α sup r > 0 , x ∈ ℝ k , + n r − λ ∫ E ( 0 , r ) T t y [ | | b , I α , γ | f | ] q ( t x ) ( y ′ ) γ d y 1 / q = t − α − Q q sup r > 0 , x ∈ ℝ k , + n r − λ ∫ E ( 0 , t r ) T y [ | | b , I α , γ | f | ] q ( x ) ( y ′ ) γ d y 1 / q = t − α − Q − λ q ∥ | b , I α , γ | f ∥ L q , λ , γ .

Since the operator | b , I α , γ | is bounded from L p , λ , γ to L q , λ , γ , we get

∥ | b , I α , γ | f ∥ L q , λ , γ ≤ c 1 t α + Q − λ q − Q − λ p ∥ b ∥ BMO γ ∥ f ∥ L p , λ , γ ,

where c 1 depends on p , q , λ , γ , k , n . If Q − λ p < Q − λ q + α , then ∥ | b , I α , γ | f ∥ L q , λ , γ = 0 for all f ∈ L p , λ , γ as t → 0 . Similarly, if Q − λ p > Q − λ q + α , then ∥ | b , I α , γ | f ∥ L q , λ , γ = 0 for all f ∈ L p , λ , γ as t → ∞ . Therefore, 1 p − 1 q = α Q − λ . The necessity follows from Theorem 3.2.1.

  1. Sufficiency: Let | b , I α , γ | bounded from L 1 , λ , γ to W L q , λ , γ . We have

∥ | b , I α , γ | f t ∥ W L q , λ , γ = sup r > 0 r sup τ > 0 , x ∈ ℝ k , + n τ − λ ∫ { y ∈ E ( 0 , τ )   :   T y | | b , I α , γ | f t | ( x ) > r } ( y ′ ) γ d y 1 / q = t − α sup r > 0 r t α sup τ > 0 , x ∈ ℝ k , + n τ − λ ∫ { y ∈ E ( 0 , τ )   :   T t y | | b , I α , γ | f | ( t x ) > r t α } ( y ′ ) γ d y 1 / q = t − α − Q q sup r > 0 r t α sup τ > 0 , x ∈ ℝ k , + n t λ ( t τ ) − λ ∫ { y ∈ E ( 0 , t τ )   :   T y | | b , I α , γ | f | ( x ) > r t α } ( y ′ ) γ d y 1 / q = t − α − Q − λ q ∥ | b , I α , γ | f ∥ W L q , λ , γ .

Since the operator | b , I α , γ | bounded from L 1 , λ , γ to W L q , λ , γ , we get

∥ | b , I α , γ | f ∥ W L q , λ , γ ≤ c 2 t α + Q − λ q   − ( Q − λ ) ∥ b ∥ BMO γ ∥ f ∥ L 1 , λ , γ ,

where c 2 depends on q , λ , γ , k , n .

If 1 < 1 q + α Q − λ , then ∥ | b , I α , γ | f ∥ W L q , λ , γ = 0 for all f ∈ L 1 , λ , γ as t → 0 . Similarly, if 1 > 1 q + α Q − λ , then we have ∥ | b , I α , γ | f ∥ W L q , λ , γ = 0 for all f ∈ L 1 , λ , γ as t → ∞ . Therefore, we have 1 = 1 q + α Q − λ . The necessity of the theorem follows from Theorem 3.2.2.□

3.3 Commutators of B-singular integrals in B-Morrey spaces

Let us consider the Calderón-Zygmund-type B-singular integral operator generated by the generalized shift operator, defined by

A γ f ( x ) = ∫ ℝ k , + n T y f ( x ) K ( y ) ( y ′ ) γ d y ,

where the kernel K ( x ) is a singular kernel and satisfying the following conditions:

∫ { x ∈ ℝ k , + n : ε < | x | < r } K ( x ) ( x ′ ) γ d x ≤ C , 0 < ε < r < ∞ , ∫ { x ∈ ℝ k , + n : r < | x | < 4 r } | K ( x ) | ( x ′ ) γ d x ≤ C , 0 < r < ∞ , ∫ { x ∈ ℝ k , + n : | x | ≥ 4 | y | } | T y K ( x ) − K ( x ) | ( x ′ ) γ d x ≤ C , | y | < 1 4 .

We know that

(3.5) A γ f ( x ) = lim ε → 0 + A ε , γ f ( x ) ,

where A ε , γ f ( x ) = ∫ { y ∈ ℝ k , + n : | y | > ε } T y f ( x ) K ( y ) ( y ′ ) γ d y , ε > 0 (see [27,34]).

Proposition 3.3.1

[35] Let ( X , d , ν ) be a space of homogeneous type, 1 ≤ p < ∞ and b ∈ BMO ( X , d , ν ) . Then, the commutator of the classical Calderón-Zygmund singular integral operator is bounded on L p ( X , d , ν ) .

The commutator of the B-singular integral operator A γ with a function b is defined by

[ b , A γ ] f = b A γ ( f ) − A γ ( b f ) .

In the next theorem, we prove the boundedness of the commutator [ b , A γ ] on B-Morrey spaces.

Theorem 3.3.2

Let 1 < p < ∞ , b ∈ BMO γ . Then, the commutator [ b , A γ ] is bounded on the B-Morrey space L p , λ , γ .

Proof

We need to introduce a specific maximal operator which is defined on a homogeneous-type space ( X , d , ν ) . That is a topological space X = ℝ n × ( 0 , ∞ ) k equipped with a continuous pseudometric d and a positive measure ν satisfying

ν ( E ( ( x , x ′ ¯ ) , 2 r ) ) ≤ C 1 ν ( E ( ( x , x ′ ¯ ) , r ) ) ,

where the constant C 1 is independent of ( x , x ′ ¯ ) and r > 0 .

Furthermore, we set E ( ( x , x ′ ¯ ) , r ) = { ( y , y ′ ¯ ) ∈ X   :   d ( ( x , x ′ ¯ ) , ( y , y ′ ¯ ) ) < r } , d ν ( y , y ′ ¯ ) = ( y ′ ¯ ) γ − 1 d y d y ′ ¯ and ( y ′ ¯ ) γ − 1 = ( y 1 ¯ ) γ 1 − 1 ⋯ ( y k ¯ ) γ k − 1 , d ( ( x , x ′ ¯ ) , ( y , y ′ ¯ ) ) = | ( x , x ′ ¯ ) − ( y , y ′ ¯ ) | = ( | x − y | 2 + ( x ′ ¯ − y ′ ¯ ) 2 ) 1 2 .

Let us define

[ b , A ν f ¯ ] ( x , x ′ ¯ ) = ∫ E ⁎ ( ( x , x ′ ¯ ) , r ) ( b ( y , y ′ ¯ ) − b ( x , x ′ ¯ ) f ¯ ( x − y , x ′ ¯ − y ′ ¯ ) K ( y , y ′ ¯ ) d ν ( y ) ,

where f ¯ ( x , x ′ ¯ ) = f x 1 2 + x ¯ 1 2 , … , x k 2 + x ¯ k 2 , x ″ .

With easy calculations, it is possible to prove the following inequality:

(3.6) [ b , A γ f ] z 1 2 + z ¯ 1 2 , … , z k 2 + z ¯ k 2 , z ″ = [ b , A ν f ¯ ] z 1 2 + z ¯ 1 2 , … , z k 2 + z ¯ k 2 , z ″ , 0 .

So,

(3.7) [ b , A γ f ] ( x ) = [ b , A ν f ¯ ] ( x , 0 ) .

Indeed, taking into account Lemma 2.5 and ( M χ E ( ( x , 0 ) , r ) ( y ) ) θ ∈ A p , γ , 0 < θ < 1 , we have

∫ ℝ k , + n T y [ | f |   ] p ( x ) ( M γ χ E ( 0 , r ) ( y ) ) θ ( y ′ ) γ d y = ∫ X f ¯ y 1 2 + y ¯ 1 2 , … , y k 2 + y ¯ k 2 , y ″ , 0 p ( M ν χ E ( ( x , 0 ) , r ) ( y , y ′ ¯ ) ) θ d ν ( y , y ′ ¯ )

and since

| E ( 0 , r ) | γ = ν E z 1 2 + z ¯ 1 2 , … , z k 2 + z ¯ k 2 , z ″ , 0 , r ,

then we have (3.6). Furthermore, taking z ¯ k = 0 in (3.6) we get (3.7). By using Lemma 2.5 and (3.6) we have

∫ E ( 0 , r ) T y [ b , A γ f ] p ( x ) ( y ′ ) γ d y ≤ ∫ ℝ k , + n T y | [ b , A γ f ] | p ( x ) ( M γ χ E ( 0 , r ) ( y ) ) θ ( y ′ ) γ d y = ∫ ℝ n × ( 0 , ∞ ) k [ b , A γ f ] z 1 2 + z ¯ 1 2 , … , z k 2 + z ¯ k 2 , z ″ p ( M γ χ E ( ( x , 0 ) , r ) ( z , z ′ ¯ ) ) θ d ν ( z , z ′ ¯ ) = ∫ X [ b , A ν f ¯ ] z 1 2 + z ¯ 1 2 , … , z k 2 + z ¯ k 2 , z ″ , 0 p ( M ν χ E ( ( x , 0 ) , r ) ( z , z ′ ¯ ) ) θ d ν ( z , z ′ ¯ ) .

Then, taking f ( y , y ′ ¯ ) = f ¯ y 1 2 + y ¯ 1 2 , … , y k 2 + y ¯ k 2 , y ″ , 0 and φ ( y , y ′ ¯ ) ≡ ( M ν χ E ( ( x , x ′ ¯ ) , r ) ( y , y ′ ¯ ) ) θ we obtain from Proposition 3.3.1 and Lemma 2.5 that

∫ E ( 0 , r ) T y | [ b , A γ f ] ( x ) | p ( y ′ ) γ d y ≤ ∫ X [ b , A ν f ¯ ] y 1 2 + y ¯ 1 2 , … , y k 2 + y ¯ k 2 , y ″ , 0 p ( M ν χ E ( ( x , 0 ) , r ) ( y , y ′ ¯ ) ) θ d ν ( y , y ′ ¯ ) ≤ C ∥ b ∥ BMO γ p ∫ X f ¯ y 1 2 + y ¯ 1 2 , … , y k 2 + y ¯ k 2 , y ″ , 0 p ( M ν χ E ( ( x , 0 ) , r ) ( y , y ′ ¯ ) ) θ d ν ( y , y ′ ¯ ) = C ∥ b ∥ BMO γ p ∫ X f y 1 2 + y ¯ 1 2 , … , y k 2 + y ¯ k 2 , y ″ p ( M ν χ E ( ( x , 0 ) , r ) ( y , y ′ ¯ ) ) θ d ν ( y , y ′ ¯ ) = C   ∥ b ∥ BMO γ p ∫ ℝ k , + n T y [ | f |   ] p ( x ) ( M γ χ E ( 0 , r ) ( y ) ) θ ( y ′ ) γ d y ≤ C ∥ b ∥ BMO γ p ∫ E ( 0 , r ) T y [ | f |   ] p ( x ) ( y ′ ) γ d y + C ∥ b ∥ BMO γ p ∑ j = 1 ∞ ∫ E ( 0 , 2 j + 1 r ) \ E ( 0 , 2 j r ) T y [ | f |   ] p ( x ) ( M γ χ E ( 0 , r ) ( y ) ) θ ( y ′ ) γ d y ≤ C ∥ b ∥ BMO γ p ∫ E ( 0 , r ) T y [ | f |   ] p ( x ) ( y ′ ) γ d y + C ∥ b ∥ BMO γ p ∑ j = 1 ∞ ∫ E ( 0 , 2 j + 1 r ) \ E ( 0 , 2 j r ) T y [ | f |   ] p ( x ) r Q θ ( | y | + r ) Q θ ( y ′ ) γ d y ≤ C ∥ b ∥ BMO γ p ∥ f ∥ L p , λ , γ p r λ + ∑ j = 1 ∞ 1 ( 2 j + 1 ) Q θ ( 2 j + 1 r ) λ ≤ C   r λ ∥ b ∥ BMO γ p ∥ f ∥ L p , λ , γ p .

Finally,

∥ [ b , A γ f ] ∥ L p , λ , γ ≤ C ∥ b ∥ BMO γ ∥ f ∥ L p , λ , γ . □

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Received: 2019-09-04
Revised: 2020-03-10
Accepted: 2020-05-15
Published Online: 2020-07-10

© 2020 Javanshir J. Hasanov et al., published by De Gruyter

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

Articles in the same Issue

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  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
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  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
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  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
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  43. Levinson-type inequalities via new Green functions and Montgomery identity
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  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
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  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
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  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
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  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
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  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
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  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
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  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
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  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
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  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
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https://doi.org/10.1515/math-2020-0033
Keywords for this article
commutator; generalized shift operator; B-Riesz potential; B-maximal function; B-Morrey space; bounded mean oscillation (BMO) space
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
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  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
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  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
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  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
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  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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