Bloom-type two-weight inequalities for commutators of maximal functions

Pu Zhang; Di Fan

doi:10.1515/agms-2025-0020

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Bloom-type two-weight inequalities for commutators of maximal functions

Pu Zhang and Di Fan

Published/Copyright: March 14, 2025

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From the journal Analysis and Geometry in Metric Spaces Volume 13 Issue 1

Abstract

We study Bloom-type two-weight inequalities for commutators of the Hardy-Littlewood maximal function and sharp maximal function. Some necessary and sufficient conditions are given to characterize the two-weight inequalities for such commutators.

Keywords: Hardy-Littlewood maximal function; sharp maximal function; commutator; weights; weighted BMO space

MSC 2010: 42B25; 42B20; 47B47

1 Introduction and main results

Let T be a classical singular integral operator. In 1976, Coifman et al. [6] studied the commutator generated by T and a locally integrable function b as follows:

(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 ) (bounded mean oscillation) (see [6] and [17] for details). In 1985, Bloom [4] considered two-weight behavior for the commutators of the Hilbert transform H . He proved that for two weights μ , λ ∈ A p , the commutator [ b , H ] = b H ( f ) − H ( b f ) is bounded from L p ( μ ) to L p ( λ ) if and only if the symbol b belongs to BMO ν with ν = ( μ λ − 1 ) 1 ⁄ p , where BMO ν is a kind of weighted BMO space introduced by Muckenhoupt and Wheeden [23].

Segovia and Torrea [24] first extended the sufficient part of Bloom’s result to general Calderón-Zygmund operators. They also characterized Bloom-type two-weight inequalities for maximal commutator of the Hardy-Littlewood maximal function. Soon afterward, García-Cuerva et al. [8] extended the sufficient part to a class of strongly singular integrals.

In recent years, Bloom-type inequalities have been at the focus of harmonic analysis. In 2017, Holmes et al. [13] extended Bloom’s result to commutators of Calderón-Zygmund operators. For some more up-to-date results related to Bloom’s result, we refer to [1,5,10,12–14,16,19–22] and references therein.

In this article, we will extend Bloom’s results to commutators of the Hardy-Littlewood maximal function and sharp maximal function. Some necessary and sufficient conditions are given to characterize the two-weight inequalities for such commutators. We first recall some notations.

Let Q be a cube in R n with sides parallel to the coordinate axes. Denote by ∣ Q ∣ the Lebesgue measure and by χ Q the characteristic function of Q . For a locally integrable function f 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 ♯ , introduced by Fefferman and Stein [7], 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 .

Let b be a locally integrable function. Similar to (1.1), we can define two different kinds of commutators of the Hardy-Littlewood maximal function as follows.

The maximal commutator of the Hardy-Littlewood maximal function 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 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 ( x ) M ♯ ( f ) ( x ) − M ♯ ( b f ) ( x ) .

Obviously, commutators M b and [ b , M ] essentially differ from each other. For example, M b is positive and sublinear, but [ b , M ] and [ b , M ♯ ] are neither positive nor sublinear.

In 2000, Bastero et al. [3] 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. In 2014, Zhang and Wu [27] extended the results to the setting of variable exponent Lebesgue spaces.

A weight will always mean a nonnegative locally integrable function. As usual, we denote by A p ( 1 ≤ p ≤ ∞ ) the Muckenhoupt weights classes (see [9] and [25] for details). Let ω be a weight. For a function f and a measurable set E , we write

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

For ω ∈ A p ( 1 < p < ∞ ) , we denote its conjugate weight by ω ′ , i.e., ω ′ = ω 1 − p ′ , where 1 ⁄ p + 1 ⁄ p ′ = 1 . It follows obviously from the definition of A p that ω ∈ A p if and only if ω ′ ∈ A p ′ .

Definition 1.1

[23] Let ω ∈ A ∞ . We say that a locally integrable function b belongs to weighted BMO class BMO ω if

‖ b ‖ BMO ω ≔ sup Q 1 ω ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ d x < ∞ .

When ω = ( μ λ − 1 ) 1 ⁄ p with μ , λ ∈ A p , BMO ω is also referred to as Bloom’s two-weight BMO space in the literature (see, e.g., [5,12,13,19,22]).

For a function b , denote by b + ( x ) = max { b ( x ) , 0 } and b − ( x ) = − min { b ( x ) , 0 } .

Let Q 0 be a cube. The Hardy-Littlewood maximal function relative to Q 0 is given by

M Q 0 ( b ) ( x ) = sup Q ∋ x Q ⊂ Q 0 1 ∣ Q ∣ ∫ Q ∣ b ( y ) ∣ d y ,

where the supremum is taken over all the cubes Q with Q ⊂ Q 0 and Q ∋ x .

Our results can be stated as follows:

Theorem 1.1

Let 1 < p < ∞ , μ , λ ∈ A p , and ν = ( μ λ − 1 ) 1 ⁄ p . Suppose that b is a locally integrable function on R n , then the following statements are equivalent:

b ∈ BMO ν and b − ⁄ ν ∈ L ∞ .
[ b , M ] is bounded from L p ( μ ) to L p ( λ ) .
There is a constant C > 0 such that
sup Q 1 μ ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ p λ ( x ) d x 1 ⁄ p ≤ C .
[ b , M ] is bounded from L p ′ ( λ ′ ) to L p ′ ( μ ′ ) , where μ ′ = μ 1 − p ′ and λ ′ = λ 1 − p ′ .
There is a constant C > 0 such that
sup Q 1 λ ′ ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ p ′ μ ′ ( x ) d x 1 ⁄ p ′ ≤ C .
There is a constant C > 0 such that
sup Q 1 ν ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ d x ≤ C .

Theorem 1.1 gives new characterizations for certain subclass of Bloom’s two-weight BMO. The unweighted case of Theorem 1.1 was obtained in [3, Proposition 4]. Moreover, for ω ∈ A p ∩ A p ′ , let μ = ω and λ = ω 1 − p in Theorem 1.1, observe that ω ∈ A p ′ implies λ = ω 1 − p ∈ A p , we obtain the following characterization for one-weight case, which seems to be new.

Corollary 1.1

Let 1 < p < ∞ and ω ∈ A p ∩ A p ′ . Suppose that b is a locally integrable function on R n , then the following statements are equivalent:

b ∈ BMO ω and b − ⁄ ω ∈ L ∞ .
[ b , M ] is bounded from L p ( ω ) to L p ( ω 1 − p ) .
There is a constant C > 0 such that
sup Q 1 ω ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ p ω ( x ) 1 − p d x 1 ⁄ p ≤ C .
There is a constant C > 0 such that
sup Q 1 ω ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ d x ≤ C .

Remark 1.1

We would like to note that [26, Theorem 2] provides a result similar to the equivalence of (1) and (2) for bilinear case under the assumption ω ∈ A 1 . Our result essentially improves the corresponding linear case of it. The equivalence of (1), (3), and (4) is new.

Taking μ = λ = ω ∈ A p in Theorem 1.1, we deduce the following result.

Corollary 1.2

Let 1 < p < ∞ and ω ∈ A p . Suppose that b is a locally integrable function on R n , then the following statements are equivalent:

b ∈ BMO and b − ∈ L ∞ .
[ b , M ] is bounded from L p ( ω ) to L p ( ω ) .
There is a constant C > 0 such that
sup Q 1 ω ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ p ω ( x ) d x 1 ⁄ p ≤ C .
There is a constant C > 0 such that
sup Q 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ d x ≤ C .

Remark 1.2

Corollary 1.2 was obtained by Aǧcayazi and Zhang in [2, Corollary 4.2]. The equivalence of (1) and (2) was also proved by Hu and Wang in [15, Theorem 1.4] independently.

On the other hand, Ho [11, Theorem 3.1] proved that for 1 ≤ p < ∞ and ω ∈ A p , then b ∈ BMO if and only if

sup Q 1 ω ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ p ω ( x ) d x 1 ⁄ p < ∞ .

Parallel to Ho’s result, statement (3) can characterize a function b ∈ BMO with b − ∈ L ∞ .

For the commutator of the sharp maximal function, we have similar results.

Theorem 1.2

Let 1 < p < ∞ , μ , λ ∈ A p , and ν = ( μ λ − 1 ) 1 ⁄ p . Suppose that b is a locally integrable function on R n , then the following statements are equivalent:

b ∈ BMO ν and b − ⁄ ν ∈ L ∞ .
[ b , M ♯ ] is bounded from L p ( μ ) to L p ( λ ) .
There is a constant C > 0 such that
sup Q 1 μ ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ p λ ( x ) d x 1 ⁄ p ≤ C .
[ b , M ♯ ] is bounded from L p ′ ( λ ′ ) to L p ′ ( μ ′ ) .
There is a constant C > 0 such that
sup Q 1 λ ′ ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ p ′ μ ′ ( x ) d x 1 ⁄ p ′ ≤ C .
There is a constant C > 0 such that
sup Q 1 ν ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ d x ≤ C .

Theorem 1.2 gives new characterizations for certain subclass of Bloom’s two-weight BMO class. When μ = ν = 1 , the result was obtained in [3, Proposition 6]. Similar to Corollary 1.1, we have new characterizations for one-weight results for commutator [ b , M ♯ ] as follows.

Corollary 1.3

Let 1 < p < ∞ and ω ∈ A p ∩ A p ′ . Suppose that b is a locally integrable function on R n , then the following statements are equivalent:

b ∈ BMO ω and b − ⁄ ω ∈ L ∞ .
[ b , M ♯ ] is bounded from L p ( ω ) to L p ( ω 1 − p ) .
There is a constant C > 0 such that
sup Q 1 ω ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ p ω ( x ) 1 − p d x 1 ⁄ p ≤ C .
There is a constant C > 0 such that
sup Q 1 ω ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ d x ≤ C .

By taking μ = λ = ω ∈ A p in Theorem 1.2, we have the following result, which was also obtained by Aǧcayazi and Zhang [2, Corollary 4.3].

Corollary 1.4

Let 1 < p < ∞ , ω ∈ A p and b be a locally integrable function in R n . Then, the following statements are equivalent:

b ∈ BMO and b − ∈ L ∞ .
[ b , M ] is bounded from L p ( ω ) to L p ( ω ) .
There is a constant C > 0 such that
sup Q 1 ω ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ p ω ( x ) d x 1 ⁄ p ≤ C .
There is a constant C > 0 such that
sup Q 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ d x ≤ C .

To compare our results with the ones for maximal commutator, we summarize some characterizations for Bloom’s two-weight BMO and the two-weight boundedness of maximal commutator of the Hardy-Littlewood maximal function.

Theorem 1.3

Let 1 < p < ∞ , μ , λ ∈ A p , and ν = ( μ λ − 1 ) 1 ⁄ p . Suppose that b is a locally integrable function on R n , then the following statements are equivalent:

b ∈ BMO ν .
M b is bounded from L p ( μ ) to L p ( λ ) .
There is a constant C > 0 such that
sup Q 1 μ ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ p λ ( x ) d x 1 ⁄ p ≤ C .
M b is bounded from L p ′ ( λ ′ ) to L p ′ ( μ ′ ) .
There is a constant C > 0 such that
sup Q 1 λ ′ ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ p ′ μ ′ ( x ) d x 1 ⁄ p ′ ≤ C .

The equivalence of (1) and (2) was proved in [24, Theorem 3] (see also [8, Theorem 2.4]). The equivalence of (1), (3), and (5) was given in [13, Theorem 4.1] for the dyadic version.

It is easy to see that the statements in Theorems 1.1 and 1.2, which can characterize a function b ∈ BMO ν with b − ⁄ ν ∈ L ∞ and two-weight boundedness of [ b , M ] and [ b , M ♯ ] , correspond to the ones in Theorem 1.3 that characterize Bloom’s two-weight BMO functions.

2 Preliminaries and lemmas

In this section, we present some lemmas that will be used in the proof of our results. The following weighted norm inequality for the Hardy-Littlewood maximal function is well known (see, for instance, [18] and [25] for details).

Lemma 2.1

Let 1 < p < ∞ . Then, M is bounded from L p ( ω ) to itself if and only if ω ∈ A p .

For Bloom’s weight, we have the following estimates (see [13, Lemma 2.7 and (2.15)]).

Lemma 2.2

[13] Let 1 < p < ∞ , μ , λ ∈ A p , and ν = ( μ λ − 1 ) 1 ⁄ p . Then, ν ∈ A 2 and for any cube Q,

∫ Q μ ( x ) d x 1 ⁄ p ∫ Q λ ( x ) − p ′ ⁄ p d x 1 ⁄ p ′ ≤ C ν ( Q ) .

The following characterization for Bloom-type two-weight inequalities for maximal commutator M b can be deduced from [24, Theorem 3] (see also [8, Theorem 2.4]).

Lemma 2.3

Let 1 < p < ∞ , μ , λ ∈ A p , and ν = ( μ λ − 1 ) 1 ⁄ p . For any locally integrable function b, the following statements are equivalent:

b ∈ BMO ν .
M b is bounded from L p ( μ ) to L p ( λ ) .

The following lemma shows that the maximal commutator M b pointwise controls the commutators [ b , M ] and [ b , M ♯ ] when b ≥ 0 (see [27, (3.1) and (3.2)] for details).

Lemma 2.4

Let b and f be locally integrable functions and b ≥ 0 a.e. in R n . Then, for any fixed x ∈ R n such that M ( f ) ( x ) < ∞ and 0 ≤ b ( x ) < ∞ , we have

(2.1) ∣ [ b , M ] ( f ) ( x ) ∣ ≤ M b ( f ) ( x ) .

and

(2.2) ∣ [ b , M ♯ ] ( f ) ( x ) ∣ ≤ 2 M b ( f ) ( x ) .

Proof

Let b and f be locally integrable and b ≥ 0 a.e. From the proof of [27, (3.1)], we see that each step in (3.1) can be applied to such x that satisfies 0 ≤ b ( x ) < ∞ and M ( f ) ( x ) < ∞ . So does (3.2). This concludes Lemma 2.4. We omit the details.□

Remark 2.1

Observe that a locally integrable function is finite a.e. and M ♯ ( f ) ( x ) ≤ 2 M ( f ) ( x ) . It follows from Lemma 2.4 that if b is locally integrable and b ≥ 0 a.e., then (2.1) and (2.2) hold for a.e. x ∈ R n , provided M ( f ) is finite a.e. in R n .

To prove our theorems, we also need the following result.

Theorem 2.1

Let 1 < p < ∞ , μ , λ ∈ A p , and ν = ( μ λ − 1 ) 1 ⁄ p . If b ∈ BMO ν and b ≥ 0 , then [ b , M ] and [ b , M ♯ ] are bounded from L p ( μ ) to L p ( λ ) .

Proof

For μ ∈ A p and f ∈ L p ( μ ) , by Lemma 2.1, we deduce that M ( f ) ( x ) is finite a.e. in R n . Since b ∈ BMO ν implies that b is locally integrable, it follows from Remark 2.1 that

∣ [ b , M ] ( f ) ( x ) ∣ ≤ M b ( f ) ( x ) , a.e. x ∈ R n ,

and

∣ [ b , M ♯ ] ( f ) ( x ) ∣ ≤ 2 M b ( f ) ( x ) , a.e. x ∈ R n .

By Lemma 2.3, we deduce that [ b , M ] and [ b , M ♯ ] are bounded from L p ( μ ) to L p ( λ ) .□

3 Proof of the theorems

Proof of Theorem 1.1

We first prove the implication ( 1 ) ⇒ ( 2 ) . Since ∣ b ( x ) ∣ − b ( x ) = 2 b − ( x ) and M ( b f ) ( x ) = M ( ∣ b ∣ f ) ( x ) , we have

(3.1) ∣ [ b , M ] ( f ) ( x ) ∣ ≤ ∣ [ b , M ] ( f ) ( x ) − [ ∣ b ∣ , M ] ( f ) ( x ) ∣ + ∣ [ ∣ b ∣ , M ] ( f ) ( x ) ∣ ≤ ∣ ( b ( x ) − ∣ b ( x ) ∣ ) M ( f ) ( x ) ∣ + ∣ M ( b f ) ( x ) − M ( ∣ b ∣ f ) ( x ) ∣ + ∣ [ ∣ b ∣ , M ] ( f ) ( x ) ∣ ≤ 2 b − ( x ) M ( f ) ( x ) + ∣ [ ∣ b ∣ , M ] ( f ) ( x ) ∣ .

Observe that ∣ b ∣ ∈ BMO ν when b ∈ BMO ν and M is bounded on L p ( μ ) since μ ∈ A p . Since b − ⁄ ν ∈ L ∞ and ν = ( μ λ − 1 ) 1 ⁄ p , it follows from (3.1) and Theorem 2.1 that

‖ [ b , M ] ( f ) ‖ L p ( λ ) ≤ 2 ∥ b − M ( f ) ∥ L p ( λ ) + ∥ [ ∣ b ∣ , M ] ( f ) ∥ L p ( λ ) ≤ 2 ‖ b − ⁄ ν ‖ L ∞ ‖ M ( f ) ‖ L p ( μ ) + ∥ [ ∣ b ∣ , M ] ( f ) ( x ) ∥ L p ( λ ) ≤ C ‖ f ‖ L p ( μ ) .

Next, we will show the implication ( 2 ) ⇒ ( 3 ) . For any cube Q ⊂ R n and any x ∈ Q , we have (see [3, p. 3331])

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

Applying statement (2), we obtain that

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

Thus, we obtain statement (3) since the constant C is independent of Q .

The proof of the implication ( 3 ) ⇒ ( 6 ) is easy. Indeed, by Hölder’s inequality, Lemma 2.2, and statement (3), we have

1 ν ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ d x = 1 ν ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ λ ( x ) 1 ⁄ p λ ( x ) − 1 ⁄ p d x ≤ 1 ν ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ p λ ( x ) d x 1 ⁄ p ∫ Q λ ( x ) − p ′ ⁄ p d x 1 ⁄ p ′ ≤ C 1 μ ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ p λ ( x ) d x 1 ⁄ p ≤ C .

This achieves statement (6) since Q is arbitrary and C is independent of Q .

Now, we prove the implication ( 6 ) ⇒ ( 1 ) . First, we want to prove b ∈ BMO ν . We use similar procedure to the proof of Proposition 4 in [3]. For any fixed cube Q , let

E = { x ∈ Q : b ( x ) ≤ b Q } and F = { x ∈ Q : b ( x ) > b Q } .

Then, (see [3, p. 3331])

∫ E ∣ b ( x ) − b Q ∣ d x = ∫ F ∣ b ( x ) − b Q ∣ d x .

Since for any x ∈ E we have b ( x ) ≤ b Q ≤ M Q ( b ) ( x ) ,

∣ b ( x ) − b Q ∣ ≤ ∣ b ( x ) − M Q ( b ) ( x ) ∣ , for any x ∈ E .

This yields

(3.2) 1 ν ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ d x = 1 ν ( Q ) ∫ E ∪ F ∣ b ( x ) − b Q ∣ d x = 2 ν ( Q ) ∫ E ∣ b ( x ) − b Q ∣ d x ≤ 2 ν ( Q ) ∫ E ∣ b ( x ) − M Q ( b ) ( x ) ∣ d x ≤ 2 ν ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ d x .

Combining (3.2) and statement (6), we obtain that

1 ν ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ d x ≤ C ,

which implies b ∈ BMO ν by Definition 1.1.

Now, we will show b − ⁄ ν ∈ L ∞ ( R n ) . For any fixed cube Q , observe that

0 ≤ b + ( x ) ≤ ∣ b ( x ) ∣ ≤ M Q ( b ) ( x ) , for a.e. x ∈ Q .

We have for almost all x ∈ Q that

(3.3) 0 ≤ b − ( x ) ≤ M Q ( b ) ( x ) − b + ( x ) + b − ( x ) = M Q ( b ) ( x ) − b ( x ) .

By Lemma 2.2, we have ν ∈ A 2 since μ , λ ∈ A p . Then, it follows from the A 2 condition that 1 ⁄ ν is locally integrable and

(3.4) 1 ∣ Q ∣ ∫ Q 1 ν ( x ) d x ≤ C ∣ Q ∣ ν ( Q ) .

By (3.4), (3.3), and statement (6), we deduce

(3.5) 1 ∣ Q ∣ ∫ Q b − ( x ) d x 1 ∣ Q ∣ ∫ Q 1 ν ( x ) d x ≤ C ν ( Q ) ∫ Q b − ( x ) d x ≤ C ν ( Q ) ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ d x ≤ C .

Applying the Lebesgue differentiation theorem, we can achieve that b − ⁄ ν ∈ L ∞ . Indeed, denote by ℒ ( b ) and ℒ ( 1 ⁄ ν ) the sets of all Lebesgue points of b and 1 ⁄ ν , and then, R n \ ( ℒ ( b ) ∩ ℒ ( 1 ⁄ ν ) ) is a set of measure zero.

For any x 0 ∈ ℒ ( b ) ∩ ℒ ( 1 ⁄ ν ) , it follows from the Lebesgue differentiation theorem that

lim ∣ Q ∣ → 0 Q ∋ x 0 1 ∣ Q ∣ ∫ Q b − ( x ) d x 1 ∣ Q ∣ ∫ Q 1 ν ( x ) d x = b − ( x 0 ) ν ( x 0 ) .

This together with (3.5) concludes that b − ( x 0 ) ⁄ ν ( x 0 ) ≤ C . Thus, we have b − ⁄ ν ∈ L ∞ .

Finally, to complete the proof of Theorem 1.1, we need to verify the implications ( 1 ) ⇒ ( 4 ) , ( 4 ) ⇒ ( 5 ) , and ( 5 ) ⇒ ( 6 ) . Observe that for μ , λ ∈ A p , we have μ ′ , λ ′ ∈ A p ′ and ν = ( μ λ − 1 ) 1 ⁄ p = ( λ ′ μ ′ − 1 ) 1 ⁄ p ′ . So, the procedure we just used on the ordered-group ( μ , λ , ν , p ) also applies to the ordered-group ( λ ′ , μ ′ , ν , p ′ ) . Thus, we can use the same procedure to conclude the implications ( 1 ) ⇒ ( 4 ) , ( 4 ) ⇒ ( 5 ) , and ( 5 ) ⇒ ( 6 ) . We omit the details.

The proof of Theorem 1.1 is complete.□

Proof of Theorem 1.2

Reasoning as the proof of Theorem 1.1, we only prove the implications ( 1 ) ⇒ ( 2 ) ⇒ ( 3 ) ⇒ ( 6 ) and ( 6 ) ⇒ ( 1 ) .

We first prove the implication ( 1 ) ⇒ ( 2 ) . By the definition of [ b , M ♯ ] , we have

∣ [ b , M ♯ ] ( f ) ( x ) − [ ∣ b ∣ , M ♯ ] ( f ) ( x ) ∣ ≤ ∣ b ( x ) M ♯ ( f ) ( x ) − ∣ b ( x ) ∣ M ♯ ( f ) ( x ) ∣ + ∣ M ♯ ( b f ) ( x ) − M ♯ ( ∣ b ∣ f ) ( x ) ∣ ≤ 2 b − ( x ) M ♯ ( f ) ( x ) + ∣ M ♯ ( ( ∣ b ∣ − b ) f ) ( x ) ∣ ≤ 2 b − ( x ) M ♯ ( f ) ( x ) + M ♯ ( 2 b − f ) ( x ) .

This gives

(3.6) ∣ [ b , M ♯ ] ( f ) ( x ) ∣ ≤ 2 b − ( x ) M ♯ ( f ) ( x ) + M ♯ ( 2 b − f ) ( x ) + ∣ [ ∣ b ∣ , M ♯ ] ( f ) ( x ) ∣ .

Since M ♯ ( f ) ( x ) ≤ 2 M ( f ) ( x ) , by Lemma 2.1, we have that M ♯ is bounded on both L p ( μ ) and L p ( λ ) . Observe that ∣ b ∣ ∈ BMO ν when b ∈ BMO ν and note that b − ⁄ ν ∈ L ∞ and ν = ( μ λ − 1 ) 1 ⁄ p . It follows from (3.6) and Theorem 2.1 that

∥ [ b , M ♯ ] ( f ) ∥ L p ( λ ) ≤ 2 ‖ b − M ♯ ( f ) ‖ L p ( λ ) + ‖ M ♯ ( 2 b − f ) ‖ L p ( λ ) + ∥ [ ∣ b ∣ , M ♯ ] ( f ) ∥ L p ( λ ) ≤ 2 ‖ b − ⁄ ν ‖ L ∞ ‖ M ♯ ( f ) ‖ L p ( μ ) + 2 ‖ b − f ‖ L p ( λ ) + C ‖ f ‖ L p ( μ ) ≤ 2 ‖ b − ⁄ ν ‖ L ∞ ‖ f ‖ L p ( μ ) + C ‖ f ‖ L p ( μ ) ≤ C ‖ f ‖ L p ( μ ) .

This concludes that [ b , M ♯ ] is bounded from L p ( μ ) to L p ( λ ) .

Next, we prove the implication ( 2 ) ⇒ ( 3 ) . For any fixed a cube Q , we have M ♯ ( χ Q ) ( x ) = 1 2 for x ∈ Q (see [3, p. 3333] for details). Note statement (2) that [ b , M ♯ ] is bounded from L p ( μ ) to L p ( λ ) , and then, we have

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

Since the constant C is independent of Q , this concludes the proof of ( 2 ) ⇒ ( 3 ) .

It is easy to prove the implication ( 3 ) ⇒ ( 6 ) . Indeed, by Hölder’s inequality, Lemma 2.2, and statement (3), we have

1 ν ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ d x = 1 ν ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ λ ( x ) 1 ⁄ p λ ( x ) − 1 ⁄ p d x ≤ 1 ν ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ p λ ( x ) d x 1 ⁄ p ∫ Q λ ( x ) − p ′ ⁄ p d x 1 ⁄ p ′ ≤ C 1 μ ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ p λ ( x ) d x 1 ⁄ p ≤ C .

Finally, to prove the implication ( 6 ) ⇒ ( 1 ) , we can argue as mentioned earlier. For any fixed cube Q , it was proved by Bastro et al. in [3, (2)] that

(3.7) ∣ b Q ∣ ≤ 2 M ♯ ( b χ Q ) ( x ) , for x ∈ Q .

Let E = { x ∈ Q : b ( x ) ≤ b Q } and F = { x ∈ Q : b ( x ) > b Q } as mentioned earlier. Then (see [3, p. 3331]),

∫ E ∣ b ( x ) − b Q ∣ d x = ∫ F ∣ b ( x ) − b Q ∣ d x .

Since b ( x ) ≤ b Q ≤ ∣ b Q ∣ ≤ 2 M ♯ ( b χ Q ) ( x ) for x ∈ E , we have

∣ b ( x ) − b Q ∣ ≤ ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ , x ∈ E .

Similar to (3.2), we have

(3.8) 1 ν ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ d x = 2 ν ( Q ) ∫ E ∣ b ( x ) − b Q ∣ d x ≤ 2 ν ( Q ) ∫ Q ∣ b ( x ) − 2 M ♯ ( b χ Q ) ( x ) ∣ d x .

Combining (3.8) and statement (6), we conclude that b ∈ BMO ν .

To complete the proof, we need to show b − ⁄ ν ∈ L ∞ . By (3.7) again, we obtain for x ∈ Q ,

2 M ♯ ( b χ Q ) ( x ) − b ( x ) ≥ ∣ b Q ∣ − b ( x ) = ∣ b Q ∣ − b + ( x ) + b − ( x ) .

Since μ , λ ∈ A p , by Lemma 2.2, one has ν ∈ A 2 . Using (3.4) again, we have

1 ν ( Q ) ∫ Q ∣ 2 M ♯ ( b χ Q ) ( x ) − b ( x ) ∣ d x = 1 ∣ Q ∣ ⋅ ∣ Q ∣ ν ( Q ) ∫ Q ∣ 2 M ♯ ( b χ Q ) ( x ) − b ( x ) ∣ d x ≥ 1 ∣ Q ∣ ∫ Q ∣ 2 M ♯ ( b χ Q ) ( x ) − b ( x ) ∣ d x 1 ∣ Q ∣ ∫ Q 1 ν ( x ) d x ≥ 1 ∣ Q ∣ ∫ Q ( 2 M ♯ ( b χ Q ) ( x ) − b ( x ) ) d x 1 ∣ Q ∣ ∫ Q 1 ν ( x ) d x ≥ 1 ∣ Q ∣ ∫ Q ( ∣ b Q ∣ − b + ( x ) + b − ( x ) ) d x 1 ∣ Q ∣ ∫ Q 1 ν ( x ) d x = ∣ b Q ∣ − 1 ∣ Q ∣ ∫ Q b + ( x ) d x + 1 ∣ Q ∣ ∫ Q b − ( x ) d x 1 ∣ Q ∣ ∫ Q 1 ν ( x ) d x .

This together with statement (6) gives

(3.9) ∣ b Q ∣ − 1 ∣ Q ∣ ∫ Q b + ( x ) d x + 1 ∣ Q ∣ ∫ Q b − ( x ) d x 1 ∣ Q ∣ ∫ Q 1 ν ( x ) d x ≤ C .

By the same reason as in the prove of Theorem 1.1, letting ∣ Q ∣ → 0 with Q ∋ x in (3.9), Lebesgue differentiation theorem assures that the limit of the left-hand side of (3.9) equals almost everywhere to

∣ b ( x ) ∣ − b + ( x ) + b − ( x ) ν ( x ) = 2 b − ( x ) ν ( x ) .

Putting this fact together with (3.9) gives that b − ⁄ ν ∈ L ∞ .

This completes the proof of Theorem 1.2.□

Before ending this article, we give another proof of the equivalence of (1), (3), and (5) in Theorem 1.3, which is different from the ones provided in [13, Theorem 4.1].

Proof of Theorem 1.3

Since the equivalence of (1) and (2) is given in [24, Theorem 3], it is sufficient to prove the implications ( 2 ) ⇒ ( 3 ) ⇒ ( 1 ) and ( 1 ) ⇒ ( 4 ) ⇒ ( 5 ) ⇒ ( 1 ) .

For any cube Q , noting statement (2) that M b is bounded from L p ( μ ) to L p ( λ ) , we have

1 μ ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ p λ ( x ) d x ≤ 1 μ ( Q ) ∫ Q 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − b ( y ) ∣ d y p λ ( x ) d x ≤ 1 μ ( Q ) ∫ Q ( M b ( χ Q ) ( x ) ) p λ ( x ) d x ≤ 1 μ ( Q ) ∥ M b ( χ Q ) ∥ L p ( λ ) p ≤ C μ ( Q ) ‖ χ Q ‖ L p ( μ ) p ≤ C ,

which concludes the implication ( 2 ) ⇒ ( 3 ) .

Now, we prove the implication ( 3 ) ⇒ ( 1 ) . Given a cube Q , by Hölder’s inequality, Lemma 2.2, and statement (3), we have

1 ν ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ d x = 1 ν ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ λ ( x ) 1 ⁄ p λ ( x ) − 1 ⁄ p d x ≤ 1 ν ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ p λ ( x ) d x 1 ⁄ p ∫ Q λ ( x ) − p ′ ⁄ p d x 1 ⁄ p ′ ≤ C 1 μ ( Q ) ∫ Q ∣ b ( x ) − b Q ∣ p λ ( x ) d x 1 ⁄ p ≤ C .

This achieves b ∈ BMO ν by Definition 1.1.

Observe that 1 < p ′ < ∞ , μ ′ , λ ′ ∈ A p ′ , and ν = ( μ λ − 1 ) 1 ⁄ p = ( λ ′ μ ′ − 1 ) 1 ⁄ p ′ . Applying the equivalence of (1) and (2) to the ordered-group ( λ ′ , μ ′ , ν , p ′ ) gives the equivalence of (1) and (4). Using the procedure in proving the implications ( 2 ) ⇒ ( 3 ) ⇒ ( 1 ) to the ordered-group ( λ ′ , μ ′ , ν , p ′ ) , we conclude the implications ( 4 ) ⇒ ( 5 ) and ( 5 ) ⇒ ( 1 ) .□

Acknowledgments

The authors would like to thank the referees for their very helpful and detailed comments.

Funding information: This work was partially supported by the Fundamental Research Funds for Education Department of Heilongjiang Province (1453ZD031) and the Scientific Research Fund of Mudanjiang Normal University (MSB201201).
Author contributions: All authors contributed equally to the writing of this article. All authors read the final manuscript and approved its submission.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: No data were used for the research described in the article.

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Received: 2024-09-23

Revised: 2025-01-06

Accepted: 2025-02-10

Published Online: 2025-03-14

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

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https://doi.org/10.1515/agms-2025-0020

Keywords for this article

Hardy-Littlewood maximal function; sharp maximal function; commutator; weights; weighted BMO space

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