Home Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
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Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces

  • Pu Zhang EMAIL logo and Di Fan
Published/Copyright: July 30, 2025
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

We study the boundedness of commutators of the Hardy-Littlewood maximal function and the sharp maximal function on weighted Morrey spaces when the symbols of the commutators belong to weighted Lipschitz spaces (weighted Morrey-Campanato spaces). Some new characterizations for weighted Lipschitz functions are obtained in terms of the boundedness of the commutators.

Keywords: Hardy-Littlewood maximal function; sharp maximal function; commutator; A p weight; weighted Lipschitz space; weighted Morrey space
MSC 2010: 42B25; 42B20; 26A16; 47B47

1 Introduction and results

Let T be the classical singular integral operator. The commutator [ b , T ] generated by T and a suitable function b is given by

(1.1) [ b , T ] ( f ) ( x ) = T ( ( b ( x ) − b ) f ) ( x ) = b ( x ) T ( f ) ( x ) − T ( b f ) ( x ) .

A well-known result states that [ b , T ] is bounded on L p ( R n ) for 1 < p < ∞ if and only if b ∈ BMO ( R n ) (Coifman et al. [1] and Janson [2]), where BMO ( R n ) = { f ∈ L loc : sup Q 1 ∣ Q ∣ ∫ Q ∣ f ( x ) − f Q ∣ d x < ∞ } . Another kind of boundedness of [ b , T ] was given by Janson [2], see also Paluszyński [3]. It was proved that [ b , T ] is bounded from L p ( R n ) to L q ( R n ) for 1 < p < q < ∞ if and only if b ∈ Lip β ( R n ) with 0 < β = n ( 1 ⁄ p − 1 ⁄ q ) < 1 , where Lip β ( R n ) is the classical Lipschitz space of order β . In 2008, Hu and Gu [4] studied the weighted boundedness of commutator [ b , T ] when b ∈ Lip β , μ , the weighted Lipschitz space (Definition 1.2), and obtained the following result.

Theorem A

[4] Let μ ∈ A 1 , 0 < β < 1 , 1 < p < q < ∞ , and 1 ⁄ q = 1 ⁄ p − β ⁄ n . Suppose that T is a singular integral operator with associated kernel K such that 1 ⁄ K can be expressed as an absolutely convergent Fourier series in a ball. Then [ b , T ] is bounded from L p ( μ ) to L p ( μ 1 − p ) if and only if b ∈ Lip β , μ .

In 2009, Komori and Shirai [5] introduced weighted Morrey space and studied the boundedness of some classical operators on such spaces, including the commutators of singular integral with BMO functions. In 2012, Wang [6] considered the boundedness of [ b , T ] on weighted Morrey spaces when the symbol b belongs to weighted BMO or weighted Lipschitz spaces.

On the other hand, the boundedness properties of commutators of the Hardy-Littlewood maximal function and sharp maximal function have been studied intensively by many authors. See [7–14], for instance.

Let Q be a cube in R n with sides parallel to the coordinate axes. We denote by ∣ Q ∣ the Lebesgue measure and by χ Q the characteristic function of Q . Let f be a locally integrable function on R n . The Hardy-Littlewood maximal function of f is defined by

M ( f ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ f ( y ) ∣ d y ,

and the sharp maximal function M ♯ is given by

M ♯ ( f ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ f ( y ) − f Q ∣ d y ,

where the supremum is taken over all cubes Q ⊂ R n containing x and f Q = ∣ Q ∣ − 1 ∫ Q f ( x ) d x .

Similar to (1.1), we define two different kinds of commutators of the Hardy-Littlewood maximal function as follows.

Let b be a locally integrable function. The maximal commutator generated by M and b is given by

M b ( f ) ( x ) = M ( ( b ( x ) − b ) f ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − b ( y ) ∣ ∣ f ( y ) ∣ d y ,

where the supremum is taken over all cubes Q ⊂ R n containing x .

The (nonlinear) commutator generated by M and b is defined by

[ b , M ] ( f ) ( x ) = b ( x ) M ( f ) ( x ) − M ( b f ) ( x ) .

Similarly, we can also define the commutator generated by M ♯ and b by

[ b , M ♯ ] ( f ) ( x ) = b M ♯ ( f ) ( x ) − M ♯ ( b f ) ( x ) .

Obviously, M b and [ b , M ] essentially differ from each other. M b is positive and sublinear, but [ b , M ] and [ b , M ♯ ] are neither positive nor sublinear. The operator [ b , M ] can be used in studying the product of a function in H 1 and a function in BMO, see [15] for example.

In 1990, Milman and Schonbek [7] showed that [ b , M ] is bounded on L p ( R n ) ( 1 < p < ∞ ) when b ∈ BMO ( R n ) and b ≥ 0 . In 2000, Bastero et al. [8] gave some characterizations for the boundedness of [ b , M ] and [ b , M ♯ ] on L p spaces. Certain BMO classes are characterized by the boundedness of the commutators. Xie [9] extended the results to the context of Morrey spaces. Zhang [10,16] considered the mapping properties of M b , [ b , M ] and [ b , M ♯ ] when the symbols b belong to Lipschitz space. Some necessary and sufficient conditions for the boundedness of the commutators on Lebesgue and Morrey spaces are given. The results were extended to variable exponent Lebesgue spaces in [16–18] and to the context of Orlicz spaces in [14,19–21].

Most recently, the boundedness of M b , [ b , M ] and [ b , M ♯ ] on weighted Lebesgue spaces are characterized when the symbols b belong to weighted Lipschitz spaces in [22], which extends the aforementioned result of Theorem A to commutators of maximal functions. Some new characterizations for weighted Lipschitz functions are also given.

In this article, we will study the mapping properties of commutators of the Hardy-Littlewood and the sharp maximal functions on weighted Morrey spaces when the symbols belong to certain weighted Lipschitz spaces. To state our results, let us recall some definitions and notations.

Definition 1.1

[23,24] A weight is a nonnegative locally integrable function on R n that takes values in ( 0 , ∞ ) almost everywhere. Let μ be a weight.

  1. We say that μ ∈ A p for 1 < p < ∞ , if there exists a constant C > 0 such that for any cube Q in R n ,

    1 ∣ Q ∣ ∫ Q μ ( x ) d x 1 ∣ Q ∣ ∫ Q μ ( x ) 1 − p ′ d x p − 1 ≤ C ,

    here and below, p ′ denotes the conjugate exponent of p , that is, 1 ⁄ p + 1 ⁄ p ′ = 1 .

  2. We say that μ ∈ A 1 if there is a constant C > 0 such that for any cube Q ⊂ R n ,

    1 ∣ Q ∣ ∫ Q μ ( y ) d y ≤ C μ ( x ) a.e. x ∈ Q .

  3. We define A ∞ = ⋃ 1 ≤ p < ∞ A p .

Let μ be a weight. For a cube Q and a function f , we write μ ( Q ) = ∫ Q μ ( x ) d x and

‖ f ‖ L p ( μ ) = ∫ R n ∣ f ( x ) ∣ p μ ( x ) d x 1 ⁄ p .

Following [25], we define the weighted Lipschitz function spaces. See also [4,26].

Definition 1.2

Let 1 ≤ p ≤ ∞ , 0 < β < 1 and μ ∈ A ∞ . The weighted Lipschitz space, denoted by Lip β , μ p , is given by

Lip β , μ p = { f ∈ L loc 1 ( R n ) : ‖ f ‖ Lip β , μ p < ∞ } ,

where

‖ f ‖ Lip β , μ p = sup Q 1 μ ( Q ) β ⁄ n 1 μ ( Q ) ∫ Q ∣ f ( x ) − f Q ∣ p μ ( x ) 1 − p d x 1 ⁄ p , for 1 ≤ p < ∞

and

‖ f ‖ Lip β , μ ∞ = sup Q 1 μ ( Q ) β ⁄ n ess sup x ∈ Q ∣ f ( x ) − f Q ∣ μ ( x ) ,

here and later, “ sup Q ” always means the supremum that is taken over all cubes Q ⊂ R n .

Obviously, Lip β , μ p is a special case of the so-called weighted Morrey-Campanato spaces studied by García-Cuerva [25] and other authors, see [26–28] and the references therein.

It is interesting to discuss the inclusions between Lip β , ω p for different p . Let μ ∈ A ∞ , 0 < β < 1 and 1 < r < s < ∞ . By Definition 1.2 and Hölder’s inequality, it is easy to see that

Lip β , μ ∞ ⊂ Lip β , μ s ⊂ Lip β , μ r ⊂ Lip β , μ 1 .

The reverse inclusions are not obvious. When ω ∈ A 1 , Tang [26] proved the following result.

Proposition 1.1

[26] Let μ ∈ A 1 and 0 < β < 1 . Then Lip β , μ p = Lip β , μ ∞ with equivalent norms, for any 1 ≤ p ≤ ∞ .

Based on Proposition 1.1, when μ ∈ A 1 and 1 < β < 1 , we simply use the notation Lip β , μ to represent Lip β , μ p ( 1 ≤ p ≤ ∞ ) in the sequel.

The study of Morrey spaces goes back to the work of Morrey [29], which investigated the local behavior of solutions to second order elliptic partial differential equations. In 2009, Komori and Shirai [5] introduced the weighted Morrey spaces as follows.

Definition 1.3

[5] Let 1 ≤ p < ∞ and 0 < κ < 1 . For two weights u and v , the weighted Morrey space L p , κ ( u , v ) is defined by

L p , κ ( u , v ) = { f ∈ L loc p ( u ) : ‖ f ‖ L p , κ ( u , v ) < ∞ } ,

where

‖ f ‖ L p , κ ( u , v ) = sup Q 1 v ( Q ) κ ∫ Q ∣ f ( x ) ∣ p u ( x ) d x 1 ⁄ p .

If u = v = μ , we write L p , κ ( u , v ) = L p , κ ( μ ) , the weighted Morrey space with one weight.

Given a cube Q 0 , we need the following locally maximal function with respect to Q 0 ,

M Q 0 ( f ) ( x ) = sup Q 0 ⊇ Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ f ( y ) ∣ d y ,

where the supremum is taken over all cubes Q ⊆ Q 0 and x ∈ Q .

We are now in a position to state our results.

Theorem 1.1

Let b be a locally integrable function, μ ∈ A 1 and 0 < β < 1 . Suppose that 1 < p < n ⁄ β , 1 ⁄ q = 1 ⁄ p − β ⁄ n , and 0 < κ < p ⁄ q . Then the following assertions are equivalent:

  1. b ∈ Lip β , μ .

  2. M b is bounded from L p , κ ( μ ) to L q , κ q ⁄ p ( μ 1 − q , μ ) .

Remark 1.1

When b belongs to classical Lipschitz space, the boundedness of M b and [ b , M ] on Morrey spaces were studied in [10]. When μ ≡ 1 , Theorem 1.1 covers [10, Theorem 1.3].

Theorem 1.2

Let b be a locally integrable function, μ ∈ A 1 and 0 < β < 1 . Suppose that 1 < p < n ⁄ β , 1 ⁄ q = 1 ⁄ p − β ⁄ n , and 0 < κ < p ⁄ q . Then the following assertions are equivalent:

  1. b ∈ Lip β , μ and b ≥ 0 a.e. in R n .

  2. [ b , M ] is bounded from L p , κ ( μ ) to L q , κ q ⁄ p ( μ 1 − q , μ ) .

  3. There exists a constant C > 0 such that

    (1.2) sup Q 1 μ ( Q ) β ⁄ n 1 μ ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ q μ ( x ) 1 − q d x 1 ⁄ q ≤ C .

Remark 1.2

When μ ≡ 1 , Theorem 1.2 covers the result of [10, Theorem 1.7].

Theorem 1.3

Let b be a locally integrable function, μ ∈ A 1 and 0 < β < 1 . Suppose that 1 < p < n ⁄ β , 1 ⁄ q = 1 ⁄ p − β ⁄ n , and 0 < κ < p ⁄ q . Then the following assertions are equivalent:

  1. b ∈ Lip β , μ and b ≥ 0 a.e. in R n .

  2. [ b , M ♯ ] is bounded from L p , κ ( μ ) to L q , κ q ⁄ p ( μ 1 − q , μ ) .

  3. There exists a constant C > 0 such that

    (1.3) sup Q 1 μ ( Q ) β ⁄ n 1 μ ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ q μ ( x ) 1 − q d x 1 ⁄ q ≤ C .

Remark 1.3

When b belongs to Lipschitz space, the boundedness of [ b , M ♯ ] on Lebesgue and variable Lebesgue spaces were studied in [16,21]. When b belongs to weighted Lipschitz spaces, the boundedness of [ b , M ♯ ] on weighted Lebesgue spaces was considered in [22].

We end this section by giving some remarks. The proofs of our theorems essentially depend on the equivalent characterizations for the spaces Lip β , μ p obtained by Tang [26, Theorem 1.1]. To achieve the characterizations in [26], the condition μ ∈ A 1 is needed (see also [27,30]). Based on the characterizations of Lip β , μ p and compared our results with Theorem A, it seems to be reasonable to assume μ ∈ A 1 .

2 Preliminaries and lemmas

In this section, we present some lemmas that will be used in the proof of our results. The first lemma is a direct consequence of Proposition 1.1 and Definition 1.2 for the case p = ∞ .

Lemma 2.1

Let 0 < β < 1 and w ∈ A 1 . If b ∈ Lip β , w , then, for any cube Q ⊂ R n , there is a constant C, such that

(2.1) ∣ b ( x ) − b Q ∣ ≤ C ‖ b ‖ Lip β , w w ( Q ) β ⁄ n w ( x ) f o r a.e. x ∈ Q .

Lemma 2.2

[5] If 1 < p < ∞ , 0 < κ < 1 , and μ ∈ A p , then the Hardy-Littlewood maximal operator M is bounded on L p , κ ( μ ) .

Lemma 2.3

If 1 < p < ∞ , 0 < κ < 1 , and μ ∈ A ∞ , then, for any cube Q, we have

‖ χ Q ‖ L p , κ ( μ ) ≤ μ ( Q ) ( 1 − κ ) ⁄ p .

Proof

For any fixed cube Q , noting that 0 < κ < 1 , we have

‖ χ Q ‖ L p , κ ( μ ) = sup Q ′ 1 μ ( Q ′ ) κ ∫ Q ′ ∣ χ Q ( x ) ∣ p μ ( x ) d x 1 ⁄ p = sup Q ′ 1 μ ( Q ′ ) κ ∫ Q ′ ∩ Q ∣ χ Q ( x ) ∣ p μ ( x ) d x 1 ⁄ p ≤ sup Q ′ 1 μ ( Q ′ ∩ Q ) κ ∫ Q ′ ∩ Q ∣ χ Q ( x ) ∣ p μ ( x ) d x 1 ⁄ p ≤ sup Q ′ μ ( Q ′ ∩ Q ) ( 1 − κ ) ⁄ p ≤ μ ( Q ) ( 1 − κ ) ⁄ p .

This proves the required conclusion.□

We need some estimates for weighted fractional maximal function.

Lemma 2.4

[6] Let 0 < β < n , 1 < p < n ⁄ β , 1 ⁄ q = 1 ⁄ p − β ⁄ n , and 0 < κ < p ⁄ q . Suppose that μ ∈ A ∞ , then for any 1 < r < p , we have

‖ M β , μ , r ( f ) ‖ L q , κ q ⁄ p ( μ ) ≤ C ‖ f ‖ L p , κ ( μ ) ,

where M β , μ , r is a weighted fractional maximal function given by

M β , μ , r ( f ) = sup Q ∋ x 1 μ ( Q ) 1 − r β ⁄ n ∫ Q ∣ f ( y ) ∣ r μ ( y ) d y 1 ⁄ r .

Lemma 2.5

[31] Let μ ∈ A 1 , 0 < β < 1 , and b ∈ Lip β , μ . Then there exists a constant C > 0 such that

  1. for any 1 ≤ r < ∞ and any cube Q ∋ x , we have

    1 ∣ Q ∣ ∫ Q ∣ f ( y ) ∣ d y ≤ C μ ( Q ) − β ⁄ n M β , μ , r ( f ) ( x ) ;

  2. for any 1 < r < ∞ and any cube Q ∋ x , we have

    1 ∣ Q ∣ ∫ Q ∣ ( b ( y ) − b Q ) f ( y ) ∣ d y ≤ C ‖ b ‖ Lip β , μ μ ( x ) M β , μ , r ( f ) ( x ) .

Lemma 2.6

Let μ ∈ A 1 , 0 < β < 1 , and b ∈ Lip β , μ . Then for any locally integrable function f and any 1 < r < ∞ , we have

M b ( f ) ( x ) ≤ C ‖ b ‖ Lip β , μ μ ( x ) M β , μ , r ( f ) ( x ) a . e . x ∈ R n .

Proof

Given any 1 < r < ∞ . For any x satisfying (2.1) and any cube Q ∋ x , by Lemmas 2.1 and 2.5, we have

1 ∣ Q ∣ ∫ Q ∣ b ( x ) − b ( y ) ∣ ∣ f ( y ) ∣ d y ≤ ∣ b ( x ) − b Q ∣ 1 ∣ Q ∣ ∫ Q ∣ f ( y ) ∣ d y + 1 ∣ Q ∣ ∫ Q ∣ b ( y ) − b Q ∣ ∣ f ( y ) ∣ d y ≤ C ‖ b ‖ Lip β , μ μ ( x ) M β , μ , r ( f ) ( x ) .

Therefore, we obtain

M b ( f ) ( x ) ≤ C ‖ b ‖ Lip β , μ μ ( x ) M β , μ , r ( f ) ( x ) for a.e. x ∈ R n ,

which completes the proof.□

We also need the following characterizations for nonnegative weighted Lipschitz function, see Corollaries 1.15 and 1.20 in [22].

Lemma 2.7

Let 0 < β < 1 , b be a locally integrable function and μ ∈ A 1 . Then the following statements are equivalent:

  1. b ∈ L i p β , μ and b ≥ 0 a.e. in R n .

  2. There exists 1 ≤ s < ∞ such that

    (2.2) sup B 1 μ ( Q ) β ⁄ n 1 μ ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ s μ ( x ) 1 − s d x 1 ⁄ s ≤ C .

  3. For all 1 ≤ s < ∞ , (2.2) holds.

  4. There exists 1 ≤ s < ∞ such that

    (2.3) sup Q 1 μ ( Q ) β ⁄ n 1 μ ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ s μ ( x ) 1 − s d x 1 ⁄ s ≤ C .

  5. For all 1 ≤ s < ∞ , (2.3) holds.

3 Proof of the theorems

Proof of Theorem 1.1

We first prove the implication ( 1 ) ⇒ ( 2 ) . For any r satisfying 1 < r < p , by Lemma 2.6, we have

‖ M b ( f ) ‖ L q , κ q ⁄ p ( μ 1 − q , μ ) = sup Q 1 μ ( Q ) κ q ⁄ p ∫ Q [ M b ( f ) ( x ) ] q μ ( x ) 1 − q d x 1 ⁄ q ≤ C ‖ b ‖ Lip β , μ sup Q 1 μ ( Q ) κ q ⁄ p ∫ Q [ M β , μ , r ( f ) ( x ) ] q μ ( x ) d x 1 ⁄ q ≤ C ‖ b ‖ Lip β , μ ‖ M β , μ , r ( f ) ‖ L q , κ q ⁄ p ( μ ) ≤ C ‖ b ‖ Lip β , μ ‖ f ‖ L p , κ ( μ ) ,

where the last step follows from Lemma 2.4. This completes the proof of ( 1 ) ⇒ ( 2 ) .

Next, we prove the implication ( 2 ) ⇒ ( 1 ) . Suppose that assertion (2) holds, it suffices to prove that there is a constant C > 0 such that, for all cubes Q ⊂ R n , we have

(3.1) 1 μ ( Q ) 1 + β ⁄ n ∫ Q ∣ b ( x ) − b Q ∣ d x ≤ C .

For any cube Q ⊂ R n , applying Hölder’s inequality, Lemma 2.3 and assertion (2), and noting that 1 ⁄ p = 1 ⁄ q + β ⁄ n , we obtain

1 μ ( Q ) 1 + β ⁄ n ∫ Q ∣ b ( x ) − b Q ∣ d x ≤ 1 μ ( Q ) 1 + β ⁄ n ∫ Q 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − b ( y ) ∣ χ Q ( y ) d y d x ≤ 1 μ ( Q ) 1 + β ⁄ n ∫ Q M b ( χ Q ) ( x ) d x = 1 μ ( Q ) 1 + β ⁄ n ∫ Q M b ( χ Q ) ( x ) μ ( x ) ( 1 − q ) ⁄ q μ ( x ) ( q − 1 ) ⁄ q d x ≤ 1 μ ( Q ) 1 + β ⁄ n ∫ Q [ M b ( χ Q ) ( x ) ] q μ ( x ) 1 − q d x 1 ⁄ q ∫ Q μ ( x ) d x 1 − 1 ⁄ q ≤ 1 μ ( Q ) 1 ⁄ p − κ ⁄ p 1 μ ( Q ) κ q ⁄ p ∫ Q [ M b ( χ Q ) ( x ) ] q μ ( x ) 1 − q d x 1 ⁄ q ≤ 1 μ ( Q ) ( 1 − κ ) ⁄ p ‖ M b ( χ Q ) ‖ L q , κ q ⁄ p ( μ 1 − q , μ ) ≤ C μ ( Q ) ( 1 − κ ) ⁄ p ‖ χ Q ‖ L p , κ ( μ ) ≤ C ,

which is nothing other than (3.1) since the constant C is independent of Q .

The proof of Theorem 1.1 is complete.□

Proof of Theorem 1.2

Observe that the equivalence of (1) and (3) follows readily from Lemma 2.7. We only need to check the implications ( 1 ) ⇒ ( 2 ) and ( 2 ) ⇒ ( 3 ) .

( 1 ) ⇒ ( 2 ) . Given x ∈ R n such that M ( f ) ( x ) < ∞ and 0 ≤ b ( x ) < ∞ . Noting that b ≥ 0 a.e. in R n , we have

∣ [ b , M ] ( f ) ( x ) ∣ = ∣ b ( x ) M ( f ) ( x ) − M ( b f ) ( x ) ∣

(3.2) = sup Q ∋ x 1 ∣ Q ∣ ∫ Q b ( x ) ∣ f ( y ) ∣ d y − sup Q ∋ x 1 ∣ Q ∣ ∫ Q b ( y ) ∣ f ( y ) ∣ d y ≤ sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − b ( y ) ∣ ∣ f ( y ) ∣ d y = M b ( f ) ( x ) .

Observe that if f ∈ L p , κ ( μ ) then, by Lemma 2.2, we have M ( f ) ∈ L p , κ ( μ ) , which implies that M ( f ) < ∞ a.e. in R n , and note that b is finite a.e. in R n since b is locally integrable. Then, we conclude that (3.2) holds for a.e. x ∈ R n .

Thus, by Theorem 1.1, we obtain that [ b , M ] is bounded from L p , κ ( μ ) to L q , κ q ⁄ p ( μ 1 − q , μ ) .

( 2 ) ⇒ ( 3 ) . For any fixed Q ⊂ R n and x ∈ Q , we have (see [8] page 3331 or (2.4) in [32])

M ( χ Q ) ( x ) = χ Q ( x ) and M ( b χ Q ) ( x ) = M Q ( b ) ( x ) .

Noting that 1 ⁄ p = 1 ⁄ q + β ⁄ n and applying assertion (2) and Lemma 2.3, we have

1 μ ( Q ) β ⁄ n 1 μ ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ q μ ( x ) 1 − q d x 1 ⁄ q = 1 μ ( Q ) β ⁄ n 1 μ ( Q ) ∫ Q ∣ b ( x ) M ( χ Q ) ( x ) − M Q ( b χ Q ) ( x ) ∣ q μ ( x ) 1 − q d x 1 ⁄ q = μ ( Q ) κ ⁄ p μ ( Q ) β ⁄ n + 1 ⁄ q 1 μ ( Q ) κ q ⁄ p ∫ Q ∣ [ b , M ] ( χ Q ) ( x ) ∣ q μ ( x ) 1 − q d x 1 ⁄ q ≤ μ ( Q ) ( κ − 1 ) ⁄ p ∥ [ b , M ] ( χ Q ) ∥ L q , κ q ⁄ p ( μ 1 − q , μ ) ≤ C μ ( Q ) ( κ − 1 ) ⁄ p ‖ χ Q ‖ L p , κ ( μ ) ≤ C .

Since the constant C is independent of Q , then we conclude assertion (3).

So the proof of Theorem 1.2 is finished.□

Proof of Theorem 1.3

Similar to the proof of Theorem 1.2, by Lemma 2.7, we need to prove the implications ( 1 ) ⇒ ( 2 ) and ( 2 ) ⇒ ( 3 ) .

( 1 ) ⇒ ( 2 ) . For any x ∈ R n such that M ♯ ( f ) ( x ) < ∞ and 0 ≤ b ( x ) < ∞ , noting that b ≥ 0 a.e. in R n , we have

(3.3) ∣ [ b , M ♯ ] ( f ) ( x ) ∣ = sup Q ∋ x b ( x ) ∣ Q ∣ ∫ Q ∣ f ( y ) − f Q ∣ d y − sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ b ( y ) f ( y ) − ( b f ) Q ∣ d y ≤ sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ b ( x ) f ( y ) − b ( x ) f Q ∣ d y − ∫ Q ∣ b ( y ) f ( y ) − ( b f ) Q ∣ d y ≤ sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ ( b ( x ) − b ( y ) ) f ( y ) + b ( x ) f Q − ( b f ) Q ∣ d y ≤ sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − b ( y ) ∣ ∣ f ( y ) ∣ d y + ∣ b ( x ) f Q − ( b f ) Q ∣ ≤ M b ( f ) ( x ) + sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − b ( z ) ∣ ∣ f ( z ) ∣ d z ≤ 2 M b ( f ) ( x ) .

For f ∈ L p , κ ( μ ) , by Lemma 2.2 again, we have M ( f ) < ∞ a.e. in R n . Then M ♯ ( f ) < ∞ a.e. in R n . Note that b is finite a.e. in R n , we obtain that (3.3) holds for a.e. x ∈ R n . Therefore, it follows from Theorem 1.1 that [ b , M ♯ ] is bounded from L p , κ ( μ ) to L q , κ q ⁄ p ( μ 1 − q , μ ) .

( 2 ) ⇒ ( 3 ) . For any fixed cube Q , we have (see [8] page 3333 or [11] page 1383)

M ♯ ( χ Q ) ( x ) = 1 2 , for all x ∈ Q .

Thus, for any x ∈ Q , we have

b ( x ) − 2 M ♯ ( b χ Q ) ( x ) = 2 1 2 b ( x ) − M ♯ ( b χ Q ) ( x ) = 2 ( b ( x ) M ♯ ( χ Q ) ( x ) − M ♯ ( b χ Q ) ( x ) ) = 2 [ b , M ♯ ] ( χ Q ) ( x ) .

Therefore, by assertion (2) and noting that 1 ⁄ q = 1 ⁄ p − β ⁄ n , we obtain

1 μ ( Q ) β ⁄ n 1 μ ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ q μ ( x ) 1 − q d x 1 ⁄ q = 2 μ ( Q ) 1 ⁄ p ∫ Q ∣ [ b , M ♯ ] ( χ Q ) ( x ) ∣ q μ ( x ) 1 − q d x 1 ⁄ q = 2 μ ( Q ) ( κ − 1 ) ⁄ p 1 μ ( Q ) κ q ⁄ p ∫ Q ∣ [ b , M ♯ ] ( χ Q ) ( x ) ∣ q μ ( x ) 1 − q d x 1 ⁄ q = 2 μ ( Q ) ( κ − 1 ) ⁄ p ∥ [ b , M ♯ ] ( χ Q ) ∥ L q , κ q ⁄ p ( μ 1 − q , μ ) ≤ C μ ( Q ) ( κ − 1 ) ⁄ p ‖ χ Q ‖ L p , κ ( μ ) ≤ C ,

where in the last step we have used Lemma 2.3. This concludes the proof of the implication that ( 2 ) ⇒ ( 3 ) , since the constant C is independent of Q .

The proof of Theorem 1.3 is finished.□

Acknowledgments

The authors wish to express their sincere gratitude to the referees for their careful reading and valuable comments.

  1. Funding information: This work was partly supported by the Fundamental Research Funds of Education Department of Heilongjiang Province (No. 1453ZD031) and the Scientific Research Fund of Mudanjiang Normal University (MSB201201).

  2. Author contributions: All authors contributed equally to this work and gave the final approval for publication.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: Not applicable.

  5. Data availability statement: Not applicable.

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Received: 2023-09-06
Revised: 2025-03-21
Accepted: 2025-05-22
Published Online: 2025-07-30

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

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

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https://doi.org/10.1515/dema-2025-0143
Keywords for this article
Hardy-Littlewood maximal function; sharp maximal function; commutator; A p weight; weighted Lipschitz space; weighted Morrey space
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
  3. Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
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  14. Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
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