Home Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
Article Open Access

Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces

  • Pu Zhang and Müjdat Ağcayazı EMAIL logo
Published/Copyright: October 15, 2024
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this work, we present necessary and sufficient conditions for the boundedness of the commutators generated by multilinear fractional maximal operators on the products of Morrey spaces when the symbol belongs to Lipschitz spaces.

Keywords: commutators; multilinear fractional maximal operators; Lipschitz spaces; Morrey spaces
MSC 2010: 47B47; 42B25; 42B35; 46E30

1 Introduction and main results

Let T be a Calderón-Zygmund singular integral operator. In 1976, Coifman et al. [1] studied the commutator generated by T and a suitable function b as follows:

(1) [ T , b ] f ( x ) ≔ T ( b f ) ( x ) − b ( x ) T f ( x ) .

A well-known result states that [ T , b ] is bounded on L p ( R n ) for 1 < p < ∞ if and only if b ∈ BMO ( R n ) . Sufficiency was proved by Coifman et al. [1] and necessity part was obtained by Janson [2].

In 1978, Janson [2] gave some characterizations of Lipschitz space Λ ˙ β ( R n ) via commutator [ T , b ] and proved that b ∈ Λ ˙ β ( R n ) ( 0 < β < 1 ) if and only if [ T , b ] is bounded from L p ( R n ) to L q ( R n ) , where 1 < p < n ⁄ β and 1 ⁄ p − 1 ⁄ q = β ⁄ n [3].

Let 0 ≤ α < n . The fractional maximal operator M α is defined by

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

where the supremum is taken over all the cubes Q ⊆ R n containing x with sides parallel to the coordinate axes and where f is any locally integrable function. When α = 0 , M 0 ≔ M is known as the Hardy-Littlewood maximal operator.

In order to investigate [ M α , b ] , we start with the consideration of the fractional maximal commutators M b , α .

Definition 1

Let 0 ≤ α < n . Given a measurable function b , the fractional maximal commutators M b , α are defined by

M b , α ( f ) ( x ) ≔ sup Q ∋ x 1 ∣ Q ∣ 1 − α ⁄ n ∫ Q ∣ b ( x ) − b ( y ) ∣ ∣ f ( y ) ∣ d y , ( x ∈ R n ) .

This type of operator plays an important role in the study of commutators of singular integral operators with bounded mean oscillation (BMO) symbols [4–6]. Segovia and Torrea [6, Theorem 3.5] proved that if b ∈ BMO ( R n ) , 0 < α < n , 1 ⁄ p − 1 ⁄ q = α ⁄ n , w ∈ A p , q , then M b , α is bounded from L p ( w p ) to L q ( w q ) . In general, M b , α fails to be weak type ( 1 , n ⁄ ( n − α ) ) when b ∈ BMO ( R n ) . Instead, an endpoint theory was provided for this operator such as weak type L ( 1 + log + L ) , see for instance [7, Theorem 3].

Similar to (1), we can recall the definition of commutators of fractional maximal operators as follows:

Definition 2

Let 0 ≤ α < n . Given a measurable function b , the commutators of the fractional maximal operators M α and b are defined by

[ M α , b ] f ( x ) ≔ M α ( b f ) ( x ) − b ( x ) M α f ( x ) .

We would like to point out that operators M b , α and [ M α , b ] are essentially different from each other. For example, M b , α is positive and sublinear, but [ M α , b ] is neither positive nor sublinear.

Mapping properties of M b , α and [ M α , b ] have been studied extensively by many authors for BMO functions and remarkable results have been obtained in Lebesgue spaces, weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, and Morrey spaces, [8–13].

Zhang et al. [14] gave some necessary and sufficient conditions for the boundedness of M b , α and [ M α , b ] on variable Lebesgue spaces L p ( ⋅ ) ( R n ) when the nonnegative symbol b belongs to Lipschitz spaces, which gives some new characterizations for certain subclasses of Lipschitz spaces. The first necessary and sufficient conditions of Lipschitz functions on Orlicz spaces L Φ ( R n ) were given by Guliyev et al. [15] when 0 ≤ α < n , and Zhang [16] obtained new characterizations of non-negative Lipschitz functions on Orlicz spaces by extending and improving the results in [15].

Multilinear Calderón-Zygmund operators (CZOs) were introduced and first studied by Coifman and Meyer [17], and later on by Grafakos and Torres [18] as a natural generalization of the linear case. In recent years, the theory of multilinear CZOs and related operators such as multilinear singular integral operators, maximal and fractional maximal type operators, and fractional integrals have attracted much attention [19–23].

Commutators of multilinear operators with BMO symbols have also been studied extensively by many authors, and some important theorems obtained in linear cases, such as boundedness and endpoint theorems, have also been obtained for these operators [24,25].

Definition 3

Let 0 ≤ α < m n , f → = ( f 1 , f 2 , … , f m ) be locally integrable functions. The multilinear fractional maximal function ℳ α is defined by

(2) ℳ α ( f → ) ( x ) = sup Q ∋ x ∏ i = 1 m 1 ∣ Q ∣ 1 − α ⁄ m n ∫ Q ∣ f i ( y i ) ∣ d y i ,

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

When α = 0 , ℳ 0 is exactly the multilinear maximal function, simply denoted by ℳ , introduced by Lerner et al. [20] to adopt to multiple weight theory of CZOs. In [20] Lerner et al. developed a multiple weight theory that adopts to the multilinear CZOs. When 0 < α < m n , ℳ α is the multilinear fractional maximal function considered by Chen and Xue [26] and Moen [22] in studying the multiple weight theory of multilinear fractional integral operator.

Similar to linear case, we can define the maximal and the nonlinear commutators of ℳ α .

For notational convenience, we will occasionally write f → = ( f 1 , … , f m ) , y → = ( y 1 , … , y m ) , and d y → = d y 1 … d y m . We also use b → = ( b 1 , … , b m ) ∈ ( Λ ˙ β ) m to stand for b j ∈ Λ ˙ β for j = 1 , … , m , where Λ β is the Lipschitz space (see Definition 5). For a measurable set E , we denote by ∣ E ∣ the Lebesgue measure of E and by χ E the characteristic function of E and write E m = ( E ) m = E × … × E ︸ m and χ → E = ( χ E , … , χ E ︸ m ) .

Definition 4

Let 0 ≤ α < m n , f → = ( f 1 , f 2 , … , f m ) , and b → = ( b 1 , b 2 … , b m ) be two collections of locally integrable functions. The maximal commutator of ℳ α with b → is given by

ℳ α , b → ( f → ) ( x ) = ∑ i = 1 m ℳ α , b → i ( f → ) ( x ) ,

where

ℳ α , b → i ( f → ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ m − α ⁄ n ∫ Q m ∣ b i ( x ) − b i ( y i ) ∣ ∏ j = 1 m ∣ f j ( y j ) ∣ d y → .

The nonlinear commutator of ℳ α with b → is defined by

[ b → , ℳ α ] ( f → ) ( x ) = ∑ i = 1 m [ b → , ℳ α ] i ( f → ) ( x ) ,

where

[ b → , ℳ α ] i ( f → ) ( x ) = b i ( x ) ℳ α ( f → ) ( x ) − ℳ α ( f 1 , … , b i f i , … , f m ) ( x ) .

When α = 0 , we simply write ℳ b → instead of ℳ 0 , b → and [ b → , ℳ ] instead of [ b → , ℳ 0 ] . We note that the operators ℳ α , b → and [ b → , ℳ α ] essentially differ from each other. For example ℳ α , b → is positive and sublinear in each entry, but [ b → , ℳ α ] is neither positive nor sublinear.

In 2015, Zhang [27] first considered mapping properties of commutators ℳ b → and [ b → , ℳ ] when the symbols belong to BMO. More clearly, the multiple weighted strong, and weak-type estimates for these operators are studied and some necessary and sufficient conditions for the boundedness of these operators are given. Wang and Wang [28] established characterizations of the weighted BMO space in terms of several different commutators of bilinear Hardy-Littlewood maximal function ℳ b → and [ b → , ℳ ] . Yu et al. [29] considered equivalent conditions for the boundedness of the commutators ℳ b → and [ b → , ℳ ] generated by the multilinear maximal function ℳ and the BMO functions on Morrey spaces. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given. Zhang [30] gave some necessary and sufficient conditions for boundedness for the maximal and nonlinear commutators of a multi-sublinear maximal function ℳ α with symbols belonging to Lipschitz spaces.

In this study, we present necessary and sufficient conditions for the boundedness of the commutators ℳ α , b → and [ b → , ℳ α ] on the product of Morrey spaces when the symbol belongs to Lipschitz functions.

Definition 5

Let β ∈ ( 0 , 1 ) . Then, a function b belongs to Lipschitz space Λ ˙ β ( R n ) if there exists a constant c > 0 such that for all x , y ∈ R n ,

∣ b ( x ) − b ( y ) ∣ ≤ c ∣ x − y ∣ β .

The smallest such constant c is called Λ ˙ β norm of b and is denoted by ∥ b ∥ Λ ˙ β ( R n ) .

We may now formulate our main results for the maximal commutators ℳ α , b → first, and then for the commutators of the maximal functions [ b → , ℳ α ] on Morrey spaces as follows.

Theorem 1

Let 0 < β < 1 , 0 ≤ α < m n , 0 < α + β < m n , and b → = ( b 1 , b 2 , … , b m ) be a collection of locally integrable functions. Suppose that 1 < p 1 , … , p m < ∞ , 1 p = 1 p 1 + … + 1 p m , p j < m n α + β , p > 1 , λ p = ∑ j = 1 m λ j p j , 1 q j = 1 p j − α + β m ( n − λ j ) , 0 ≤ λ j < n − ( α + β ) p j m , and 1 q = 1 p − α + β n − λ . Then, the following assertions are equivalent:

  1. b → ∈ ( Λ ˙ β ) m .

  2. ℳ α , b → is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to L q , λ ( R n ) .

Theorem 2

Let 0 < β < 1 , 0 ≤ α < m n , 0 < α + β < m n , and b → = ( b 1 , b 2 , … , b m ) be a collection of locally integrable functions. Suppose that 1 < p 1 , … , p m < ∞ , 1 = 1 p 1 + … + 1 p m , p j < m n α + β , q > 1 , λ = ∑ j = 1 m λ j p j , 1 q j = 1 p j − α + β m ( n − λ j ) , 0 ≤ λ j < n − ( α + β ) p j m , and 1 q = 1 − α + β n − λ . Then, the following assertions are equivalent:

  1. b → ∈ ( Λ ˙ β ) m .

  2. ℳ α , b → is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to W L q , λ ( R n ) .

Theorem 3

Let 0 < β < 1 , 0 ≤ α < m n , 0 < α + β < m n , and b → = ( b 1 , b 2 , … , b m ) be a collection of locally integrable functions. Suppose that 1 < p 1 , … , p m < ∞ , 1 p = 1 p 1 + … + 1 p m , p j < m n α + β , p > 1 , λ p = ∑ j = 1 m λ j p j , 1 q j = 1 p j − α + β m ( n − λ j ) , 0 ≤ λ j < n − ( α + β ) p j m , and 1 q = 1 p − α + β n − λ .

  1. If b → ∈ ( Λ ˙ β ) m and b j ≥ 0 for j = 1 , 2 , … , m , then [ b → , ℳ α ] is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to L q , λ ( R n ) .

  2. If q ≥ 1 and [ b → , ℳ α ] j is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to L q , λ ( R n ) , then b j ∈ Λ ˙ β ( R n ) and b j ≥ 0 .

Theorem 4

Let 0 < β < 1 , 0 ≤ α < m n , 0 < α + β < m n , and b → ∈ ( Λ ˙ β ) m and b j ≥ 0 for j = 1 , 2 , … , m . Suppose that 1 < p 1 , … , p m < ∞ , 1 = 1 p 1 + … + 1 p m , p j < m n α + β , q > 1 , λ = ∑ j = 1 m λ j p j , 1 q j = 1 p j − α + β m ( n − λ j ) , 0 ≤ λ j < n − ( α + β ) p j m , and 1 q = 1 − α + β n − λ . Then [ b → , ℳ α ] is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to W L q , λ ( R n ) .

Remark 1

When λ = 0 , since Morrey space reduces to Lebesgue space, note that Theorems 1–4 cover Zhang’s results in [30].

The rest of the work is organized as follows: Section 2 contains some lemmas and definitions necessary to prove the main theorems. In Section 3, we prove Theorems 1–2 and the proof of Theorems 3–4 will be given in the the final section.

2 Notations and preliminaries

Throughout the work, we always denote by c a positive constant, which is independent of main parameters but it may vary from line to line. By a ≲ b , we mean that a ≤ c b , where c > 0 depends on inessential parameters.

As usual, for a locally integrable function f and a cube Q , f Q denotes the usual average of f over Q f Q = 1 ∣ Q ∣ ∫ Q f .

Let β ∈ ( 0 , 1 ) and 1 ≤ q < ∞ . We can characterize the Lipschitz (Campanato) space Λ ˙ β , q ( R n ) . Recall that the Lipschitz (Campanato) space Λ ˙ β , q ( R n ) is the set of all f ∈ L l o c p ( R n ) for which the quantity

sup Q ∣ Q ∣ − β ⁄ n 1 ∣ Q ∣ ∫ Q ∣ b ( y ) − b Q ∣ q d y 1 ⁄ q ≤ c < ∞ .

Then, for all β ∈ ( 0 , 1 ) and 1 ≤ q < ∞ , Λ ˙ β , q ( R n ) = Λ ˙ β ( R n ) with equivalent norms [3,31,32]. We know also that there exists a constant c β , q such that

(3) ∥ b ∥ Λ ˙ β , 1 ( R n ) ≤ ∥ b ∥ Λ ˙ β , q ( R n ) ≤ c β , q ∥ b ∥ Λ ˙ β , 1 ( R n ) .

The works of Campanato and Meyers led to this equivalence ([33, p. 183] and [34, p. 718], where both authors showed that these norms are equivalent to the Lip θ norm). For an account of these facts, see [35, p. 72] and also [36, Theorem 3.1].

Definition 6

For a fixed cube Q 0 , the Hardy-Littlewood maximal function with respect to Q 0 of a function f is given by

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

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

The following characterization for nonnegative Lipschitz functions can be deduced from the proof of Theorem 1.4 in [37] [14, Lemma 2.2].

Lemma 1

[14, Lemma 2.2] Let b be a locally integrable function and β ∈ ( 0 , 1 ) . Then, the following statements are equivalent:

  1. b ∈ Λ ˙ β ( R n ) and b ≥ 0 .

  2. For all 1 ≤ s < ∞ ,

    (4) sup Q 1 ∣ Q ∣ β ⁄ n 1 ∣ Q ∣ ∫ Q ∣ b ( x ) − M Q ( b ) ( x ) ∣ s d x 1 ⁄ s < ∞ .

  3. (4) holds for some 1 ≤ s < ∞ .

In the theory of partial differential equations (PDEs), Morrey spaces play an important role which was introduced by Morrey [38] in 1938 in connection with certain problems in elliptic PDEs and calculus of variations.

Definition 7

Let 1 ≤ p < ∞ , 0 ≤ λ ≤ n . We denote by L p , λ = L p , λ ( R n ) the Morrey space, and by W L p , λ = W L p , λ ( R n ) weak Morrey space, the sets of locally integrable functions f , with the finite norms

‖ f ‖ L p , λ ( R n ) ≔ sup Q 1 ∣ Q ∣ λ ⁄ n ∫ Q ∣ f ( y ) ∣ p d y 1 ⁄ p < ∞ , ‖ f ‖ W L p , λ ( R n ) ≔ sup r > 0 r sup Q 1 ∣ Q ∣ λ ⁄ n ∣ { y ∈ Q : ∣ f ( y ) ∣ > r } ∣ 1 ⁄ p < ∞ ,

respectively.

Wang and Si [39] obtained the necessary and sufficient conditions on the parameters for the boundedness of the multilinear fractional maximal operator ℳ Ω , α and the multilinear fractional integral operator ℐ Ω , α with rough kernels on Morrey spaces.

Lemma 2

[39, Theorem 1.1]  0 < α < m n , 1 < s < ∞ , s ′ = s ⁄ ( s − 1 ) , Ω ∈ ( S m n − 1 ) . Suppose that λ p = ∑ j = 1 m λ j p j , 1 q j = 1 p j − α m ( n − λ j ) , and 0 ≤ λ j < n − α p j m .

  1. p > s ′ and λ q = ∑ j = 1 m λ j q j , then the condition 1 q = 1 p − α n − λ is necessary and sufficient condition for the boundedness of the operator ℳ Ω , α from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to L q , λ ( R n ) .

  2. p = s ′ and ∑ j = 1 m λ j p j q j = λ ∑ j = 1 m 1 p j q j , then the condition 1 q = 1 p − α n − λ is necessary and sufficient condition for the boundedness of the operator ℳ Ω , α from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to W L q , λ ( R n ) .

When Ω ≡ 1 , ℳ Ω , α is the multilinear fractional maximal operator ℳ α defined in (2).

Remark 2

Lemma 2 also holds for s = ∞ . We leave the details to the reader.

3 Proof of Theorems 1 and 2

As we have mentioned in Section 1, we will prove Theorems 1 and 2.

Proof of the Theorem 1

(i) ⇒ (ii) If b → ∈ ( Λ ˙ β ) m , then for any x ∈ R n , we have

(5) ℳ α , b → ( f → ) ( x ) ≲ ∑ i = 1 m ‖ b i ‖ Λ ˙ β ℳ α + β ( f → ) ( x ) .

Indeed,

(6) ℳ α , b → i ( f → ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ m − α ⁄ n ∫ Q m ∣ b i ( x ) − b i ( y i ) ∣ ∏ j = 1 m ∣ f j ( y j ) ∣ d y → ≲ ‖ b i ‖ Λ ˙ β sup Q ∋ x 1 ∣ Q ∣ m − ( α + β ) ⁄ n ∫ Q m ∏ j = 1 m ∣ f j ( y j ) ∣ d y → ≲ ‖ b i ‖ Λ ˙ β ℳ α + β ( f → ) ( x ) .

Thus,

(7) ℳ α , b → ( f → ) ( x ) = ∑ i = 1 m ℳ α , b → i ( f → ) ( x ) ≲ ∑ i = 1 m ‖ b i ‖ Λ ˙ β ℳ α + β ( f → ) ( x ) .

Then, using lattice property of the norm, inequality (7), and Lemma 2 (i), respectively, we obtain

‖ ℳ α , b → ( f → ) ‖ L q , λ ( R n ) ≲ ‖ b → ‖ Λ ˙ β ‖ ℳ α + β ( f → ) ‖ L q , λ ( R n ) ≲ ‖ b → ‖ Λ ˙ β ∏ j = 1 m ‖ f j ‖ L p j , λ j ( R n ) .

(ii) ⇒ (i) We may assume j = 1 and m = 2 , without loss of generality, and ℳ α , b → is bounded from L p 1 , λ 1 ( R n ) × L p 2 , λ 2 ( R n ) to L q , λ ( R n ) . Then, we obtain

1 ∣ Q ∣ 1 + β n ∫ Q ∣ b 1 ( x ) − ( b 1 ) Q ∣ d x ≤ 1 ∣ Q ∣ 2 + β n ∫ Q ∫ Q ∣ b 1 ( x ) − b 1 ( y ) ∣ d y 1 d x = 1 ∣ Q ∣ 1 + α + β n ∫ Q 1 ∣ Q ∣ 2 − α n ∫ Q ∣ b 1 ( x ) − b 1 ( y ) ∣ χ Q ( y 1 ) d y 1 ∫ Q χ Q ( y 2 ) d y 2 d x ≤ 1 ∣ Q ∣ 1 + α + β n ∫ Q ℳ α , b 1 ( χ Q , χ Q ) ( x ) d x ≤ 1 ∣ Q ∣ 1 + α + β n 1 ∣ Q ∣ λ n ∫ Q ℳ α , b 1 ( χ Q , χ Q ) q ( x ) d x 1 ⁄ q ∣ Q ∣ λ n q + 1 q ′ ≲ ∣ Q ∣ λ n q + 1 q ′ − 1 − α + β n ‖ χ Q ‖ L p 1 , λ 1 ( R n ) ‖ χ Q ‖ L p 2 , λ 2 ( R n ) ≲ ∣ Q ∣ λ n q + 1 q ′ − 1 − α + β n + 1 p 1 − λ 1 n p 1 + 1 p 2 − λ 2 n p 2 = c ,

which proves that b 1 ∈ Λ ˙ β ( R n ) . This completes the proof.□

Proof of the Theorem 2

(i) ⇒ (ii) By using inequality (5), lattice property of the norm, and Lemma 2, respectively, we obtain

‖ ℳ α , b → ( f → ) ‖ W L q , λ ( R n ) ≲ ‖ b → ‖ Λ ˙ β ‖ ℳ α + β ( f → ) ‖ W L q , λ ( R n ) ≲ ‖ b → ‖ Λ ˙ β ∏ j = 1 m ‖ f j ‖ L p j , λ j ( R n ) .

(ii) ⇒ (i) For any fixed cube Q 0 and any x ∈ Q 0 , we obtain

ℳ α , b → i ( χ → Q 0 ) ( x ) = sup Q ∋ x 1 ∣ Q ∣ m − α ⁄ n ∫ Q m ∣ b i ( x ) − b i ( y i ) ∣ ∏ j = 1 m ∣ χ Q 0 ( y j ) ∣ d y → ≥ 1 ∣ Q 0 ∣ m − α ⁄ n ∫ ( Q 0 ) m ∣ b i ( x ) − b i ( y i ) ∣ ∏ j = 1 m ∣ χ Q 0 ( y j ) ∣ d y → = 1 ∣ Q 0 ∣ 1 − α ⁄ n ∫ Q 0 ∣ b i ( x ) − b i ( y i ) ∣ ∣ χ Q 0 ( y i ) ∣ d y i ≥ ∣ Q 0 ∣ α n ∣ b i ( x ) − ( b i ) Q 0 ∣

[30, p. 10]. Along with the definition of ℳ α , b → gives

∣ b i ( x ) − ( b i ) Q 0 ∣ ≤ ∣ Q 0 ∣ − α n ℳ α , b → i ( χ → Q 0 ) ( x ) ≤ ∣ Q 0 ∣ − α n ℳ α , b → ( χ → Q 0 ) ( x ) .

Since ℳ α , b → is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to W L q , λ ( R n ) , we have

(8) ∣ { x ∈ Q 0 : ∣ b i ( x ) − ( b i ) Q 0 ∣ > λ } ∣ ≤ { x ∈ Q 0 : ∣ Q 0 ∣ − α n ℳ α , b → ( χ → Q 0 ) ( x ) > λ } ≲ ∣ Q 0 ∣ λ n λ − 1 ∣ Q 0 ∣ − α n ∏ j = 1 m ‖ χ Q 0 ‖ L p j , λ j ( R n ) q ≲ λ − q ∣ Q 0 ∣ q ( λ n q − λ n p + 1 p − α n ) ,

where 1 ⁄ p = 1 ⁄ p 1 + … + 1 ⁄ p m .

Let t > 0 be a constant to be determined later. Then, by (8) and noting that q > 1 , we obtain

1 ∣ Q 0 ∣ 1 + β n ∫ Q 0 ∣ b i ( x ) − ( b i ) Q 0 ∣ d x = 1 ∣ Q 0 ∣ 1 + β n ∫ 0 ∞ ∣ { x ∈ Q 0 : ∣ b i ( x ) − ( b i ) Q 0 ∣ > u } ∣ d u = 1 ∣ Q 0 ∣ 1 + β n ∫ 0 t + ∫ t ∞ ∣ { x ∈ Q 0 : ∣ b i ( x ) − ( b i ) Q 0 ∣ > u } ∣ d u ≲ t ∣ Q 0 ∣ 1 + β n ∣ Q 0 ∣ + 1 ∣ Q 0 ∣ 1 + β n ∫ t ∞ u − q ∣ Q 0 ∣ q ( λ n q − λ n p + 1 p − α n ) d u .

If we take t = ∣ Q 0 ∣ β n and note that 1 ⁄ q = 1 ⁄ p − ( α + β ) ⁄ ( n − λ ) , we obtain

1 ∣ Q 0 ∣ 1 + β n ∫ Q 0 ∣ b i ( x ) − ( b i ) Q 0 ∣ d x ≤ 1 + ∣ Q 0 ∣ − 1 − β n ∣ Q 0 ∣ q ( λ n q − λ n p + 1 p − α n ) ∣ Q 0 ∣ β n − q β n ≤ c .

It follows from (3) that b i ∈ Λ ˙ β ( R n ) since Q 0 is arbitrary. The proof is completed.□

4 Proof of Theorems 3 and 4

To prove Theorems 3 and 4, we need some auxiliary definitions and lemmas. One of these is the pointwise estimate such that nonlinear commutator [ b → , ℳ α ] can be controlled by maximal commutator ℳ α , b → when b → satisfies some suitable assumptions.

Lemma 3

[30, Lemma 4.1] Let 0 ≤ α < m n and b → = ( b 1 , b 2 , … , b m ) be a collection of nonnegative locally integrable functions. Then, for all f → = ( f 1 , f 2 , … , f m ) with locally integrable function f j , j = 1 , … , m ,

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

Definition 8

Let 0 ≤ α < m n and f → = ( f 1 , f 2 , … , f m ) be a collection of locally integrable functions, the multi-sublinear maximal function with respect to Q is defined by

(9) ℳ α , Q ( f → ) ( x ) = sup Q ⊇ Q ′ ∋ x ∏ j = 1 m 1 ∣ Q ′ ∣ m − α ⁄ n ∫ Q ′ ∣ f j ( y j ) ∣ d y j , x ∈ R n ,

where the supremum is taken over all the cubes Q ′ such that Q ⊇ Q ′ ∋ x . When α = 0 , we simply write ℳ Q = ℳ 0 , Q .

Lemma 4

[30, Lemma 4.2] Let 0 ≤ α < m n , f be a locally integrable function, and Q 0 be a cube in R n . Then, for any 1 ≤ j ≤ m and all x ∈ Q 0 ,

(10) ℳ α , Q 0 ( χ Q 0 , … , χ Q 0 ︸ j − 1 , f χ Q 0 , χ Q 0 , … , χ Q 0 ︸ m − j ) ( x ) = ℳ α ( χ Q 0 , … , χ Q 0 ︸ j − 1 , f χ Q 0 , χ Q 0 , … , χ Q 0 ︸ m − j ) ( x )

and

(11) ℳ α , Q 0 ( χ Q 0 , … , χ Q 0 ︸ m ) ( x ) = ℳ α ( χ Q 0 , … , χ Q 0 ︸ m ) ( x ) = ∣ Q 0 ∣ α ⁄ n .

Lemma 5

Let 0 < β < 1 , 0 ≤ α < m n , 0 < α + β < m n , and b → = ( b 1 , b 2 , … , b m ) be a collection of locally integrable function. Suppose that 1 < p 1 , … , p m < ∞ , 1 p = 1 p 1 + … + 1 p m , q > 1 , λ p = ∑ i = 1 m λ i p i , 1 q i = 1 p i − α + β m ( n − λ i ) , 0 ≤ λ i < n − ( α + β ) p i m , and 1 q = 1 p − α + β n − λ . If [ b → , ℳ α ] i is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to L q , λ ( R n ) , then

sup Q ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q b i ( x ) − ∣ Q ∣ − α n ℳ α , Q ( b i ∘ χ → Q i ) ( x ) q d x 1 ⁄ q < ∞ ,

where ( b i ∘ χ → Q i ) = ( χ Q , … , χ Q ︸ i − 1 , b i χ Q , χ Q , … , χ Q ︸ m − i ) as above.

Proof

For any fixed cube Q and all x ∈ Q , the following equalities are valid:

ℳ α ( b i ∘ χ → Q i ) ( x ) = ℳ α , Q ( b i ∘ χ → Q i ) ( x ) , ℳ α ( χ → Q ) ( x ) = ∣ Q ∣ α n

[30, p. 14]. Since [ b → , ℳ α ] i is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to L q , λ ( R n ) and note that 1 ⁄ q = 1 ⁄ p − ( α + β ) ⁄ ( n − λ ) , then

∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q b i ( x ) − ∣ Q ∣ − α n ℳ α , Q ( b i ∘ χ → Q i ) ( x ) q d x 1 ⁄ q = ∣ Q ∣ − 1 q − α + β n ∫ Q b i ( x ) ∣ Q ∣ α n − ℳ α , Q ( b i ∘ χ → Q i ) ( x ) q d x 1 ⁄ q = ∣ Q ∣ − 1 q − α + β n ∫ Q ∣ b i ( x ) ℳ α ( χ → Q ) ( x ) − ℳ α ( b i ∘ χ → Q i ) ( x ) ∣ q d x 1 ⁄ q = ∣ Q ∣ − 1 q − α + β n ∫ Q ∣ [ b → , ℳ α ] i ( χ → Q ) ( x ) ∣ q d x 1 ⁄ q = ∣ Q ∣ − 1 q − α + β n − λ n q 1 ∣ Q ∣ λ n ∫ R n ∣ [ b → , ℳ α ] i ( χ → Q ) ( x ) ∣ q d x 1 ⁄ q ≤ ∣ Q ∣ − 1 q − α + β n − λ n q ‖ [ b → , ℳ α ] i ( χ → Q ) ( x ) ‖ L q , λ ( R n ) ≲ ∣ Q ∣ − 1 q − α + β n − λ n q ∏ i = 1 m ‖ ( χ → Q ) ‖ L p i , λ i ( R n ) ≲ ∣ Q ∣ − 1 q − α + β n − λ n q + n − λ n p ≤ c .

By taking the supremum of both sides because the constant c is independent of f , we are done.□

Lemma 6

[30, Lemma 4.4] Let 0 < β < 1 , 0 ≤ α < m n , 0 < α + β < m n , and 1 ≤ s < ∞ . If f is a locally integrable function and satisfies, for some 1 ≤ i ≤ m ,

sup Q ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ f ( x ) − ∣ Q ∣ − α n ℳ α , Q ( f ∘ χ → Q i ) ( x ) ∣ s d x 1 ⁄ s < ∞ ,

then f ∈ Λ ˙ β ( R n ) , where ( f ∘ χ → Q i ) = ( χ Q , … , χ Q ︸ i − 1 , f χ Q , χ Q , … , χ Q ︸ m − i ) as above.

Proof of the Theorem 3

The first part of the theorem is obvious from Lemma 3, (7), and Lemma 2 (i). The second part will be proven. By the hypothesis,

(12) ‖ [ b → , ℳ α ] i ( f → ) ‖ L q , λ ( R n ) ≤ c ∏ i = 1 m ‖ f i ‖ L p i , λ i ( R n )

holds for the constant c independent of f . By Lemma 1, it suffices to prove that

(13) sup Q ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ b i ( x ) − M Q ( b i ) ( x ) ∣ d x < ∞ .

Case 1. Let α = 0 . Then, from [30, Lemma 4.2], for any fixed cube Q and any x ∈ Q , we obtain

ℳ ( b i ∘ χ → Q i ) ( x ) = ℳ Q ( b i ∘ χ → Q i ) ( x ) = M Q ( b i ) ( x ) , ℳ ( χ → Q ) ( x ) = 1 ,

where χ → Q = ( χ Q , … , χ Q ︸ m ) and ( f ∘ χ → Q i ) = ( χ Q , … , χ Q ︸ i − 1 , f χ Q , χ Q , … , χ Q ︸ m − i ) as above.

Since 1 ⁄ q = 1 ⁄ p 1 + … + 1 ⁄ p m − ( α + β ) ⁄ ( n − λ ) in this case, it follows from (12) that

∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ b i ( x ) − M Q ( b i ) ( x ) ∣ d x = ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ b i ( x ) ℳ ( χ → Q ) ( x ) − ℳ ( b i ∘ χ → Q i ) ( x ) ∣ d x = ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ [ b → , ℳ ] i ( χ → Q ) ( x ) ∣ d x ≤ ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ [ b → , ℳ ] i ( χ → Q ) ( x ) ∣ q d x 1 ⁄ q ≲ ∣ Q ∣ − β n − 1 q + λ n q ∣ Q ∣ λ − n n p = c

which implies (13), since Q is arbitrary cube in R n and the constant c is independent of f .

Case 2. Let 0 < α < m n . For any fixed cube Q , we have

(14) ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ b i ( x ) − M Q ( b i ) ( x ) ∣ d x ≤ ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q b i ( x ) − ∣ Q ∣ − α n ℳ α , Q ( b i ∘ χ → Q i ) ( x ) d x + ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ Q ∣ − α n ℳ α , Q ( b i ∘ χ → Q i ) ( x ) − M Q ( b i ) ( x ) d x ≔ I 1 + I 2 .

For I 1 , since [ b → , ℳ α ] i is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to L q , λ ( R n ) , it follows from Hölder’s inequality and Lemma 5 that

(15) I 1 ≤ ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q b i ( x ) − ∣ Q ∣ − α n ℳ α , Q ( b i ∘ χ → Q i ) ( x ) q d x 1 ⁄ q ≤ c .

Next we consider I 2 . For all x ∈ Q , by Lemma 4, we obtain

∣ Q ∣ − α n ℳ α , Q ( b i ∘ χ → Q i ) ( x ) − M Q ( b i ) ( x ) ≤ ∣ Q ∣ − α n ℳ α , Q ( b i ∘ χ → Q i ) ( x ) − ∣ b i ( x ) ∣ ∣ Q ∣ α n + ∣ ∣ b i ( x ) ∣ − M Q ( b i ) ( x ) ∣ = ∣ Q ∣ − α n ∣ ℳ α ( b i ∘ χ → Q i ) ( x ) − ∣ b i ( x ) ∣ ℳ α ( χ → Q ) ( x ) ∣ + ∣ ∣ b i ( x ) ∣ ℳ ( χ → Q ) ( x ) − ℳ ( b i ∘ χ → Q i ) ( x ) ∣ = ∣ Q ∣ − α n ∣ [ ∣ b i ∣ , ℳ α ] i ( χ → Q ) ( x ) ∣ + ∣ [ ∣ b i ∣ , ℳ ] i ( χ → Q ) ( x ) ∣ .

Then,

I 2 = ∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ Q ∣ − α n ℳ α , Q ( b i ∘ χ → Q i ) ( x ) − M Q ( b i ) ( x ) d x ≲ ∣ Q ∣ − 1 q − α + β n ∫ Q ∣ [ ∣ b i ∣ , ℳ α ] i ( χ → Q ) ( x ) ∣ q d x 1 q + ∣ Q ∣ − 1 q − β n ∫ Q ∣ [ ∣ b i ∣ , ℳ ] i ( χ → Q ) ( x ) ∣ q d x 1 q ≤ c

because of the fact that [ b → , ℳ α ] i is bounded from L p 1 , λ 1 ( R n ) × … × L p m , λ m ( R n ) to L q , λ ( R n ) , then it follows from Lemmas 5 and 6, b i ∈ Λ ˙ β ( R n ) which implies ∣ b i ∣ ∈ Λ ˙ β ( R n ) . Thus, we can apply Lemma 3 for each j entry and (6) to [ ∣ b i ∣ , ℳ α ] i and [ ∣ b i ∣ , ℳ ] i , since ∣ b i ∣ ∈ Λ ˙ β ( R n ) and ∣ b i ∣ ≥ 0 . Then, by (11), we have

∣ [ ∣ b i ∣ , ℳ α ] i ( χ → Q ) ( x ) ∣ ≲ ℳ α + β ( χ → Q ) ( x ) = c ∣ Q ∣ α + β n

and

∣ [ ∣ b i ∣ , ℳ ] i ( χ → Q ) ( x ) ∣ ≲ ℳ β ( χ → Q ) ( x ) = c ∣ Q ∣ β n .

Therefore I 2 ≤ c . This along with (14) and (15) gives

∣ Q ∣ − β n 1 ∣ Q ∣ ∫ Q ∣ b i ( x ) − M Q ( b i ) ( x ) ∣ d x ≤ c ,

which implies b i ∈ Λ ˙ β ( R n ) and b i ≥ 0 by Lemma 1. The proof of Theorem 3 is completed.□

Proof of the Theorem 4

b → ∈ ( Λ ˙ β ) m and b i ≥ 0 for i = 1 , 2 , … , m , then by Lemma 3 and (7), we have

∣ [ b → , ℳ α ] ( f → ) ( x ) ∣ ≤ ℳ α , b → ( f → ) ( x ) ≲ ‖ b → ‖ ( Λ ˙ β ) m ℳ α + β ( f → ) ( x ) .

Obviously, Theorem 4 can be obtained from Lemma 2 (ii).□

Acknowledgements

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

  1. Funding information: Pu Zhang was supported by the Fundamental Research Funds for Education Department of Heilongjiang Province (No. 1453ZD031).

  2. Author contributions: All authors contributed equally to this work. All authors read and approved the final manuscript.

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

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

References

[1] R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), no. 3, 611–635, DOI: https://doi.org/10.2307/1970954. 10.2307/1970954Search in Google Scholar

[2] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263–270, DOI: https://doi.org/10.1007/BF02386000. 10.1007/BF02386000Search in Google Scholar

[3] M. Paluszyński, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J. 44 (1995), no. 1, 1–17, DOI: http://www.jstor.org/stable/24898462. 10.1512/iumj.1995.44.1976Search in Google Scholar

[4] J. Garciiia-Cuerva, E. Harboure, C. Segovia, and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991), no. 4, 1397–1420, DOI: https://www.jstor.org/stable/24896301. 10.1512/iumj.1991.40.40063Search in Google Scholar

[5] C. Segovia and J. L. Torrea. Vector-valued commutators and applications, Indiana Univ. Math. J. 38 (1989), no. 4, 959–971, DOI: http://www.jstor.org/stable/24895375. 10.1512/iumj.1989.38.38044Search in Google Scholar

[6] C. Segovia and J. L. Torrea, Weighted inequalities for commutators of fractional and singular integrals, Conference on Mathematical Analysis (El Escorial, 1989), Publ. Mat. 35 (1991), no. 1, 209–235.10.5565/PUBLMAT_35191_09Search in Google Scholar

[7] Y. Ding, S. Lu, and P. Zhang, Weak estimates for commutators of fractional integral operators, Sci. China Ser. A. 44 (2001), no. 7, 877–888, DOI: https://doi.org/10.1007/BF02880137. 10.1007/BF02880137Search in Google Scholar

[8] A. Almeida, Maximal commutators and commutators of potential operators in new vanishing Morrey spaces, Nonlinear Anal. 192, (2020), 111684, DOI: https://doi.org/10.1016/j.na.2019.111684. 10.1016/j.na.2019.111684Search in Google Scholar

[9] V. S. Guliyev, F. Deringoz, and S. G. Hasanov, Commutators of fractional maximal operator on Orlicz spaces, Math. Notes 104 (2018), no. 4, 498–507, DOI: https://doi.org/10.1134/S0001434618090171. 10.1134/S0001434618090171Search in Google Scholar

[10] V. S. Guliyev and P. S. Shukurov, On the boundedness of the fractional maximal operator, Riesz potential and their commutators in generalized Morrey spaces, in: A. Almeida, L. Castro, F.O. Speck (Eds.), Advances in Harmonic Analysis and Operator Theory, Operator Theory: Advances and Applications, vol. 229. Birkhäuser, Basel, 2013, pp. 175–199, DOI: https://doi.org/10.1007/978-3-0348-0516-2_10. 10.1007/978-3-0348-0516-2_10Search in Google Scholar

[11] M. Milman and T. Schonbek, Second order estimates in interpolation theory and applications, Proc. Amer. Math. Soc. 110 (1990), no. 4, 961–969, DOI: https://doi.org/10.1090/S0002-9939-1990-1075187-4. 10.1090/S0002-9939-1990-1075187-4Search in Google Scholar

[12] P. Zhang and J. Wu, Commutators of fractional maximal functions, Acta Math. Sinica (Chinese Ser.) 52 (2009), no. 6, 1235–1238. Search in Google Scholar

[13] P. Zhang and J. Wu, Commutators of fractional maximal functions on variable exponent Lebesgue spaces, Czechoslovak Math. J. 64 (2014), no. 1, 183–197, DOI: https://doi.org/10.1007/s10587-014-0093-x. 10.1007/s10587-014-0093-xSearch in Google Scholar

[14] P. Zhang, Z. Si, and J. Wu, Some notes on commutators of the fractional maximal function on variable Lebesgue spaces, J. Inequal. Appl. 2019 (2019), 9, DOI: https://doi.org/10.1186/s13660-019-1960-7. 10.1186/s13660-019-1960-7Search in Google Scholar

[15] V. S. Guliyev, F. Deringoz, and S. G. Hasanov, Fractional maximal function and its commutators on Orlicz spaces, Anal. Math. Phys. 9 (2019), no. 1, 165–179, DOI: https://doi.org/10.1007/s13324-017-0189-1. 10.1007/s13324-017-0189-1Search in Google Scholar

[16] P. Zhang, J. Wu, and J. Sun, Commutators of some maximal functions with Lipschitz function on Orlicz spaces, Mediterr. J. Math. 15 (2018), 216, DOI: https://doi.org/10.1007/s00009-018-1263-0. 10.1007/s00009-018-1263-0Search in Google Scholar

[17] R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315–331, DOI: https://doi.org/10.1090/S0002-9947-1975-0380244-8. 10.2307/1998628Search in Google Scholar

[18] L. Grafakos and R. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164, DOI: https://doi.org/10.1006/aima.2001.2028. 10.1006/aima.2001.2028Search in Google Scholar

[19] W. Damián, A. Lerner, and C. Pérez, Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators, J. Fourier Anal. Appl. 21 (2015), no. 1, 161–181, DOI: https://doi.org/10.1007/s00041-014-9364-z. 10.1007/s00041-014-9364-zSearch in Google Scholar

[20] A. Lerner, S. Ombrosi, C. Pérez, R. Torres, and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222–1264, DOI: https://doi.org/10.1016/j.aim.2008.10.014. 10.1016/j.aim.2008.10.014Search in Google Scholar

[21] G. Lu and P. Zhang, Multilinear Calderón-Zygmund operators with kernels of Dini’s type and applications, Nonlinear Anal. 107 (2014), 92–117, DOI: https://doi.org/10.1016/j.na.2014.05.005. 10.1016/j.na.2014.05.005Search in Google Scholar

[22] K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math. 60 (2009), no. 2, 213–238, DOI: https://doi.org/10.1007/BF03191210. 10.1007/BF03191210Search in Google Scholar

[23] C. Pérez and R. Torres, Sharp maximal function estimates for multilinear singular integrals, Contemp. Math. 320 (2003), 323–331, DOI: https://doi.org/10.1090/conm/320. 10.1090/conm/320/05615Search in Google Scholar

[24] C. Pérez, G. Pradolini, R. Torres, and R. Trujillo-González, End-point estimates for iterated commutators of multilinear singular integrals, Bull. Lond. Math. Soc. 46 (2014), no. 1, 26–42, DOI: https://doi.org/10.1112/blms/bdt065. 10.1112/blms/bdt065Search in Google Scholar

[25] C. Pérez and R. Trujillo-González, Sharp weighted estimates for multilinear commutators, J. Lond. Math. Soc. 65 (2002), no. 3, 672–692, DOI: https://doi.org/10.1112/S0024610702003174. 10.1112/S0024610702003174Search in Google Scholar

[26] X. Chen and Q. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl. 362 (2010), no. 2, 355–373, DOI: https://doi.org/10.1016/j.jmaa.2009.08.022. 10.1016/j.jmaa.2009.08.022Search in Google Scholar

[27] P. Zhang, Multiple weighted estimates for commutators of multilinear maximal function, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 6, 973–994, DOI: https://doi.org/10.1007/s10114-015-4293-6. 10.1007/s10114-015-4293-6Search in Google Scholar

[28] D. Wang and G. Wang, Necessary and sufficient conditions for boundedness of commutators of bilinear Hardy-Littlewood maximal function, Math. Inequal. Appl. 25 (2022), no. 3, 789–807, DOI: https://doi.org/10.7153/mia-2022-25-50. 10.7153/mia-2022-25-50Search in Google Scholar

[29] X. Yu, P. Zhang, and H. Li, Some estimates for the commutators of multilinear maximal function on Morrey-type space, Open Math. 19 (2021), no. 1, 515–530, DOI: https://doi.org/10.1515/math-2021-0031. 10.1515/math-2021-0031Search in Google Scholar

[30] P. Zhang, Commutators of multi-sublinear maximal functions with Lipschitz functions, Results Math. 74 (2019), 49, DOI: https://doi.org/10.1007/s00025-019-0971-5. 10.1007/s00025-019-0971-5Search in Google Scholar

[31] R. A. DeVore and R. C. Sharpley, Maximal functions measuring smoothness, Memoirs of the American Mathematical Society, vol. 293, American Mathematical Society (AMS), Providence, RI, 1984. 10.1090/memo/0293Search in Google Scholar

[32] S. Janson, M. Taibleson, and G. Weiss, Elementary characterizations of the Morrey-Campanato spaces, Harmonic analysis, Proceedings of a Conference, Cortona/Italy 1982, Lecture Notes on Mathematics, vol. 992, 1983, DOI: https://doi.org/10.1007/BFb0069154. 10.1007/BFb0069154Search in Google Scholar

[33] S. Campanato, Proprieta di Hölderianita di alcune classi di funzioni, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 17 (1963), no. 1–2, 175–188, http://www.numdam.org/item/ASNSP_1963_3_17_1-2_175_0/. Search in Google Scholar

[34] N. G. Meyers, Mean oscillation over cubes and Hölder continuity, Proc. Amer. Math. Soc. 15 (1964), no. 5, 717–721, DOI: https://doi.org/10.2307/2034586. 10.1090/S0002-9939-1964-0168712-3Search in Google Scholar

[35] J. Peetre, On the theory of ℒp,λ spaces, J. Funct. Anal. 4 (1969), no. 1, 71–87, DOI: https://doi.org/10.1016/0022-1236(69)90022-6. 10.1016/0022-1236(69)90022-6Search in Google Scholar

[36] E. Nakai, A generalization of Hardy spaces Hp by using atoms, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 8, 1243–1268, DOI: https://doi.org/10.1007/s10114-008-7626-x. 10.1007/s10114-008-7626-xSearch in Google Scholar

[37] P. Zhang, Characterization of Lipschitz spaces via commutators of the Hardy-Littlewood maximal function, C. R. Math. Acad. Sci. Paris 355 (2017), no. 3, 336–344, DOI: https://doi.org/10.1016/j.crma.2017.01.022. 10.1016/j.crma.2017.01.022Search in Google Scholar

[38] C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166, DOI: https://doi.org/10.1090/S0002-9947-1938-1501936-8. 10.1090/S0002-9947-1938-1501936-8Search in Google Scholar

[39] Z. Wang and Z. Si, Necessary and sufficient conditions for the boundedness of rough multilinear fractional operators on Morrey-type spaces, J. Inequal. Appl. 2015 (2015), 104, DOI: https://doi.org/10.1186/s13660-015-0627-2. 10.1186/s13660-015-0627-2Search in Google Scholar
Received: 2024-05-25
Revised: 2024-09-10
Accepted: 2024-09-11
Published Online: 2024-10-15

© 2024 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0070
Keywords for this article
commutators; multilinear fractional maximal operators; Lipschitz spaces; Morrey spaces
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 13.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2024-0070/html
Scroll to top button