Home Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces
Article Open Access

Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces

  • Pu Zhang and Jianglong Wu EMAIL logo
Published/Copyright: July 5, 2023
Download Article (PDF)
Analysis and Geometry in Metric Spaces
From the journal Analysis and Geometry in Metric Spaces Volume 11 Issue 1

Abstract

The main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.

Keywords: multilinear commutator; multilinear fractional integral operator; Lipschitz function space; variable exponent
MSC 2010: 47B47; 42B20; 42B35

1 Introduction and main results

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

[ b , T ] f = b T ( f ) − T ( b f ) .

The first result for the commutator [ b , T ] was established by Coifman et al. in [7], and the authors proved that the bounded mean oscillations (BMO) is characterized by the boundedness of the singular integral operators’ commutator [ b , T ] . In 1978, Janson [24] generalized the results of the work by Coifman et al. [7] to functions belonging to a Lipschitz functional space and provided a characterization in terms of the boundedness of the commutators of singular integral operators with Lipschitz functions. In 1982, Chanillo [5] proved that BMO can be characterized by means of the boundedness between Lebesgue spaces of the commutators of fractional integral operators with BMO functions. In 1995, Paluszyński [34] gave some results in the spirit of [5] for the functions belonging to Lipschitz function spaces. In 2017, Pradolini and Ramos [39] obtained characterizations of a variable version of Lipschitz spaces via the commutators of Calderón-Zygmund and fractional-type operators in variable Lebesgue spaces.

The multilinear Calderón-Zygmund theory was first studied by Coifman and Meyer in [6,8]. This theory was then further investigated by many authors in the last few decades, see, e.g., [20,21,28], for the theory of multilinear Calderón-Zygmund operators with kernels satisfying the standard estimates. Multilinear fractional integral operators were first studied by Grafakos [19], followed by Kenig and Stein [26]. The importance of fractional integral operators is owing to the fact that they are smooth operators and have been extensively used in various areas, such as potential analysis, harmonic analysis, and partial differential equations.

Let R n be an n -dimensional Euclidean space and ( R n ) m = R n × ⋯ × R n be an m -fold product space ( m ∈ N ). We denote by S ( R n ) the space of all Schwartz functions on R n and by S ′ ( R n ) its dual space, the set of all tempered distributions on R n . Let C c ∞ ( R n ) denote the set of smooth functions with compact support in R n .

The following notations can be found from [22,42,43].

Let 0 ≤ α < m n , 0 < η ≤ 1 . Consider a kernel function K α ( x , y → ) defined in ( R n ) m + 1 , away from the diagonal x = y 1 = ⋯ = y m , satisfying the following fractional conditions:

  1. Size estimate:

    (1.1) ∣ K α ( x , y → ) ∣ ≤ A ( ∣ x − y 1 ∣ + ⋯ + ∣ x − y m ∣ ) m n − α

    for some A > 0 and all ( x , y 1 , … , y m ) ∈ ( R n ) m + 1 with x ≠ y j for some j ∈ { 1 , 2 , … , m } .

  2. Smoothness estimate: assume that for each j ∈ { 1 , 2 , … , m } , there are regularity conditions

    (1.2) ∣ K α ( x , y → ) − K α ( x ′ , y → ) ∣ ≤ A ∣ x − x ′ ∣ η ( ∣ x − y 1 ∣ + ⋯ + ∣ x − y m ∣ ) m n + η − α

    when ∣ x − x ′ ∣ ≤ 1 2 max 1 ≤ j ≤ m ∣ x − y j ∣ . For each fixed j with 1 ≤ j ≤ m ,

    (1.3) ∣ K α ( x , y 1 , … , y j , … , y m ) − K α ( x , y 1 , … , y j ′ , … , y m ) ∣ ≤ C ∣ y j − y j ′ ∣ η ∑ j = 1 m ∣ x − y j ∣ m n + η − α

    when ∣ y j − y j ′ ∣ ≤ 1 2 max 1 ≤ j ≤ m ∣ x − y j ∣ .

We say K α is a standard m -linear fractional Calderón-Zygmund kernel if it satisfies equations (1.1)–(1.3). The kernel K α is homogeneous of degree − ( m n − α ) , i.e.,

K α ( λ ( x , y → ) ) = ∣ λ ∣ − ( m n − α ) K α ( x , y → ) ,

K α ∈ C ∞ ( S n − 1 ) , and K α ≢ 0 .

We say T α is an m -linear fractional singular integral operator with an m -linear fractional Calderón-Zygmund kernel, K α ( x , y 1 , … , y m ) , if

(1.4) T α ( f → ) ( x ) = ∫ ( R n ) m K α ( x , y 1 , … , y m ) ∏ i = 1 m f i ( y i ) d y → ,

when x ∉ ⋂ j = 1 m supp f j and each f j ∈ C c ∞ ( R n ) , j = 1 , … , m .

If T α is a bounded linear operator from L p 1 ( R n ) × L p 2 ( R n ) × ⋯ × L p m ( R n ) to L q ( R n ) with 1 < p 1 , … , p m < ∞ , 1 p = 1 p 1 + 1 p 2 + ⋯ + 1 p m , and 1 < p ≤ q < ∞ , then T α is called as a standard m -linear fractional Calderón-Zygmund operator.

Let b → = ( b 1 , b 2 , … , b m ) be a collection of locally integrable functions, then the m -linear commutator of T α with b → is defined as follows:

T α , Σ b → ( f → ) ( x ) = T α , Σ b → ( f 1 , … , f m ) ( x ) = ∑ j = 1 m T α , b j ( f → ) ( x ) ,

where each term is the commutator of b j and T α in the jth entry of T α , i.e.,

T α , b j ( f → ) ( x ) = [ b j , T α ] ( f → ) ( x ) = b j ( x ) T α ( f 1 , … , f j , … , f m ) ( x ) − T α ( f 1 , … , b j f j , … , f m ) ( x )

for every j = 1 , 2 , … , m . This definition coincides with the linear commutator [ b , T α ] when m = 1 . To clarify the notation, the commutators can be formally written as follows:

T α , Σ b → ( f → ) ( x ) = ∑ j = 1 m ∫ ( R n ) m ( b j ( x ) − b j ( y j ) ) K α ( x , y → ) ∏ i = 1 m f i ( y i ) d y → .

Denote by L ( δ ) and L ( δ ( ⋅ ) ) the Lipschitz spaces and the variable Lipschitz spaces (see Definition 2.10), respectively. Let b → = ( b 1 , b 2 , … , b m ) be a collection of locally integrable functions. Motivated by the works mentioned above, the main aim of this article is to establish some new characterizations of the (variable) Lipschitz spaces via the boundedness of the commutator of multilinear fractional Calderón-Zygmund integral operators with (variable) Lipschitz functions in the context of the variable exponent Lebesgue spaces. The necessary and sufficient conditions for b j ( j = 1 , 2 , … , m ) belonging to L ( δ ) or L ( δ ( ⋅ ) ) are given by the aid of the boundedness of a multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The key tools in obtaining the results are the Fourier series applied by Jansen [24], the multilinear fractional integral operator, and certain generalizations of the classical sharp maximal operator.

Our main results can be stated as follows. Some notations can refer to Section 2, such as p − , p + , C log ( R n ) , and P ( R n ) .

Theorem 1.1

Let T α be an m-linear fractional Calderón-Zygmund operator given in (1.4). Suppose that 0 < δ < 1 , 0 < α < α + δ < m n , p 1 ( ⋅ ) , p 2 ( ⋅ ) , … , p m ( ⋅ ) ∈ P log ( R n ) satisfy 1 p ( x ) = 1 p 1 ( x ) + 1 p 2 ( x ) + ⋯ + 1 p m ( x ) and m n α + δ + n < ( p j ) − < ( p j ) + < m n α + δ ( j = 1 , 2 , … , m ) . Define the variable exponent q ( ⋅ ) by

1 p ( x ) − α n − 1 q ( x ) = δ n .

  1. Let b → = ( b 1 , b 2 , … , b m ) be a collection of locally integrable functions. Assume that the associated kernel K α of T α is a homogeneous of degree − ( m n − α ) and the Fourier series of 1 K α is absolutely convergent on some ball B ⊂ R m n . If T α , b j ( j = 1 , 2 , … , m ) is bounded from L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) × ⋯ × L p m ( ⋅ ) ( R n ) to L q ( ⋅ ) ( R n ) , then b → ∈ L ( δ ) × L ( δ ) × ⋯ × L ( δ ) .

  2. If b → ∈ L ( δ ) × L ( δ ) × ⋯ × L ( δ ) , then T α , b j : L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) × ⋯ × L p m ( ⋅ ) ( R n ) → L q ( ⋅ ) ( R n ) ( j = 1 , 2 , … , m ) .

Remark 1.2

  1. The above result gives a characterization of the Lipschitz spaces L ( δ ) in terms of the boundedness of T α , Σ b → between variable Lebesgue spaces.

  2. In order to prove the first part of Theorem 1.1, the authors employ some techniques and ideas of Fourier series applied by Janson [24] and modify it to adapt to the multilinear setting.

  3. When α = 0 , the above result coincides with Theorem 1.1 in [46].

Theorem 1.3

Let T α be an m-linear fractional Calderón-Zygmund operator given in (1.4), and let η ∈ ( η 0 , 1 ] be expressed as in (1.2), where η 0 is closely related to k 0 in (2.5). Suppose that 0 < α < m n , 0 < δ ( ⋅ ) < 1 , 0 < γ < ε < min { 1 , n / ( m n − α ) } , and r ( ⋅ ) , p 1 ( ⋅ ) , p 2 ( ⋅ ) , … , p m ( ⋅ ) ∈ P log ( R n ) satisfy 1 p ( x ) = 1 p 1 ( x ) + 1 p 2 ( x ) + ⋯ + 1 p m ( x ) , r ( x ) ≥ r ∞ , for almost every x ∈ R n and 1 < β ≤ r − such that 0 < 1 p ( x ) − α n − 1 q ( x ) = δ ( x ) n = 1 β − 1 r ( x ) < 1 .

  1. Let b → = ( b 1 , b 2 , … , b m ) be a collection of locally integrable functions. Assume that the associated kernel K α of T α is a homogeneous of degree − ( m n − α ) and the Fourier series of 1 K α is absolutely convergent on some ball B ⊂ R m n . If T α , b j ( j = 1 , 2 , … , m ) is bounded from L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) × ⋯ × L p m ( ⋅ ) ( R n ) to L q ( ⋅ ) ( R n ) , then b → = ( b 1 , b 2 , … , b m ) ∈ L ( δ ( ⋅ ) ) × L ( δ ( ⋅ ) ) × ⋯ × L ( δ ( ⋅ ) ) .

  2. If b → ∈ L ( δ ( ⋅ ) ) × L ( δ ( ⋅ ) ) × ⋯ × L ( δ ( ⋅ ) ) and ‖ T α , b j f → ‖ q ( ⋅ ) < ∞ for f → = ( f 1 , f 2 , … , f m ) ∈ L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) × ⋯ × L p m ( ⋅ ) ( R n ) , then T α , b j : L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) × ⋯ × L p m ( ⋅ ) ( R n ) → L q ( ⋅ ) ( R n ) ( j = 1 , 2 , … , m ) .

Remark 1.4

  1. The above result characterizes the variable Lipschitz spaces L ( δ ( ⋅ ) ) in terms of the boundedness of T α , Σ b → between variable Lebesgue spaces. The authors do so by applying the Fourier series technique and certain generalizations of the classical sharp maximal operator with variable exponent.

  2. When α = 0 , the above result coincides with Theorem 1.2 in [46].

The following theorem gives a new necessary condition on the symbol b → = ( b 1 , b 2 , … , b m ) belonging to the products of variable Lipschitz spaces.

Theorem 1.5

Let 0 < α < m n and r ( ⋅ ) , p 1 ( ⋅ ) , p 2 ( ⋅ ) , … , p m ( ⋅ ) ∈ P log ( R n ) satisfy 1 p ( x ) = 1 p 1 ( x ) + 1 p 2 ( x ) + ⋯ + 1 p m ( x ) , r ( x ) ≥ r ∞ , and ( p j ) ∞ ≤ p j ( x ) ≤ ( p j ) + < m n / α ( j = 1 , 2 , … , m ) . Let 1 < β < r − such that 0 < δ ( x ) n = 1 β − 1 r ( x ) < 1 and 0 < δ ˜ ( x ) = δ ( x ) + α − n p ( x ) < 1 . Suppose that there exits η such that max { η 0 , δ ˜ + } < η ≤ 1 . If b → = ( b 1 , b 2 , … , b m ) ∈ L ( δ ( ⋅ ) ) × L ( δ ( ⋅ ) ) × ⋯ × L ( δ ( ⋅ ) ) , then T α , b j : L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) × ⋯ × L p m ( ⋅ ) ( R n ) → L ( δ ˜ ( ⋅ ) ) ( j = 1 , 2 , … , m ).

Throughout this article, the letter C always stands for a constant independent of the main parameters involved and whose value may differ from line to line. A cube Q ⊂ R n always means a cube whose sides are parallel to the coordinate axes and denote its side length by l ( Q ) . For some t > 0 , the notation t Q stands for the cube with the same center as Q and with side length l ( t Q ) = t l ( Q ) . Denote by ∣ S ∣ the Lebesgue measure and by χ S the characteristic function for a measurable set S ⊂ R n . B ( x , r ) means the ball is centered at x and of radius r , and B 0 = B ( 0 , 1 ) . For any index 1 < q ( x ) < ∞ , we denote by q ′ ( x ) its conjugate index, namely, q ′ ( x ) = q ( x ) q ( x ) − 1 . We will occasionally use the notational f → = ( f 1 , … , f m ) , T ( f → ) = T ( f 1 , … , f m ) , d y → = d y 1 ⋯ d y m , and ( x , y → ) = ( x , y 1 , … , y m ) for convenience. For a set E and a positive integer m , we will use the notation ( E ) m = E × ⋯ × E ︸ m sometimes.

2 Preliminaries

Over the last three decades, the study of variable exponent function spaces has drawn many authors’ attention (see [10,12,14,16]). In fact, many classical operators have been studied in variable exponent function spaces (see [10,12,16]).

To prove the main theorems, we recall some known results and definitions.

Definition 2.1

Let q ( ⋅ ) : R n → [ 1 , ∞ ) be a measurable function.

  1. The variable exponent Lebesgue space L q ( ⋅ ) ( R n ) is defined as follows:

    L q ( ⋅ ) ( R n ) = { f is measurable function : F q ( f / η ) < ∞ for some constant η > 0 } ,

    where F q ( f ) ≔ ∫ R n ∣ f ( x ) ∣ q ( x ) d x is a convex functional modular. The Lebesgue space L q ( ⋅ ) ( R n ) is a Banach function space with respect to the Luxemburg-type norm

    ‖ f ‖ L q ( ⋅ ) ( R n ) = inf η > 0 : F q ( f / η ) = ∫ R n ∣ f ( x ) ∣ η q ( x ) d x ≤ 1 .

  2. The space L loc q ( ⋅ ) ( R n ) is defined as follows:

    L loc q ( ⋅ ) ( R n ) = { f is measurable : f ∈ L q ( ⋅ ) ( E ) for all compact subsets E ⊂ R n } .

Next, we define some classes of variable exponent functions. Given a function f ∈ L loc 1 ( R n ) , the Hardy-Littlewood maximal operator M is defined as follows:

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

Definition 2.2

Given a measurable function q ( ⋅ ) defined on R n . For E ⊂ R n , we write

q − ( E ) ≔ ess inf x ∈ E q ( x ) , q + ( E ) ≔ ess sup x ∈ E q ( x ) ,

and write q − ( R n ) = q − and q + ( R n ) = q + simply.

  1. q − ′ = ess inf x ∈ R n q ′ ( x ) = q + q + − 1 , q + ′ = ess sup x ∈ R n q ′ ( x ) = q − q − − 1 .

  2. Denote by P 0 ( R n ) the set of all measurable functions q ( ⋅ ) : R n → ( 0 , ∞ ) such that

    0 < q − ≤ q ( x ) ≤ q + < ∞ , x ∈ R n .

  3. Denote by P 1 ( R n ) the set of all measurable functions q ( ⋅ ) : R n → [ 1 , ∞ ) such that

    1 ≤ q − ≤ q ( x ) ≤ q + < ∞ , x ∈ R n .

  4. Denote by P ( R n ) the set of all measurable functions q ( ⋅ ) : R n → ( 1 , ∞ ) such that

    1 < q − ≤ q ( x ) ≤ q + < ∞ , x ∈ R n .

  5. The set B ( R n ) consists of all measurable functions q ( ⋅ ) ∈ P ( R n ) , satisfying that the Hardy-Littlewood maximal operator M is bounded on L q ( ⋅ ) ( R n ) .

Now, we introduce the log -Hölder continuity.

Definition 2.3

( log -Hölder continuity) Let q ( ⋅ ) be a real-valued function on R n .

  1. Denote by C loc log ( R n ) the set of all local log -Hölder continuous functions q ( ⋅ ) that satisfy

    ∣ q ( x ) − q ( y ) ∣ ≤ − C ln ( ∣ x − y ∣ ) , ∣ x − y ∣ ≤ 1 / 2 , x , y ∈ R n ,

    where C denotes a universal positive constant that may differ from line to line, and C does not depend on x , y .

  2. The set C ∞ log ( R n ) consists of all log -Hölder continuous functions q ( ⋅ ) at infinity satisfies

    ∣ q ( x ) − q ∞ ∣ ≤ C ∞ ln ( e + ∣ x ∣ ) , x ∈ R n ,

    where q ∞ = lim ∣ x ∣ → ∞ q ( x ) .

  3. Denote by C log ( R n ) ≔ C loc log ( R n ) ∩ C ∞ log ( R n ) the set of all global log -Hölder continuous functions q ( ⋅ ) .

Remark 2.4

  1. The C ∞ log ( R n ) condition is equivalent to the uniform continuity condition

    ∣ q ( x ) − q ( y ) ∣ ≤ C ln ( e + ∣ x ∣ ) , ∣ y ∣ ≥ ∣ x ∣ , x , y ∈ R n .

    The C ∞ log ( R n ) condition was originally defined in this form in [9].

  2. In what follows, we denote C log ( R n ) ∩ P ( R n ) by P log ( R n ) .

2.1 Auxiliary propositions and lemmas

Next, we give some auxiliary propositions and lemmas we need.

Lemma 2.5

Let p ( ⋅ ) ∈ P ( R n ) .

  1. If p ( ⋅ ) ∈ C log ( R n ) then we have p ( ⋅ ) ∈ B ( R n ) .

  2. The following conditions are equivalent:

    1. p ( ⋅ ) ∈ B ( R n ) .

    2. p ′ ( ⋅ ) ∈ B ( R n ) .

    3. p ( ⋅ ) / p 0 ∈ B ( R n ) for some 1 < p 0 < p − .

    4. ( p ( ⋅ ) / p 0 ) ′ ∈ B ( R n ) for some 1 < p 0 < p − .

The first part in Lemma 2.5 is independently due to Cruz-Uribe et al. [9] and Nekvinda [33], respectively. The second part of Lemma 2.5 belongs to Diening [15] (see Theorem 8.1 or Theorem 1.2 in [10]).

As the classical Lebesgue norm, the (quasi-)norm of variable exponent Lebesgue space is also homogeneous in the exponent. Precisely, we have the following result (see Proposition 2.18 in [12], Lemma 2.3 in [13], or Lemma 3.2.6 in [16]).

Lemma 2.6

Let p ( ⋅ ) ∈ P 0 ( R n ) and s > 0 be such that s p − ≥ 1 . Then,

‖ ∣ f ∣ s ‖ p ( ⋅ ) = ‖ f ‖ s p ( ⋅ ) s .

The next lemma is known as the generalized Hölder’s inequality on Lebesgue spaces with variable exponent, and the proof can also be found in [27] or [12] (see p. 27–30).

Lemma 2.7

(Generalized Hölder’s inequality)

  1. (See p. 81–82, Lemma 3.2.20 in [16]). Let p ( ⋅ ) , q ( ⋅ ) , r ( ⋅ ) ∈ P 0 ( R n ) satisfy the condition

    1 r ( x ) = 1 p ( x ) + 1 q ( x ) f o r a . e . x ∈ R n .

    1. Then, for all f ∈ L p ( ⋅ ) ( R n ) and g ∈ L q ( ⋅ ) ( R n ) , one has

      (2.1) ‖ f g ‖ r ( ⋅ ) ≤ C ‖ f ‖ p ( ⋅ ) ‖ g ‖ q ( ⋅ ) .

    2. When r = 1 , then p ′ ( ⋅ ) = q ( ⋅ ) , hence, for all f ∈ L p ( ⋅ ) ( R n ) and g ∈ L p ′ ( ⋅ ) ( R n ) , one has

      (2.2) ∫ R n ∣ f g ∣ ≤ C ‖ f ‖ p ( ⋅ ) ‖ g ‖ p ′ ( ⋅ ) .

  2. The generalized Hölder’s inequality in Orlicz space (see [28,35,36] for details and the more general cases).

    1. Let r 1 , … , r m ≥ 1 with 1 r = 1 r 1 + ⋯ + 1 r m and Q be a cube in R n . Then,

      1 ∣ Q ∣ ∫ Q ∣ f 1 ( x ) ⋯ f m ( x ) g ( x ) ∣ d x ≤ C ‖ f 1 ‖ exp L r 1 , Q ⋯ ‖ f m ‖ exp L r m , Q ‖ g ‖ L ( log L ) 1 / r , Q .

    2. Let t ≥ 1 , then

      (2.3) 1 ∣ Q ∣ ∫ Q ∣ f ( x ) g ( x ) ∣ d x ≤ C ‖ f ‖ exp L t , Q ‖ g ‖ L ( log L ) 1 / t , Q .

  3. (See Lemma 9.2 in [30]) Let q ( ⋅ ) , q 1 ( ⋅ ) , … , q m ( ⋅ ) ∈ P ( R n ) satisfy the condition

    1 q ( x ) = 1 q 1 ( x ) + ⋯ + 1 q m ( x ) f o r a . e . x ∈ R n .

    Then, for any f j ∈ L q j ( ⋅ ) ( R n ) , j = 1 , … , m , one has

    ‖ f 1 ⋯ f m ‖ q ( ⋅ ) ≤ C ‖ f 1 ‖ q 1 ( ⋅ ) ⋯ ‖ f m ‖ q m ( ⋅ ) .

The following results are also needed.

Lemma 2.8

(Norms of characteristic functions)

  1. Let q ( ⋅ ) ∈ P log ( R n ) and q ( x ) ≤ q ∞ for a.e. x ∈ R n . Then, there exists a positive constant C such that the inequality

    ‖ χ Q ‖ q ( ⋅ ) ≤ C ∣ Q ∣ 1 / q ( x )

    holds for every cube Q ⊂ R n and a.e. x ∈ Q (see Lemma 4.4 in [39] or p. 126, Corollary 4.5.9 in [16]).

  2. Let q ( ⋅ ) ∈ P ( R n ) . 1 q Q = 1 ∣ Q ∣ ∫ Q 1 q ( y ) d y is the harmonic mean of q ( ⋅ ) . Then, the following conditions are equivalent (see Theorem 4.5.7 in [16] Proposition 4.66 in [12]):

    1. ‖ χ Q ‖ q ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ≈ ∣ Q ∣ uniformly for all cubes Q ⊂ R n .

    2. ‖ χ Q ‖ q ( ⋅ ) ≈ ∣ Q ∣ 1 q Q and ‖ χ Q ‖ q ′ ( ⋅ ) ≈ ∣ Q ∣ 1 q Q ′ uniformly for all cubes Q ⊂ R n .

  3. Let q + < ∞ . Then, the following conditions are equivalent (see p. 101, Lemma 4.1.6 in [16], Lemma 4.2 in [39], or Corollary 3.24 in [12]).

    1. The function q ( ⋅ ) ∈ C 0 log ( R n ) .

    2. For every cube Q ⊂ R n , there exists a positive constant C such that

      ∣ Q ∣ q − ( Q ) − q + ( Q ) ≤ C .

    3. For all cube Q ⊂ R n and all x ∈ Q , there exists a positive constant C such that

      ∣ Q ∣ q − ( Q ) − q ( x ) ≤ C .

    4. For all cube Q ⊂ R n and all x ∈ Q , there exists a positive constant C such that

      ∣ Q ∣ q ( x ) − q + ( Q ) ≤ C .

  4. Given a cube Q = Q ( x 0 , r ) , with center in x 0 and diameter r .

    1. If r < 1 , there exist two positive constants a 1 and a 2 such that

      a 1 ∣ Q ∣ 1 / q − ( Q ) ≤ ‖ χ Q ‖ q ( ⋅ ) ≤ a 2 ∣ Q ∣ 1 / q + ( Q ) ;

    2. If r > 1 , there exist two positive constants c 1 and c 2 such that

      c 1 ∣ Q ∣ 1 / q + ( Q ) ≤ ‖ χ Q ‖ q ( ⋅ ) ≤ c 2 ∣ Q ∣ 1 / q − ( Q ) .

    Therefore, ‖ χ Q ‖ q ( ⋅ ) ≤ max { ∣ Q ∣ 1 / q + ( Q ) , ∣ Q ∣ 1 / q − ( Q ) } (see p. 25–26, Corollary 2.23 in [12], or [17,39]).

  5. If p ( ⋅ ) ∈ P log ( R n ) and β = n / α with p + < n β ( n − β ) + , then there exists a number a > 1 such that

    (2.4) ‖ χ Q ( x , a r ) ‖ p ′ ( ⋅ ) ≤ a n − n / β + 1 2 ‖ χ Q ( x , r ) ‖ p ′ ( ⋅ )

    for every r > 0 and x ∈ R n , where Q ( x , r ) denotes a cube centered at x and with diameter r (It is easy to check that the result above can be obtained for every β > 1 , for more information, see Lemma 2.17 in [40] or [39]).

Let k 0 ∈ N satisfy that

(2.5) a k 0 − 1 < 2 < a k 0 ,

where a is given in equation (2.4), and let η 0 = 1 / k 0 (which will be used in the following sections).

Note that, if p ( ⋅ ) ∈ P log ( R n ) , the estimates (2.4) and (2.5) imply the doubling condition for the functional a ( Q ) ≔ ‖ χ Q ‖ p ( ⋅ ) , i.e.,

‖ χ 2 Q ‖ p ( ⋅ ) ≤ C ‖ χ Q ‖ p ( ⋅ )

for every cube Q ⊂ R n .

Set 0 < γ < n and p ( ⋅ ) , q ( ⋅ ) ∈ P ( R n ) such that 1 / q ( x ) = 1 / p ( x ) − γ / n with p + < n / γ . Then, a weight ω ∈ A p ( ⋅ ) , q ( ⋅ ) γ ( R n ) if there exists a positive constant C such that for every cube Q , the inequality

(2.6) ‖ ω χ Q ‖ q ( ⋅ ) ‖ ω − 1 χ Q ‖ p ′ ( ⋅ ) ≤ C ∣ Q ∣ 1 − γ / n

holds.

When γ = 0 , the inequality above is the A p ( ⋅ ) ( R n ) class given by Cruz-Uribe et al. in [11], which characterizes the boundedness of the Hardy-Littlewood maximal operator on L ω p ( ⋅ ) ( R n ) , i.e., the measurable functions f such that f ω ∈ L p ( ⋅ ) ( R n ) .

The first part of the following results was proved in [3] (see Lemma 4.1) and gives a relation between the A p ( ⋅ ) ( R n ) and A p ( ⋅ ) , q ( ⋅ ) γ ( R n ) classes (see also Lemma 4.14 in [39]). On the basis of the first part, note that if q ( ⋅ ) ∈ P log ( R n ) , then s ( ⋅ ) ∈ P log ( R n ) . Since M is continuous on L s ( ⋅ ) ( R n ) from Lemma 2.5, Pradolini and Ramos obtained the second part result (see Lemma 4.15 in [39]).

Lemma 2.9

Let 0 ≤ γ < n and p ( ⋅ ) , q ( ⋅ ) ∈ P ( R n ) such that 1 / q ( x ) = 1 / p ( x ) − γ / n with p + < n / γ .

  1. Suppose that s ( ⋅ ) ∈ P ( R n ) and s ( x ) = ( 1 − γ / n ) q ( x ) . Set ω be a weight. Then, ω ∈ A p ( ⋅ ) , q ( ⋅ ) γ ( R n ) if and only if ω n n − γ ∈ A s ( ⋅ ) ( R n ) .

  2. Assume that p ( ⋅ ) ∈ C log ( R n ) . Then, 1 ∈ A p ( ⋅ ) , q ( ⋅ ) γ ( R n ) .

Some notations of Lipschitz-type function spaces are stated as follows.

Definition 2.10

(Lipschitz-type spaces)

  1. Let 0 < δ < 1 . The space Λ δ of the Lipschitz continuous functions with order δ is defined as follows:

    Λ δ ( R n ) = { f : ∣ f ( x ) − f ( y ) ∣ ≤ C ∣ x − y ∣ δ for a.e. x , y ∈ R n } ,

    where f is the locally integrable function on R n , and the smallest constant C > 0 will be denoted as the Lipschitz norm by ‖ f ‖ Λ δ .

  2. Let 0 ≤ δ < 1 . The space L ( δ ) is defined to be the set of all locally integrable functions f , i.e., there exists a positive constant C > 0 such that

    sup Q 1 ∣ Q ∣ 1 + δ / n ∫ Q ∣ f ( y ) − f Q ∣ d y < C ,

    where the supremum is taken over every cube Q ⊂ R n and f Q = 1 ∣ Q ∣ ∫ Q f ( z ) d z . The least constant C will be denoted by ‖ f ‖ L ( δ ) .

  3. [40] Let 0 < α < n and an exponent function p ( ⋅ ) ∈ P 1 ( R n ) . We say that a locally integrable function f belongs to L α , p ( ⋅ ) = L α , p ( ⋅ ) ( R n ) if there exists a constant C such that

    1 ∣ B ∣ α / n ‖ χ B ‖ p ′ ( ⋅ ) ∫ B ∣ f ( y ) − f B ∣ d y < C

    for every ball B ⊂ R n , with f B = 1 ∣ B ∣ ∫ B f ( z ) d z . The least constant C will be denoted by ‖ f ‖ L α , p ( ⋅ ) .

  4. Let r ( ⋅ ) ∈ P log ( R n ) such that 1 < β ≤ r − ≤ r ( x ) ≤ r + < n β ( n − β ) + , and set δ ( x ) n = 1 β − 1 r ( x ) . The space L ( δ ( ⋅ ) ) is defined by the set of the measurable functions f such that (see [40] for more details)

    ‖ f ‖ L ( δ ( ⋅ ) ) = sup B 1 ∣ B ∣ 1 / β ‖ χ B ‖ r ′ ( ⋅ ) ∫ B ∣ f ( y ) − f B ∣ d y < ∞ .

  5. (Weighted Lipschitz integral spaces L w ( δ ) , see [37] or [38]) Let w be a weight and 0 ≤ δ < 1 , we say that a locally integrable function f belongs to L w ( δ ) if there exists a positive constant C such that the inequality

    ‖ w χ B ‖ ∞ ∣ B ∣ 1 + δ / n ∫ B ∣ f ( y ) − f B ∣ d y < C

    holds for every ball B ⊂ R n . The least constant C will be denoted by ‖ f ‖ L w ( δ ) .

Remark 2.11

  1. In (1) of Definition 2.10, it is well known that the space Λ δ coincides with the space L ( δ ) (see [23, 39]).

  2. In (2) of Definition 2.10, it is not difficult to see that, for δ = 0 , the space L ( δ ) coincides with the space of bounded mean oscillation functions BMO (see [25]).

  3. In (3) of Definition 2.10, it is easy to see that the average f B can be replaced by a constant in the following sense (see [40]):

    1 2 ‖ f ‖ L α , p ( ⋅ ) ≤ sup B ∈ R n inf c ∈ R 1 ∣ B ∣ α n ‖ χ B ‖ p ′ ( ⋅ ) ∫ B ∣ f ( y ) − c ∣ d y ≤ ‖ f ‖ L α , p ( ⋅ ) .

  4. In (4) of Definition 2.10, denote max { z , 0 } by z + (see [4]). In addition, when r ( x ) = r is constant, L ( δ ( ⋅ ) ) coincides with the space L ( n / β − n / r ) .

  5. In (5) of Definition 2.10, it is not difficult to see that, for δ = 0 , the space L w ( δ ) coincides with one of the versions of weighted bounded mean oscillation spaces (see [32]). Moreover, for the case w ≡ 1 , the space L w ( δ ) is the known Lipschitz integral space for 0 < δ < 1 .

Lemma 2.12

[39] Let p ( ⋅ ) ∈ P log ( R n ) and 1 < β ≤ p − . Then, the functional

a ( Q ) = ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ )

satisfies the T ∞ condition, i.e., there exists a positive constant C such that a ( Q ′ ) ≤ C a ( Q ) for each cube Q and each cube Q ′ ⊂ Q .

Lemma 2.13

[29] Let 1 ≤ r < ∞ and a ∈ T ∞ . Then,

sup Q 1 a ( Q ) 1 ∣ Q ∣ ∫ Q ∣ f ( x ) − ( f ) Q ∣ r d x 1 / r ≈ sup Q 1 a ( Q ) 1 ∣ Q ∣ ∫ Q ∣ f ( x ) − ( f ) Q ∣ d x .

The following inequalities are also necessary (see (2.16) in [28], Lemma 2.3 in [47], Lemma 4.6 in [30], or p. 485 in [18]).

Lemma 2.14

(Kolmogorov’s inequality) Let 0 < p < q < ∞ , cube Q ⊂ R n . Using L q , ∞ ( Q ) denotes the weak space with norm ‖ f ‖ L q , ∞ ( Q ) = sup t > 0 t ∣ { x ∈ Q : ∣ f ( x ) ∣ > t } ∣ 1 / q .

  1. Then, there is a positive constant C = C p , q such that for any measurable function f, we have

    ∣ Q ∣ − 1 / p ‖ f ‖ L p ( Q ) ≤ C ∣ Q ∣ − 1 / q ‖ f ‖ L q , ∞ ( Q ) .

  2. If 0 < α < n and 1 / q = 1 / p − α / n , then there is a positive constant C = C p , q such that for any measurable function f, we have

    ‖ f ‖ L p ( Q ) ≤ C ∣ Q ∣ α / n ‖ f ‖ L q , ∞ ( Q ) .

The following results can be obtained from [26] or [45].

Lemma 2.15

Let 0 < α < m n and T α be an m-linear fractional Calderón-Zygmund operator. Suppose 1 ≤ p 1 , p 2 , … , p m ≤ ∞ and 1 / q = 1 / p 1 + 1 / p 2 + ⋯ + 1 / p m − α / n > 0 .

  1. If each p j > 1 , j = 1 , … , m , then

    ‖ T α ( f → ) ‖ L q ( R n ) ≤ C ∏ j = 1 m ‖ f j ‖ L p j ( R n ) .

  2. If at least one p j = 1 for some j , then

    ‖ T α ( f → ) ‖ L q , ∞ ( R n ) ≤ C ∏ j = 1 m ‖ f j ‖ L p j ( R n ) .

The following definition of the multilinear fractional integral operator was considered by several authors (see [19,26,31,44]).

Definition 2.16

(Multilinear fractional integral operator) Let 0 ≤ α < m n and f → = ( f 1 , f 2 , … , f m ) . The multilinear fractional integral is defined as follows:

ℐ α ( f → ) ( x ) = ∫ ( R n ) m f 1 ( y 1 ) ⋯ f m ( y m ) ( ∣ x − y 1 ∣ + ⋯ + ∣ x − y m ∣ ) m n − α d y → ,

where the integral is convergent if f → ∈ S ( R n ) × ⋯ × S ( R n ) .

If we take m = 1 , then ℐ α is the classical fractional integral operator.

The following lemma for multilinear fractional integral operators in variable Lebesgue spaces is needed, and its proof can be found in [44]. In addition, the weighted inequalities for multilinear fractional integral operators has been established by Moen in classical function spaces [31].

Lemma 2.17

Suppose that 0 < α < m n and p 1 ( ⋅ ) , p 2 ( ⋅ ) , … , p m ( ⋅ ) ∈ C log ( R n ) ∩ P ( R n ) satisfy 1 p ( x ) = 1 p 1 ( x ) + 1 p 2 ( x ) + ⋯ + 1 p m ( x ) and ( p j ) + < m n α ( j = 1 , 2 , … , m ) . Define the variable exponent q ( ⋅ ) as follows:

1 q ( x ) = 1 p ( x ) − α n .

Then, there exists a positive constant C such that

‖ ℐ α ( f → ) ‖ q ( ⋅ ) ≤ C ∏ i = 1 m ‖ f i ‖ p i ( ⋅ ) .

Lemma 2.18

Set p ( ⋅ ) ∈ P log ( R n ) , 1 < β ≤ p − such that 0 ≤ δ ( ⋅ ) / n = 1 / β − 1 / p ( ⋅ ) ≤ 1 and b ∈ L ( δ ( ⋅ ) ) .

  1. (See Lemma 3.7 in [39]) Then, there exists a positive constant C such that

    sup Q ‖ b − b Q ‖ exp L , Q ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ≤ C ‖ b ‖ L ( δ ( ⋅ ) ) .

  2. (See Lemma 3.8 in [39]) Then, there is a positive constant C such that for every j ∈ N , we have

    ∣ b a k 0 ( j + 1 ) Q − b a k 0 Q ∣ ≤ C j ‖ b ‖ L ( δ ( ⋅ ) ) ∣ a k 0 ( j + 1 ) Q ∣ 1 / β − 1 ‖ χ a k 0 ( j + 1 ) Q ‖ p ′ ( ⋅ ) .

The following pointwise results can be founded in [41].

Lemma 2.19

Let p ( ⋅ ) ∈ P log ( R n ) .

  1. If 1 < β < p − , then, for every s > 0 and x ∈ R n , there exists a positive constant C such that

    ∫ 0 s ‖ χ Q ( x , t ) ‖ p ′ ( ⋅ ) t n − n / β d t t ≤ C ‖ χ Q ( x , s ) ‖ p ′ ( ⋅ ) s n − n / β .

  2. Suppose that 1 < β ≤ p − such that 0 ≤ δ ( ⋅ ) / n = 1 / β − 1 / p ( ⋅ ) . Then, for every function b ∈ L ( δ ( ⋅ ) ) and x , y ∈ R n , there is a positive constant C such that

    ∣ b ( x ) − b ( y ) ∣ ≤ C ‖ b ‖ L ( δ ( ⋅ ) ) ∫ 0 2 ∣ x − y ∣ ‖ χ Q ( x , t ) ‖ p ′ ( ⋅ ) + ‖ χ Q ( y , t ) ‖ p ′ ( ⋅ ) t n − n / β d t t .

2.2 A pointwise estimate

The following notations can be founded in [39] or [4].

Definition 2.20

Let f be a locally integrable function defined on R n .

  1. Set 0 ≤ δ < 1 . The δ -sharp maximal operator is defined as follows:

    f δ ♯ ( x ) = sup Q ∋ x 1 ∣ Q ∣ 1 + δ / n ∫ Q ∣ f ( y ) − f Q ∣ d y ,

    where the supremum is taken over all cube Q ⊂ R n containing x , and f Q = ∣ Q ∣ − 1 ∫ Q f ( z ) d z denotes the average of f over the cube Q ⊂ R n .

  2. Let 0 ≤ δ ( ⋅ ) < 1 , p ( ⋅ ) ∈ P ( R n ) , and 1 < β ≤ p − such that δ ( ⋅ ) / n = 1 / β − 1 / p ( ⋅ ) .

    1. The δ ( ⋅ ) -sharp maximal operator is defined as follows:

      f δ ( ⋅ ) ♯ ( x ) = sup Q ∋ x 1 ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) 1 ∣ Q ∣ ∫ Q ∣ f ( y ) − f Q ∣ d y .

    2. For any γ > 0 , the generalization of the operator f δ ( ⋅ ) ♯ is defined as follows:

      f δ ( ⋅ ) , γ ♯ ( x ) = sup Q ∋ x 1 ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) 1 ∣ Q ∣ ∫ Q ∣ ∣ f ( y ) ∣ γ − ( ∣ f ∣ γ ) Q ∣ d y 1 / γ .

  3. Let 0 ≤ α < n and ε > 0 , define the following operators via

    1. M ε f ( x ) = [ M ( ∣ f ∣ ε ) ( x ) ] 1 / ε = sup Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ f ( y ) ∣ ε d y 1 / ε .

    2. M α , L ( log L ) f ( x ) = sup Q ∋ x ∣ Q ∣ α / n ‖ f ‖ L ( log L ) , Q , where ‖ ⋅ ‖ L ( log L ) , Q is the Luxemburg-type average defined as follows:

      ‖ f ‖ L ( log L ) , Q = inf λ > 0 : 1 ∣ Q ∣ ∫ Q ∣ f ( x ) ∣ λ log ( e + ∣ f ∣ / λ ) d x ≤ 1 .

Definition 2.21

(Multilinear fractional maximal functions) For all locally integrable functions f → = ( f 1 , f 2 , … , f m ) and x ∈ R n , 0 ≤ α < m n ,

  1. the multilinear maximal fractional functions ℳ α and ℳ r , α are defined as follows:

    ℳ α ( f → ) ( x ) = sup Q ∋ x ∣ Q ∣ α / n ∏ j = 1 m 1 ∣ Q ∣ ∫ Q ∣ f j ( y j ) ∣ d y j = sup Q ∋ x ∏ j = 1 m 1 ∣ Q ∣ 1 − α / ( n m ) ∫ Q ∣ f j ( y j ) ∣ d y j

    and

    ℳ α , r ( f → ) ( x ) = sup Q ∋ x ∣ Q ∣ α / n ∏ j = 1 m 1 ∣ Q ∣ ∫ Q ∣ f j ( y j ) ∣ r d y j 1 / r , for r > 1 ,

  2. the multilinear maximal fractional functions related to Young function Φ ( t ) = t ( 1 + log + t ) are defined as follows:

    ℳ α , L ( log L ) i ( f → ) ( x ) = sup Q ∋ x ∣ Q ∣ α / n ‖ f i ‖ L ( log L ) , Q ∏ j = 1 j ≠ i m 1 ∣ Q ∣ ∫ Q ∣ f j ( y j ) ∣ d y j

    and

    ℳ α , L ( log L ) ( f → ) ( x ) = sup Q ∋ x ∣ Q ∣ α / n ∏ j = 1 m ‖ f j ‖ L ( log L ) , Q ,

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

Remark 2.22

  1. If we take f ≡ 1 in equation (2.3) with t = 1 , it follows that, for every α ∈ [ 0 , m n ) , the inequality

    (2.7) ℳ α ( f → ) ( x ) ≤ C ℳ L ( log L ) i ( f → ) ( x ) ≤ C 1 ℳ α , L ( log L ) ( f → ) ( x ) .

  2. In [2], the authors prove that M α ( M k ) ≈ M α , L ( log L ) k with k ∈ N , where M k is the iteration of the Hardy-Littlewood maximal operator k times. Particularly, for α = 0 and k = 1 , one have M L ( log L ) ≈ M 2 = M ∘ M (see also [35] or [1]).

Lemma 2.23

(See Lemma 4.11 in [39]) Let 0 < γ < 1 and r ( ⋅ ) , p ( ⋅ ) ∈ P log ( R n ) , and let q ( ⋅ ) , β , δ ( ⋅ ) such that 0 ≤ δ ( ⋅ ) / n = 1 / p ( ⋅ ) − 1 / q ( ⋅ ) = 1 / β − 1 / r ( ⋅ ) ≤ 1 / n . If ‖ f ‖ q ( ⋅ ) < ∞ , then there is a positive constant C such that

‖ f ‖ q ( ⋅ ) ≤ C ‖ f δ ( ⋅ ) , γ ♯ ‖ p ( ⋅ ) .

The following result is a generalization to the variable context of a pointwise estimate of commutators.

Lemma 2.24

Let m ≥ 2 , 0 < α < m n , 0 < γ < ε < min { 1 , n / ( m n − α ) } , η 0 < η ≤ 1 , p ( ⋅ ) ∈ P log ( R n ) , and 1 < β ≤ p − such that 0 ≤ δ ( ⋅ ) / n = 1 / β − 1 / p ( ⋅ ) ≤ 1 and b → = ( b 1 , b 2 , … , b m ) ∈ L ( δ ( ⋅ ) ) × L ( δ ( ⋅ ) ) × ⋯ × L ( δ ( ⋅ ) ) . Then, there exists a positive constant C such that

( T α , b j ( f → ) ) δ ( ⋅ ) , γ ♯ ( x ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ( M ε ( T α f → ) ( x ) + ℳ α , L ( log L ) ( f → ) ( x ) ) ( j = 1 , 2 , … , m ) .

Furthermore,

( T α , Σ b → ( f → ) ) δ ( ⋅ ) , γ ♯ ( x ) ≤ C ∑ j = 1 m ‖ b j ‖ L ( δ ( ⋅ ) ) ( M ε ( T α ( f → ) ) ( x ) + ℳ α , L ( log L ) ( f → ) ( x ) ) .

Proof

Let Q ⊂ R n and x ∈ Q . Due to the fact ∣ ∣ a ∣ γ − ∣ c ∣ γ ∣ ≤ ∣ a − c ∣ γ for 0 < γ < 1 , it is enough to show that, for some constant C Q , there exists a positive constant C such that

1 ∣ Q ∣ ∫ Q ∣ T α , b j ( f → ) ( z ) − C Q ∣ γ d z 1 / γ ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ( M ε ( T α ( f → ) ) ( x ) + ℳ α , L ( log L ) ( f → ) ( x ) ) .

For each j , we decompose f j = f j 0 + f j ∞ with f j 0 = f j χ a k 0 Q , where a and k 0 are defined as in equations (2.4) and (2.5), respectively. Then,

∏ j = 1 m f j ( y j ) = ∏ j = 1 m ( f j 0 ( y j ) + f j ∞ ( y j ) ) = ∑ ρ 1 , … , ρ m ∈ { 0 , ∞ } f 1 ρ 1 ( y 1 ) ⋯ f m ρ m ( y m ) = ∏ j = 1 m f j 0 ( y j ) + ∑ ( ρ 1 , … , ρ m ) ∈ ρ f 1 ρ 1 ( y 1 ) ⋯ f m ρ m ( y m ) ,

where ρ = { ( ρ 1 , … , ρ m ) : there is at least one ρ j ≠ 0 } . Let λ be some positive constant to be chosen. It is easy to see that

T α , b j ( f → ) ( x ) = ( b j ( x ) − λ ) T α ( f → ) ( x ) − T α ( f 1 0 , … , ( b j − λ ) f j 0 , … , f m 0 ) ( x ) − ∑ ( ρ 1 , … , ρ m ) ∈ ρ T α ( f 1 ρ 1 , … , ( b j − λ ) f j ρ j , … , f m ρ m ) ( x ) .

By taking λ = ( b j ) a k 0 Q and C Q = ∑ ( ρ 1 , … , ρ m ) ∈ ρ ( T α ( f 1 ρ 1 , … , ( b j − λ ) f j ρ j , … , f m ρ m ) ) Q , we obtain that

1 ∣ Q ∣ ∫ Q ∣ T α , b j ( f → ) ( z ) − C Q ∣ γ d z 1 / γ ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ≤ C ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ( I + I I + I I I ) ,

where

I = 1 ∣ Q ∣ ∫ Q ∣ ( b j ( z ) − λ ) T α ( f → ) ( z ) ∣ γ d z 1 / γ I I = 1 ∣ Q ∣ ∫ Q ∣ T α ( f 1 0 , … , ( b j − λ ) f j 0 , … , f m 0 ) ( z ) ∣ γ d z 1 / γ I I I = ∑ ( ρ 1 , … , ρ m ) ∈ ρ 1 ∣ Q ∣ ∫ Q ∣ T α ( f 1 ρ 1 , … , ( b j − λ ) f j ρ j , … , f m ρ m ) ( z ) − ( T α ( f 1 ρ 1 , … , ( b j − λ ) f j ρ j , … , f m ρ m ) ) Q ∣ γ d z 1 / γ .

Let us first estimate I . By taking 1 < r < ε / γ and using Hölder’s inequality, we obtain that

I ≤ C 1 ∣ Q ∣ ∫ Q ∣ ( b j ( z ) − ( b j ) a k 0 Q ) ∣ r ′ γ d z 1 / ( r ′ γ ) 1 ∣ Q ∣ ∫ Q ∣ T α ( f → ) ( z ) ∣ r γ d z 1 / ( r γ ) .

It is known from Lemma 2.12 that the functional a ( Q ) = ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) satisfies T ∞ condition. Then, by Lemma 2.13, it can obtain that

I ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ≤ C 1 ∣ Q ∣ ∫ Q ∣ ( b j ( z ) − ( b j ) a k 0 Q ) ∣ r ′ γ d z 1 / ( r ′ γ ) ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) 1 ∣ Q ∣ ∫ Q ∣ T α ( f → ) ( z ) ∣ r γ d z 1 / ( r γ ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) M r γ ( T α ( f → ) ) ( x ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) M ε ( T α ( f → ) ) ( x ) .

To estimate I I , note that 0 < γ < n / ( m n − α ) , by using L 1 × L 1 × ⋯ × L 1 to L n / ( m n − α ) , ∞ boundedness of T α (see Lemma 2.15), Kolmogorov’s inequality (see Lemma 2.14), we have

∣ Q ∣ II γ = ‖ T α ( f 1 0 , … , ( b j − λ ) f j 0 , … , f m 0 ) ‖ L γ ( Q ) γ ≤ C ∣ Q ∣ 1 − γ ( m − α n ) ‖ ( b j − ( b j ) a k 0 Q ) f j 0 ‖ L 1 ( R n ) γ ∏ k = 1 k ≠ j m ‖ f k 0 ‖ L 1 ( R n ) γ ,

then, using the generalized Hölder’s inequality in Orlicz space (see equation (2.3) with t = 1 ) and equation (2.7), we obtain

I I ≤ C ∣ a k 0 Q ∣ α n ∣ a k 0 Q ∣ ∫ a k 0 Q ∣ b j ( z ) − ( b j ) a k 0 Q ∣ ∣ f j ( z ) ∣ d z ∏ k = 1 k ≠ j m 1 ∣ a k 0 Q ∣ ∫ a k 0 Q ∣ f k ( z ) ∣ d z ≤ C ‖ b j − ( b j ) a k 0 Q ‖ exp L , a k 0 Q ∣ a k 0 Q ∣ α n ‖ f j ‖ L ( log L ) , a k 0 Q ∏ k = 1 k ≠ j m 1 ∣ a k 0 Q ∣ ∫ a k 0 Q ∣ f k ( z ) ∣ d z ≤ C ‖ b j − ( b j ) a k 0 Q ‖ exp L , a k 0 Q ℳ α , L ( log L ) ( f → ) ( x ) .

Thus, by Lemmas 2.12 and 2.18 and doubling condition implied in equations (2.4) and (2.5), we obtain that

I I ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ≤ C ‖ b j − ( b j ) a k 0 Q ‖ exp L , a k 0 Q ∣ a k 0 Q ∣ 1 / β − 1 ‖ χ a k 0 Q ‖ p ′ ( ⋅ ) ℳ α , L ( log L ) ( f → ) ( x ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ℳ α , L ( log L ) ( f → ) ( x ) .

To estimate I I I , we first consider the case when ρ 1 = ⋯ = ρ m = ∞ . For any z ∈ Q , we have

∣ T α ( f 1 ∞ , … , ( b j − ( b j ) a k 0 Q ) f j ∞ , … , f m ∞ ) ( z ) − ( T α ( f 1 ∞ , … , ( b j − ( b j ) a k 0 Q ) f j ∞ , … , f m ∞ ) ) Q ∣ ≤ 1 ∣ Q ∣ ∫ Q ∫ ( R n \ a k 0 Q ) m ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ K α ( z , y → ) − K α ( w , y → ) ∣ ∏ i = 1 m ∣ f i ∞ ( y i ) ∣ d y → d w ≤ 1 ∣ Q ∣ ∑ k = 1 ∞ ∫ Q ∫ ( Q k ) m ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ K α ( z , y → ) − K α ( w , y → ) ∣ ∏ i = 1 m ∣ f i ∞ ( y i ) ∣ d y → d w ,

where Q k = ( a k 0 ( k + 1 ) Q ) \ ( a k 0 k Q ) for k = 1 , 2 , … . Note that, for w , z ∈ Q and any ( y 1 , … , y m ) ∈ ( Q k ) m , we have

a k 0 k l ( Q ) ≤ ∣ z − y i ∣ < a k 0 ( k + 1 ) l ( Q ) and ∣ z − w ∣ ≤ a k 0 l ( Q ) ,

and applying equation (1.2), we have

(2.8) ∣ K α ( z , y → ) − K α ( w , y → ) ∣ ≤ A ∣ z − w ∣ η ( ∣ z − y 1 ∣ + ⋯ + ∣ z − y m ∣ ) m n + η − α ≤ C a − k 0 k η ∣ a k 0 k Q ∣ m − α / n .

Then

(2.9) ∣ T α ( f 1 ∞ , … , ( b j − ( b j ) Q * ) f j ∞ , … , f m ∞ ) ( z ) − ( T α ( f 1 ∞ , … , ( b j − ( b j ) a k 0 Q ) f j ∞ , … , f m ∞ ) ) Q ∣ ≤ C ∣ Q ∣ ∑ k = 1 ∞ a − k 0 k η ∫ Q ∫ ( Q k ) m ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ 1 ∣ a k 0 k Q ∣ m − α / n ∏ i = 1 m ∣ f i ∞ ( y i ) ∣ d y → d w ≤ C ∑ k = 1 ∞ a − k 0 k η ∫ ( Q k ) m ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ 1 ∣ a k 0 k Q ∣ m − α / n ∏ i = 1 m ∣ f i ∞ ( y i ) ∣ d y → ≤ C ∑ k = 1 ∞ a − k 0 k η 1 ∣ a k 0 ( k + 1 ) Q ∣ ∫ a k 0 ( k + 1 ) Q ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ f j ( y j ) ∣ d y j ∣ a k 0 ( k + 1 ) Q ∣ α / n ∏ i = 1 i ≠ j m 1 ∣ a k 0 ( k + 1 ) Q ∣ ∫ a k 0 ( k + 1 ) Q ∣ f i ( y i ) ∣ d y i .

Therefore, by Lemma 2.18 and the generalized Hölder’s inequality in Orlicz space (see equation (2.3) with t = 1 ), we have

(2.10) 1 ∣ a k 0 ( k + 1 ) Q ∣ ∫ a k 0 ( k + 1 ) Q ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ f j ( y j ) ∣ d y j ≤ 1 ∣ a k 0 ( k + 1 ) Q ∣ ∫ a k 0 ( k + 1 ) Q ∣ b j ( y j ) − ( b j ) a k 0 ( k + 1 ) Q ∣ ∣ f j ( y j ) ∣ d y j + ∣ ( b j ) a k 0 ( k + 1 ) Q − ( b j ) a k 0 Q ∣ ∣ a k 0 ( k + 1 ) Q ∣ ∫ a k 0 ( k + 1 ) Q ∣ f j ( y j ) ∣ d y j ≤ C ( ‖ b j − ( b j ) a k 0 ( k + 1 ) Q ‖ exp L , a k 0 ( k + 1 ) Q + ∣ ( b j ) a k 0 ( k + 1 ) Q − ( b j ) a k 0 Q ∣ ) ‖ f j ‖ L ( log L ) , a k 0 ( k + 1 ) Q ≤ C ( k + 1 ) ‖ b j ‖ L ( δ ( ⋅ ) ) ∣ a k 0 ( k + 1 ) Q ∣ 1 / β − 1 ‖ χ a k 0 ( k + 1 ) Q ‖ p ′ ( ⋅ ) ‖ f j ‖ L ( log L ) , a k 0 ( k + 1 ) Q .

Since 0 < γ < 1 , by Hölder’s inequality, Lemma 2.18, doubling condition implied in equations (2.4), (2.5), (2.7), (2.9), and (2.10), and the fact that 1 k 0 = η 0 < η ≤ 1 , we have

(2.11) I I I ∞ ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ≤ C ∣ Q ∣ ∫ Q ∑ k = 1 ∞ a − k 0 k η 1 ∣ a k 0 ( k + 1 ) Q ∣ ∫ a k 0 ( k + 1 ) Q ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ f j ( y j ) ∣ d y j × ∣ a k 0 ( k + 1 ) Q ∣ α / n ∏ i = 1 i ≠ j m 1 ∣ a k 0 ( k + 1 ) Q ∣ ∫ a k 0 ( k + 1 ) Q ∣ f i ( y i ) ∣ d y i d z 1 ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ∑ k = 1 ∞ a − k 0 k η ( k + 1 ) ∣ a k 0 ( k + 1 ) Q ∣ 1 / β − 1 ‖ χ a k 0 ( k + 1 ) Q ‖ p ′ ( ⋅ ) × ∣ a k 0 ( k + 1 ) Q ∣ α / n ‖ f j ‖ L ( log L ) , a k 0 ( k + 1 ) Q ∏ i = 1 i ≠ j m 1 ∣ a k 0 ( k + 1 ) Q ∣ ∫ a k 0 ( k + 1 ) Q ∣ f i ( y i ) ∣ d y i ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ℳ α , L ( log L ) ( f → ) ( x ) ∑ k = 1 ∞ a − k 0 k η ( k + 1 ) a k 0 k n ( 1 / β − 1 ) a k 0 k ( n − n / β + 1 ) 2 k ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ℳ α , L ( log L ) ( f → ) ( x ) ∑ k = 1 ∞ ( k + 1 ) a k 0 ( 1 − η ) 2 k ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ℳ α , L ( log L ) ( f → ) ( x ) .

From equations (2.9)–(2.11), there holds the following inequality, which will be used later,

(2.12) 1 ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ∑ k = 1 ∞ a − k 0 k η ∫ ( a k 0 ( k + 1 ) Q ) m ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ f j ( y j ) ∣ ∣ a k 0 ( k + 1 ) Q ∣ m − α / n ∏ i = 1 i ≠ j m ∣ f i ( y i ) ∣ d y → ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ℳ α , L ( log L ) ( f → ) ( x ) .

Now, for ( ρ 1 , … , ρ m ) ∈ ρ , let us consider the terms I I I ρ 1 , … , ρ m such that at least one ρ j = 0 and one ρ i = ∞ . Without loss of generality, we assume that ρ 1 = ⋯ = ρ l = 0 and ρ l + 1 = ⋯ = ρ m = ∞ with 1 ≤ l < m . For any z ∈ Q , set Q k = ( a k 0 ( k + 1 ) Q ) \ ( a k 0 k Q ) as above, when l + 1 ≤ j ≤ m , applying equation (2.8), we obtain that

∣ T α ( f 1 ρ 1 , … , ( b j − ( b j ) a k 0 Q ) f j ρ j , … , f m ρ m ) ( z ) − ( T α ( f 1 ρ 1 , … , ( b j − ( b j ) a k 0 Q ) f j ρ j , … , f m ρ m ) ) Q ∣ ≤ 1 ∣ Q ∣ ∫ Q ∫ ( a k 0 Q ) l ∏ i = 1 l ∣ f i 0 ( y i ) ∣ ∑ k = 1 ∞ ∫ ( Q k ) m − l ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ K α ( z , y → ) − K α ( w , y → ) ∣ ∏ i = l + 1 m ∣ f i ∞ ( y i ) ∣ d y → d w ≤ C ∣ Q ∣ ∫ Q ∫ ( a k 0 Q ) l ∏ i = 1 l ∣ f i 0 ( y i ) ∣ ∑ k = 1 ∞ a − k 0 k η ∫ ( Q k ) m − l ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ a k 0 k Q ∣ m − α / n ∏ i = l + 1 m ∣ f i ∞ ( y i ) ∣ d y → d w ≤ C ∑ k = 1 ∞ a − k 0 k η ∫ ( a k 0 ( k + 1 ) Q ) m ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ f j ( y j ) ∣ ∣ a k 0 ( k + 1 ) Q ∣ m − α / n ∏ i = 1 i ≠ j m ∣ f i ( y i ) ∣ d y → .

When 1 ≤ j ≤ l , similar to the above, we have that

∣ T α ( f 1 ρ 1 , … , ( b j − ( b j ) a k 0 Q ) f j ρ j , … , f m ρ m ) ( z ) − ( T α ( f 1 ρ 1 , … , ( b j − ( b j ) a k 0 Q ) f j ρ j , … , f m ρ m ) ) Q ∣ ≤ C ∑ k = 1 ∞ a − k 0 k η ∫ ( a k 0 ( k + 1 ) Q ) m ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ f j ( y j ) ∣ ∣ a k 0 ( k + 1 ) Q ∣ m − α / n ∏ i = 1 i ≠ j m ∣ f i ( y i ) ∣ d y → .

Since 0 < γ < 1 , by Hölder’s inequality and equation (2.12), we have

I I I ρ 1 , … , ρ m ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ≤ 1 ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) C ∣ Q ∣ ∫ Q ∣ T α ( f 1 ρ 1 , … , ( b j − ( b j ) a k 0 Q ) f j ρ j , … , f m ρ m ) × ( z ) ( T α ( f 1 ρ 1 , … , ( b j − ( b j ) a k 0 Q ) f j ρ j , … , f m ρ m ) ) Q ∣ d z ) ≤ C ∣ Q ∣ 1 / β − 1 ‖ χ Q ‖ p ′ ( ⋅ ) ∑ k = 1 ∞ a − k 0 k η ∫ ( a k 0 ( k + 1 ) Q ) m ∣ b j ( y j ) − ( b j ) a k 0 Q ∣ ∣ f j ( y j ) ∣ ∣ a k 0 ( k + 1 ) Q ∣ m − α / n ∏ i = 1 i ≠ j m ∣ f i ( y i ) ∣ d y → ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ℳ α , L ( log L ) ( f → ) ( x ) .

Combining the above estimates we obtain the desired result. The proof is completed.□

3 Proofs of the main results

3.1 Proof of Theorem 1.1

Proof of Theorem 1.1

Without loss of generality, we only consider the case that m = 2 . Actually, similar procedure work for all m ∈ N .

(1) We first prove b → = ( b 1 , b 2 ) ∈ L ( δ ) × L ( δ ) . Assume that T α , b j ( j = 1 , 2 ) maps L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) into L q ( ⋅ ) ( R n ) . Note that the homogeneity of K α , set z → = ( z 1 , z 2 ) ∈ R n × R n , we may assume, without loss of generality, that ∣ z j − z 0 j ∣ < ε n ( j = 1 , 2 ) with z → 0 = ( z 01 , z 02 ) ∈ R n \ { 0 } × R n \ { 0 } and ε > 0 , which implies that ∣ z → − z → 0 ∣ < ε 2 n . Then, 1 / K α ( z → ) can be represented in the ball B ( z → 0 , ε 2 n ) ⊂ R 2 n as an absolutely convergent Fourier series

1 K α ( z → ) = 1 K α ( z 1 , z 2 ) = ∑ k a k e i v → k ⋅ z → = ∑ k a k e i ( v k 1 , v k 2 ) ⋅ ( z 1 , z 2 ) ,

where the individual vectors v → k = ( v k 1 , v k 2 ) ∈ R n × R n do not play any significant role in the proof.

Set z → ∗ = ( z ∗ 1 , z ∗ 2 ) . If z ∗ j = ε − 1 z 0 j , then ∣ z j − z ∗ j ∣ < n implies that ∣ ε z j − z 0 j ∣ < ε n ( j = 1 , 2 ) . Thus, by the homogeneity of K α , for all z → ∈ B ( z → ∗ , 2 n ) ⊂ R 2 n , we can obtain

1 K α ( z → ) = ε α − 2 n K α ( ε z → ) = ε α − 2 n ∑ k a k e i ε v → k ⋅ z → .

Let Q = Q ( x 0 , l ) be an arbitrary cube in R n with sides parallel to the coordinate axes, diameter l , and center x 0 , and set y 0 j = x 0 − l z ∗ j and Q ∗ j = Q ( y 0 j , l ) ⊂ R n ( j = 1 , 2 ) . By taking x ∈ Q , y j ∈ Q ∗ j ( j = 1 , 2 ) , we have

x − y j l − z ∗ j = x − x 0 + x 0 − y 0 j + y 0 j − y j l − z ∗ j ≤ x − x 0 l + y j − y 0 j l ≤ n .

So we have ∣ ( x − y 1 l , x − y 2 l ) − z → ∗ ∣ = ∣ ( x − y 1 l , x − y 2 l ) − ( z ∗ 1 , z ∗ 2 ) ∣ ≤ 2 n , i.e., ( x − y 1 l , x − y 2 l ) ∈ B ( z → ∗ , 2 n ) , which means that ( x − y 1 , x − y 2 ) is bounded away from the singularity of K α .

Without loss of generality, let s 1 ( x ) = sgn ( b 1 ( x ) − ( b 1 ) Q ∗ 1 ) , then we have

∣ b 1 ( x ) − ( b 1 ) Q ∗ 1 ∣ = ( b 1 ( x ) − ( b 1 ) Q ∗ 1 ) s 1 ( x ) = 1 ∣ Q ∣ 2 ∫ R n ∫ R n ( b 1 ( x ) − b 1 ( y 1 ) ) χ Q ∗ 1 ( y 1 ) χ Q ∗ 2 ( y 2 ) s 1 ( x ) d y 1 d y 2 = l − 2 n ∫ R n ∫ R n ( b 1 ( x ) − b 1 ( y 1 ) ) l 2 n − α K α ( x − y 1 , x − y 2 ) K α ( x − y 1 l , x − y 2 l ) χ Q ∗ 1 ( y 1 ) χ Q ∗ 2 ( y 2 ) ( x ) s 1 ( x ) d y 1 d y 2 = ε α − 2 n ∣ Q ∣ α / n ∑ k a k s 1 ( x ) e i ε l ( v k 1 + v k 2 ) x ∫ ( R n ) 2 ( b 1 ( x ) − b 1 ( y 1 ) ) K α ( x − y 1 , x − y 2 ) e − i ε l v k 1 y 1 χ Q ∗ 1 ( y 1 ) e − i ε l v k 2 y 2 χ Q ∗ 2 ( y 2 ) d y → .

Set f k 1 ( y 1 ) = e − i ε v k 1 y 1 / l χ Q ∗ 1 ( y 1 ) , f k 2 ( y 2 ) = e − i ε v k 2 y 2 / l χ Q ∗ 2 ( y 2 ) , h k ( x ) = e i ε ( v k 1 + v k 2 ) x / l χ Q ( x ) , then

(3.1) ∫ Q ∣ b 1 ( x ) − ( b 1 ) Q ∗ 1 ∣ d x = ε α − 2 n ∣ Q ∣ α / n ∑ k a k ∫ R n T α , b 1 ( f k 1 , f k 2 ) ( x ) h k ( x ) s 1 ( x ) d x ≤ C ε α − 2 n ∣ Q ∣ α / n ∑ k ∣ a k ∣ ∫ R n ∣ T α , b 1 ( f k 1 , f k 2 ) ( x ) ∣ ∣ h k ( x ) ∣ ∣ s 1 ( x ) ∣ d x ≤ C ε α − 2 n ∣ Q ∣ α / n ∑ k ∣ a k ∣ ∫ Q ∣ T α , b 1 ( f k 1 , f k 2 ) ( x ) ∣ d x .

Note that f k j ( y j ) ≤ χ Q ∗ j ( y j ) for every y j ∈ Q ∗ j , which implies that f k j ∈ L p j ( ⋅ ) ( R n ) ( j = 1 , 2 ) for every k ∈ N . Then, from the generalized Hölder’s inequality (2.2) and the hypothesis, we obtain

∫ Q ∣ T α , b 1 ( f k 1 , f k 2 ) ( x ) ∣ d x ≤ C ‖ T α , b 1 ( f k 1 , f k 2 ) ‖ q ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ≤ C ‖ f k 1 ‖ p 1 ( ⋅ ) ‖ f k 2 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) .

Substituting equation (3.1), since Q , Q ∗ j ⊂ Q 0 j = Q ( x 0 , ( ∣ z ∗ j ∣ + 1 ) l ) ( j = 1 , 2 ) , then, we have

∫ Q ∣ b 1 ( x ) − ( b 1 ) Q ∗ 1 ∣ d x ≤ C ε α − 2 n ∣ Q ∣ α / n ∑ k ∣ a k ∣ ‖ χ Q ∗ 1 ‖ p 1 ( ⋅ ) ‖ χ Q ∗ 2 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ≤ C ε α − 2 n ∣ Q ∣ α / n ∑ k ∣ a k ∣ ‖ χ Q 01 ‖ p 1 ( ⋅ ) ‖ χ Q 02 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) .

Since 1 p ( ⋅ ) − α n − 1 q ( ⋅ ) = δ n = 1 p 1 ( ⋅ ) + 1 p 2 ( ⋅ ) − α n − 1 q ( ⋅ ) , and 2 n α + δ + n < ( p j ) − < ( p j ) + < 2 n α + δ ( j = 1 , 2 ) , 1 q ′ ( ⋅ ) = 1 + α + δ n − 1 p 1 ( ⋅ ) − 1 p 2 ( ⋅ ) = ( α + δ + n 2 n − 1 p 1 ( ⋅ ) ) + ( α + δ + n 2 n − 1 p 2 ( ⋅ ) ) . Hence, by applying the generalized Hölder’s inequality, we have

‖ χ Q ‖ q ′ ( ⋅ ) ≤ ‖ χ Q ‖ ( α + δ + n 2 n − 1 p 1 ( ⋅ ) ) − 1 ‖ χ Q ‖ ( α + δ + n 2 n − 1 p 2 ( ⋅ ) ) − 1 ≤ ‖ χ Q 01 ‖ ( α + δ + n 2 n − 1 p 1 ( ⋅ ) ) − 1 ‖ χ Q 02 ‖ ( α + δ + n 2 n − 1 p 2 ( ⋅ ) ) − 1 .

For any j = 1 , 2 , let h j ′ = ( α + δ + n 2 n − 1 p j ( ⋅ ) ) − 1 , we obtain that 1 h j ′ = α + δ + n 2 n − 1 p j ( ⋅ ) , i.e., 1 p j ( ⋅ ) = 1 h j − ( 1 − α + δ + n 2 n ) = 1 h j − ( n − α + δ + n 2 ) / n . Thus, since p 1 ( ⋅ ) , p 2 ( ⋅ ) ∈ P log ( R n ) , using Lemma 2.9, we obtain that 1 ∈ A [ ( α + δ + n 2 n − 1 p j ( ⋅ ) ) − 1 ] ′ , p j ( ⋅ ) n − α + δ + n 2 ( R n ) ( j = 1 , 2 ) . Then, from equation (2.6), doubling condition implied in equations (2.4) and (2.5), we have

‖ χ Q 01 ‖ p 1 ( ⋅ ) ‖ χ Q 02 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ≤ ‖ χ Q 01 ‖ p 1 ( ⋅ ) ‖ χ Q 01 ‖ ( α + δ + n 2 n − 1 p 1 ( ⋅ ) ) − 1 ‖ χ Q 02 ‖ p 2 ( ⋅ ) ‖ χ Q 02 ‖ ( α + δ + n 2 n − 1 p 2 ( ⋅ ) ) − 1 ≤ C ∣ Q 01 ∣ α + δ + n 2 n ∣ Q 02 ∣ α + δ + n 2 n ≤ C ∣ Q ∣ α + δ + n n .

Thus,

∫ Q ∣ b 1 ( x ) − ( b 1 ) Q ∗ 1 ∣ d x ≤ C ε α − 2 n ∣ Q ∣ α / n ∑ k ∣ a k ∣ ‖ χ Q 01 ‖ p 1 ( ⋅ ) ‖ χ Q 02 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ≤ C ∣ Q ∣ δ + n n ∑ k ∣ a k ∣ ≤ C ∣ Q ∣ δ + n n .

Since z → ∗ = ( z ∗ 1 , z ∗ 2 ) is fixed, by taking supremum over every Q ⊂ R n , we obtain that b 1 ∈ L ( δ ) . Similar to the argument above, we can obtain that b 2 ∈ L ( δ ) . Therefore, b → = ( b 1 , b 2 ) ∈ L ( δ ( ⋅ ) ) × L ( δ ( ⋅ ) ) .

(2) We now prove the second part. Assume that b → = ( b 1 , b 2 ) ∈ L ( δ ) × L ( δ ) and f → = ( f 1 , f 2 ) ∈ L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) .

Then, by the definition of Lipschitz’s space (see (1) of Definition 2.10), equation (1.1), the monotonically increasing of function t δ for any t > 0 , and taking into account that ‖ f ‖ Λ δ ≈ ‖ f ‖ L ( δ ) , we can obtain

∣ T α , b 1 ( f → ) ( x ) ∣ ≤ C ‖ b 1 ‖ L ( δ ) ∫ ( R n ) 2 ∣ x − y 1 ∣ δ ( ∣ x − y 1 ∣ + ∣ x − y 2 ∣ ) 2 n − α ∣ f 1 ( y 1 ) ∣ ∣ f 2 ( y 2 ) ∣ d y → ≤ C ‖ b 1 ‖ L ( δ ) ∫ ( R n ) 2 ∣ f 1 ( y 1 ) ∣ ∣ f 2 ( y 2 ) ∣ ( ∣ x − y 1 ∣ + ∣ x − y 2 ∣ ) 2 n − α − δ d y → = C ‖ b 1 ‖ L ( δ ) ℐ α + δ ( ∣ f 1 ∣ , ∣ f 2 ∣ ) ( x ) .

Since 1 p 1 ( x ) + 1 p 2 ( x ) − 1 q ( x ) = α + δ n , 0 < α + δ < 2 n , p 1 ( ⋅ ) , p 2 ( ⋅ ) ∈ P log ( R n ) , and ( p 1 ) + , ( p 2 ) + < 2 n α + δ . Thus, using Lemma 2.17, we have

‖ T α , b 1 ( f → ) ‖ q ( ⋅ ) ≤ C ‖ b 1 ‖ L ( δ ) ‖ ℐ α + δ ( ∣ f 1 ∣ , ∣ f 2 ∣ ) ‖ q ( ⋅ ) ≤ C ‖ b 1 ‖ L ( δ ) ‖ f 1 ‖ p 1 ( ⋅ ) ‖ f 2 ‖ p 2 ( ⋅ ) .

Similar to the argument above, we can obtain that ‖ T α , b 2 ( f → ) ‖ q ( ⋅ ) ≤ C ‖ b 2 ‖ L ( δ ) ‖ f 1 ‖ p 1 ( ⋅ ) ‖ f 2 ‖ p 2 ( ⋅ ) .

Combining these estimates, the proof is completed.□

3.2 Proof of Theorem 1.3

Proof of Theorem 1.3

Without loss of generality, we only consider the case that m = 2 . Actually, similar procedure work for all m ∈ N .

(1) We first prove b → = ( b 1 , b 2 ) ∈ L ( δ ( ⋅ ) ) × L ( δ ( ⋅ ) ) . Assume that T α , b j ( j = 1 , 2 ) maps L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) into L q ( ⋅ ) ( R n ) . By proceeding as in equation (3.1) with Q = Q ( x 0 , l ) and Q ∗ j = Q ( y 0 j , l ) ⊂ R n ( j = 1 , 2 ) . By taking x ∈ Q , y j ∈ Q ∗ j ( j = 1 , 2 ) , we have

(3.2) ∫ Q ∣ b 1 ( x ) − ( b 1 ) Q ∗ 1 ∣ d x ≤ C ε α − 2 n ∣ Q ∣ α / n ∑ k ∣ a k ∣ ∫ Q ∣ T α , b 1 ( f k 1 , f k 2 ) ( x ) ∣ d x .

Note that f k j ( y j ) ≤ χ Q ∗ j ( y j ) for every y j ∈ Q ∗ j , which implies that f k j ∈ L p j ( ⋅ ) ( R n ) ( j = 1 , 2 ) for every k ∈ N . Then, from the generalized Hölder’s inequality (2.2) and the hypothesis, we obtain that

∫ Q ∣ T α , b 1 ( f k 1 , f k 2 ) ( x ) ∣ d x ≤ C ‖ T α , b 1 ( f k 1 , f k 2 ) ‖ q ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ≤ C ‖ f k 1 ‖ p 1 ( ⋅ ) ‖ f k 2 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) .

Substituting equation (3.2), since Q , Q ∗ j ⊂ Q 0 j = Q ( x 0 , ( ∣ z ∗ j ∣ + 1 ) l ) ( j = 1 , 2 ) , we then have

∫ Q ∣ b 1 ( x ) − ( b 1 ) Q ∗ 1 ∣ d x ≤ C ε α − 2 n ∣ Q ∣ α / n ∑ k ∣ a k ∣ ‖ χ Q ∗ 1 ‖ p 1 ( ⋅ ) ‖ χ Q ∗ 2 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ≤ C ε α − 2 n ∣ Q ∣ α / n ∑ k ∣ a k ∣ ‖ χ Q 01 ‖ p 1 ( ⋅ ) ‖ χ Q 02 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) .

Since 1 p ( x ) − α n − 1 q ( x ) = δ ( x ) n = 1 β − 1 r ( x ) , then 1 q ′ ( ⋅ ) = 1 r ′ ( ⋅ ) + ( 1 β + α n − 1 p ( ⋅ ) ) = 1 r ′ ( ⋅ ) + ( 1 2 β + α 2 n − 1 p 1 ( ⋅ ) ) + ( 1 2 β + α 2 n − 1 p 2 ( ⋅ ) ) . Hence, by applying the generalized Hölder’s inequality, we have

‖ χ Q ‖ q ′ ( ⋅ ) ≤ ‖ χ Q ‖ r ′ ( ⋅ ) ‖ χ Q ‖ ( 1 2 β + α 2 n − 1 p 1 ( ⋅ ) ) − 1 ‖ χ Q ‖ ( 1 2 β + α 2 n − 1 p 2 ( ⋅ ) ) − 1 ≤ ‖ χ Q 01 ‖ ( 1 2 β + α 2 n − 1 p 1 ( ⋅ ) ) − 1 ‖ χ Q 02 ‖ ( 1 2 β + α 2 n − 1 p 2 ( ⋅ ) ) − 1 ‖ χ Q ‖ r ′ ( ⋅ ) .

For any j = 1 , 2 , let h j ′ = ( 1 2 β + α 2 n − 1 p j ( ⋅ ) ) − 1 , we obtain that 1 h j ′ = 1 2 β + α 2 n − 1 p j ( ⋅ ) , i.e., 1 p j ( ⋅ ) = 1 h j − ( 1 − 1 2 β − α 2 n ) = 1 h j − ( n − n 2 β − α 2 ) / n . Thus, since r ( ⋅ ) , p 1 ( ⋅ ) , p 2 ( ⋅ ) ∈ P log ( R n ) , using Lemma 2.9, we obtain that 1 ∈ A [ ( 1 2 β + α 2 n − 1 p j ( ⋅ ) ) − 1 ] ′ , p j ( ⋅ ) n − n 2 β − α 2 ( R n ) ( j = 1 , 2 ) . Then, from equation (2.6), Lemma 2.12, and doubling condition implied in equations (2.4) and (2.5), we have

‖ χ Q 01 ‖ p 1 ( ⋅ ) ‖ χ Q 02 ‖ p 2 ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ≤ ‖ χ Q 01 ‖ p 1 ( ⋅ ) ‖ χ Q 01 ‖ ( 1 2 β + α 2 n − 1 p 1 ( ⋅ ) ) − 1 ‖ χ Q 02 ‖ p 2 ( ⋅ ) ‖ χ Q 02 ‖ ( 1 2 β + α 2 n − 1 p 2 ( ⋅ ) ) − 1 ‖ χ Q ‖ r ′ ( ⋅ ) ≤ C ∣ Q ∣ 1 / β + α / n ‖ χ Q ‖ r ′ ( ⋅ ) .

Thus,

∫ Q ∣ b 1 ( x ) − ( b 1 ) Q ∗ 1 ∣ d x ≤ C ∣ Q ∣ 1 / β ‖ χ Q ‖ r ′ ( ⋅ ) .

Since z → ∗ = ( z ∗ 1 , z ∗ 2 ) is fixed, by taking supremum over every Q ⊂ R n , we obtain that b 1 ∈ L ( δ ( ⋅ ) ) . Similar to the argument above, we can obtain that b 2 ∈ L ( δ ( ⋅ ) ) .

(2) Now, we prove the second part. Assume that b → = ( b 1 , b 2 ) ∈ L ( δ ( ⋅ ) ) × L ( δ ( ⋅ ) ) and f → = ( f 1 , f 2 ) ∈ L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) such that ‖ T α , b j ( f → ) ‖ q ( ⋅ ) < ∞ ( j = 1 , 2 ) .

Let 1 u ( ⋅ ) = 1 p ( ⋅ ) − α n , then 1 u ( ⋅ ) − 1 q ( ⋅ ) = δ ( ⋅ ) n . From Lemma 2.23 and the hypothesis, for any given γ ∈ ( 0 , 1 ) , it follows that

‖ T α , b j ( f → ) ‖ q ( ⋅ ) ≤ C ‖ ( T α , b j ( f → ) ) δ ( ⋅ ) , γ ♯ ‖ u ( ⋅ ) ( j = 1 , 2 ) .

Since 1 k 0 = η 0 < η ≤ 1 and 0 < γ < ε < min { 1 , n / ( 2 n − α ) } , from Lemma 2.24, we have

‖ T α , b j ( f → ) ‖ q ( ⋅ ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ‖ M ε ( T α f → ) + ℳ α , L ( log L ) ( f → ) ‖ u ( ⋅ ) ( j = 1 , 2 ) .

Note that the following fact

ℳ α , L ( log L ) ( f → ) ( x ) ≤ ∏ j = 1 2 ( sup Q ∋ x ∣ Q ∣ α / n ‖ f j ‖ L ( log L ) , Q ) = ∏ j = 1 2 M α , L ( log L ) ( f j ) ( x ) .

Using Lemmas 2.5–2.7 and 2.15, the relation between p ( ⋅ ) and q ( ⋅ ) , and the pointwise equivalence between M α ( M ) and M α , L ( log L ) (see (b) in Remark 2.22), it can conclude that

‖ T α , b j ( f → ) ‖ q ( ⋅ ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ‖ [ M ( ∣ T α f → ∣ ε ) ] 1 / ε ‖ u ( ⋅ ) + ∏ i = 1 2 M α , L ( log L ) ( f i ) u ( ⋅ ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ‖ ∣ T α f → ∣ ε ‖ u ( ⋅ ) / ε 1 / ε + ∏ i = 1 2 M α ( M ( f i ) ) u ( ⋅ ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ‖ T α f → ‖ u ( ⋅ ) + ∏ i = 1 2 M ( f i ) p ( ⋅ ) ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ,

where j = 1 , 2 .

Combining these estimates, the proof is completed.□

3.3 Proof of Theorem 1.5

Proof of Theorem 1.5

Without loss of generality, we only consider the case that m = 2 . In fact, similar procedure applies to all m ∈ N .

Assume that b → = ( b 1 , b 2 ) ∈ L ( δ ( ⋅ ) ) × L ( δ ( ⋅ ) ) and f → = ( f 1 , f 2 ) ∈ L p 1 ( ⋅ ) ( R n ) × L p 2 ( ⋅ ) ( R n ) . Let x ∈ Q ⊂ R n , it is enough to show that, for some constant C Q , there exists a positive constant C such that

1 ∣ Q ∣ 1 / β + α / n ‖ χ Q ‖ q ′ ( ⋅ ) ∫ Q ∣ T α , b j ( f → ) ( x ) − C Q ∣ d x ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ,

where 1 q ( ⋅ ) = 1 r ( ⋅ ) + 1 p ( ⋅ ) , j = 1 , 2 .

In fact, for each j , we decompose f j = f j 1 + f j 2 with f j 1 = f j χ a k 0 Q , where a and k 0 are defined as in equations (2.4) and (2.5), respectively. Then,

∏ j = 1 2 f j ( y j ) = ∏ j = 1 2 ( f j 1 ( y j ) + f j 2 ( y j ) ) = f 11 ( y 1 ) f 21 ( y 2 ) + f 12 ( y 1 ) f 22 ( y 2 ) + f 11 ( y 1 ) f 22 ( y 2 ) + f 12 ( y 1 ) f 21 ( y 2 ) .

Without loss of generality, it is easy to see that

T α , b 1 ( f → ) ( x ) = T α , b 1 ( f 11 , f 21 ) ( x ) + T α , b 1 ( f 12 , f 22 ) ( x ) + T α , b 1 ( f 11 , f 22 ) ( x ) + T α , b 1 ( f 12 , f 21 ) ( x ) .

By taking

C Q = ( T α , b 1 ( f 12 , f 22 ) ) Q + ( T α , b 1 ( f 11 , f 22 ) ) Q + ( T α , b 1 ( f 12 , f 21 ) ) Q ,

we obtain that

∫ Q ∣ T α , b 1 ( f → ) ( x ) − C Q ∣ d x ≤ C ∫ Q ∣ T α , b 1 ( f 11 , f 21 ) ( x ) ∣ d x + C ∫ Q ∣ T α , b 1 ( f 12 , f 22 ) ( x ) − ( T α , b 1 ( f 12 , f 22 ) ) Q ∣ d x + C ∫ Q ∣ T α , b 1 ( f 11 , f 22 ) ( x ) − ( T α , b 1 ( f 11 , f 22 ) ) Q ∣ d x + C ∫ Q ∣ T α , b 1 ( f 12 , f 21 ) ( x ) − ( T α , b 1 ( f 12 , f 21 ) ) Q ∣ d x = C ( I 1 + I 2 + I 3 + I 4 ) .

To estimate I 1 , let us first choose u 1 ( ⋅ ) and v 1 ( ⋅ ) such that

1 v ( x ) − α n − 1 u ( x ) = δ ( x ) n = 1 β − 1 r ( x ) .

Using the generalized Hölder’s inequality (see equation (2.2)) and Theorem 1.3, we obtain

I 1 ≤ C ‖ T α , b 1 ( f 11 , f 21 ) ‖ u ( ⋅ ) ‖ χ Q ‖ u ′ ( ⋅ ) ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ‖ ∏ i = 1 2 f i 1 ‖ v ( ⋅ ) ‖ χ Q ‖ u ′ ( ⋅ ) .

Since δ ( ⋅ ) n = δ ˜ ( ⋅ ) n − α n + 1 p ( ⋅ ) , then 1 v ( ⋅ ) = 1 p ( ⋅ ) + ( δ ˜ ( ⋅ ) n + 1 u ( ⋅ ) ) . Hence, by applying the generalized Hölder’s inequality (see equation (2.1)) and doubling condition implied in equations (2.4) and (2.5), we have

I 1 ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ‖ χ Q ‖ ( δ ˜ ( ⋅ ) n + 1 u ( ⋅ ) ) − 1 ‖ χ Q ‖ u ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

For 1 q ( ⋅ ) = 1 r ( ⋅ ) + 1 p ( ⋅ ) , we have δ ˜ ( ⋅ ) n = α n + 1 β − 1 q ( ⋅ ) and δ ˜ ( ⋅ ) n + 1 u ( ⋅ ) = 1 q ′ ( ⋅ ) + ( α n + 1 β − 1 u ′ ( ⋅ ) ) . Applying again the generalized Hölder’s inequality (see equation (2.1)) to obtain

I 1 ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ‖ χ Q ‖ q ′ ( ⋅ ) ‖ χ Q ‖ ( α n + 1 β − 1 u ′ ( ⋅ ) ) − 1 ‖ χ Q ‖ u ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

Let h ′ = ( α n + 1 β − 1 u ′ ( ⋅ ) ) − 1 , we obtain that 1 h ′ = α n + 1 β − 1 u ′ ( ⋅ ) , that is, 1 u ′ ( ⋅ ) = 1 h − ( 1 − 1 β − α n ) = 1 h − ( n − α − n β ) / n . Using Lemma 2.9, we obtain that 1 ∈ A [ ( α n + 1 β − 1 u ′ ( ⋅ ) ) − 1 ] ′ , u ′ ( ⋅ ) n − α − n β ( R n ) . Then, from equation (2.6), Lemma 2.12, and the doubling condition implied in equations (2.4) and (2.5), we have

I 1 ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n ‖ χ Q ‖ q ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

In order to estimate I 2 , we first estimate the difference ∣ T α , b 1 ( f 12 , f 22 ) ( x ) − ( T α , b 1 ( f 12 , f 22 ) ) Q ∣ . If x , z ∈ Q , then

∣ T α , b 1 ( f 12 , f 22 ) ( x ) − ( T α , b 1 ( f 12 , f 22 ) ) Q ∣ ≤ 1 ∣ Q ∣ ∫ Q ∣ ∫ ( R n \ a k 0 Q ) 2 ( b 1 ( x ) − b 1 ( y 1 ) ) K α ( x , y → ) ∏ i = 1 2 f i ( y i ) d y → − ∫ ( R n \ a k 0 Q ) 2 ( b 1 ( z ) − b 1 ( y 1 ) ) K α ( z , y → ) ∏ i = 1 2 f i ( y i ) d y → ∣ d z ≤ 1 ∣ Q ∣ ∫ Q ∫ ( R n \ a k 0 Q ) 2 ∣ b 1 ( x ) − b 1 ( y 1 ) ∣ ∣ K α ( x , y → ) − K α ( z , y → ) ∣ ∏ i = 1 2 ∣ f i ( y i ) ∣ d y → d z + 1 ∣ Q ∣ ∫ Q ∫ ( R n \ a k 0 Q ) 2 ∣ b 1 ( x ) − b 1 ( z ) ∣ ∣ K α ( z , y → ) ∣ ∏ i = 1 2 ∣ f i ( y i ) ∣ d y → d z .

Since 1 < β < r − such that 0 < δ ( ⋅ ) n = 1 β − 1 r ( ⋅ ) , then, for every x and y 1 , applying Lemma 2.19 and the hypothesis, we obtain

∣ b 1 ( x ) − b 1 ( y 1 ) ∣ ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∫ 0 2 ∣ x − y 1 ∣ ‖ χ Q ( x , t ) ‖ r ′ ( ⋅ ) + ‖ χ Q ( y 1 , t ) ‖ r ′ ( ⋅ ) t n − n / β d t t ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ‖ χ Q ( x , 2 ∣ x − y 1 ∣ ) ‖ r ′ ( ⋅ ) ∣ x − y 1 ∣ n − n / β + ‖ χ Q ( y 1 , 2 ∣ x − y 1 ∣ ) ‖ r ′ ( ⋅ ) ∣ x − y 1 ∣ n − n / β .

Since r ( ⋅ ) ≥ r ∞ implies that r ′ ( ⋅ ) ≤ r ∞ ′ , then, for every τ ∈ Q ( x , 2 ∣ x − y 1 ∣ ) ∩ Q ( y 1 , 2 ∣ x − y 1 ∣ ) , from (1) in Lemma 2.8 (norms of characteristic functions), we obtain

∣ b 1 ( x ) − b 1 ( y 1 ) ∣ ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ x − y 1 ∣ n / r ′ ( τ ) ∣ x − y 1 ∣ n − n / β = C ‖ b ‖ L ( δ ( ⋅ ) ) ∣ x − y 1 ∣ δ ( τ ) .

In particular,

(3.3) ∣ b 1 ( x ) − b 1 ( y 1 ) ∣ ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ x − y 1 ∣ δ ( x ) .

Therefore, using equations (1.1) and (1.2) to deduce

∣ T α , b 1 ( f 12 , f 22 ) ( x ) − ( T α , b 1 ( f 12 , f 22 ) ) Q ∣ ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ ∫ Q ∫ ( R n \ a k 0 Q ) 2 ∣ x − y 1 ∣ δ ( x ) ∣ K α ( x , y → ) − K α ( z , y → ) ∣ ∏ i = 1 2 ∣ f i ( y i ) ∣ d y → d z + C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ ∫ Q ∫ ( R n \ a k 0 Q ) 2 ∣ x − z ∣ δ ( x ) ∣ K α ( z , y → ) ∣ ∏ i = 1 2 ∣ f i ( y i ) ∣ d y → d z ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ ∫ Q ∫ ( R n \ a k 0 Q ) 2 ∣ x − y 1 ∣ δ ( x ) ∣ x − z ∣ η ( ∣ x − y 1 ∣ + ∣ x − y 2 ∣ ) 2 n + η − α ∏ i = 1 2 ∣ f i ( y i ) ∣ d y → d z + C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ ∫ Q ∫ ( R n \ a k 0 Q ) 2 ∣ x − z ∣ δ ( x ) ( ∣ z − y 1 ∣ + ∣ z − y 2 ∣ ) 2 n − α ∏ i = 1 2 ∣ f i ( y i ) ∣ d y → d z = C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ ∫ Q ( I 21 + I 22 ) d z .

By taking Q k = ( a k 0 ( k + 1 ) Q ) \ ( a k 0 k Q ) for k = 1 , 2 , … . Note that, for x , z ∈ Q and any ( y 1 , y 2 ) ∈ ( Q k ) 2 , we have

∣ a k 0 k Q ∣ 1 / n ≤ ∣ x − y 1 ∣ < ∣ a k 0 ( k + 1 ) Q ∣ 1 / n and ∣ x − z ∣ ≤ ∣ Q ∣ 1 / n ,

and applying the generalized Hölder’s inequality, equations (1.1) and (1.2) to deduce

I 21 ≤ C ∑ k = 1 ∞ ∫ ( Q k ) 2 ∣ a k 0 k Q ∣ δ ( x ) / n ∣ Q ∣ η / n ∣ a k 0 k Q ∣ 2 + η / n − α / n ∏ i = 1 2 ∣ f i ( y i ) ∣ d y → ≤ C ∑ k = 1 ∞ a k 0 k ( δ ( x ) − 2 n − η + α ) ∣ Q ∣ δ ( x ) / n + α / n − 2 ∏ i = 1 2 ( ‖ f i ‖ p i ( ⋅ ) ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) ) , I 22 ≤ C ∑ k = 1 ∞ ∫ ( Q k ) 2 ∣ Q ∣ δ ( x ) / n ∣ a k 0 k Q ∣ 2 − α / n ∏ i = 1 2 ∣ f i ( y i ) ∣ d y → ≤ C ∑ k = 1 ∞ a − k 0 k ( 2 n − α ) ∣ Q ∣ δ ( x ) / n + α / n − 2 ∏ i = 1 2 ( ‖ f i ‖ p i ( ⋅ ) ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) ) .

If 1 q ( ⋅ ) = 1 r ( ⋅ ) + 1 p ( ⋅ ) = 1 r ( ⋅ ) + 1 p 1 ( x ) + 1 p 2 ( x ) , then α n + δ ( ⋅ ) n − 1 p ( ⋅ ) = δ ˜ ( ⋅ ) n = α n + 1 β − ( 1 r ( ⋅ ) + 1 p ( ⋅ ) ) = α n + 1 β − 1 q ( ⋅ ) . From the fact that p i ( ⋅ ) ≥ ( p i ) ∞ implies that p i ′ ( ⋅ ) ≤ ( p i ′ ) ∞ ( i = 1 , 2 ) , by using (1) in Lemma 2.8 and the doubling condition to obtain

∣ Q ∣ δ ( ⋅ ) / n + α / n − 2 ∏ i = 1 2 ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) ≤ C ∣ Q ∣ δ ( ⋅ ) / n + α / n − 2 ‖ χ a k 0 k Q ‖ p 1 ′ ( ⋅ ) ‖ χ a k 0 k Q ‖ p 2 ′ ( ⋅ ) ≤ C a k 0 k n / p 1 ′ ( ⋅ ) + k 0 k n / p 2 ′ ( ⋅ ) ∣ Q ∣ δ ( ⋅ ) / n + α / n − 2 ∣ Q ∣ 1 / p 1 ′ ( ⋅ ) + 1 / p 2 ′ ( ⋅ ) ≤ C a k 0 k n / p 1 ′ ( ⋅ ) + k 0 k n / p 2 ′ ( ⋅ ) ∣ Q ∣ 1 / β + α / n − 1 ∣ Q ∣ 1 / q ′ ( ⋅ ) .

If ∣ Q ∣ > 1 , then from (3) and (4) in Lemma 2.8, we obtain

∣ Q ∣ 1 q ′ ( ⋅ ) = ∣ Q ∣ 1 q ′ ( ⋅ ) − 1 q + ′ ( Q ) + 1 q + ′ ( Q ) ≤ C ‖ χ Q ‖ q ′ ( ⋅ ) ∣ Q ∣ 1 q ′ ( ⋅ ) − 1 q + ′ ( Q ) ≤ C ‖ χ Q ‖ q ′ ( ⋅ ) ∣ Q ∣ 1 q − ′ ( Q ) − 1 q + ′ ( Q ) ≤ C ‖ χ Q ‖ q ′ ( ⋅ ) .

If ∣ Q ∣ < 1 , then from (3) and (4) in Lemma 2.8, we obtain

∣ Q ∣ 1 q ′ ( ⋅ ) = ∣ Q ∣ 1 q ′ ( ⋅ ) − 1 q − ′ ( Q ) + 1 q − ′ ( Q ) ≤ C ‖ χ Q ‖ q ′ ( ⋅ ) ∣ Q ∣ 1 q ′ ( ⋅ ) − 1 q − ′ ( Q ) ≤ C ‖ χ Q ‖ q ′ ( ⋅ ) ∣ Q ∣ 1 q + ′ ( Q ) − 1 q − ′ ( Q ) ≤ C ‖ χ Q ‖ q ′ ( ⋅ ) .

Combining the above estimates to obtain

(3.4) ∣ Q ∣ δ ( ⋅ ) / n + α / n − 2 ∏ i = 1 2 ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) ≤ C a k 0 k ( n / p 1 ′ ( ⋅ ) + n / p 2 ′ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n − 1 ‖ χ Q ‖ q ′ ( ⋅ ) .

Thus, we have

I 21 + I 22 ≤ C ∣ Q ∣ δ ( x ) / n + α / n − 2 ∑ k = 1 ∞ ( a k 0 k ( δ ( x ) − 2 n − η + α ) + a − k 0 k ( 2 n − α ) ) ∏ i = 1 2 ( ‖ f i ‖ p i ( ⋅ ) ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) ) ≤ C ‖ χ Q ‖ q ′ ( ⋅ ) ∣ Q ∣ 1 / β + α / n − 1 ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ∑ k = 1 ∞ ( a k 0 k ( δ ( x ) + α − n / p 1 ( ⋅ ) − n / p 2 ( ⋅ ) − η ) + a k 0 k ( α − n / p 1 ( ⋅ ) − n / p 2 ( ⋅ ) ) ) = C ‖ χ Q ‖ q ′ ( ⋅ ) ∣ Q ∣ 1 / β + α / n − 1 ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ∑ k = 1 ∞ ( a k 0 k ( δ ˜ ( x ) − η ) + a k 0 k ( α − n / p ( ⋅ ) ) ) .

Since ( p j ) ∞ ≤ p j ( x ) ≤ ( p j ) + < 2 n / α ( j = 1 , 2 ) implies that p ( ⋅ ) < n / α and max { η 0 , δ ˜ + } < η ≤ 1 . Consequently, it results that

∣ T α , b 1 ( f 12 , f 22 ) ( x ) − ( T α , b 1 ( f 12 , f 22 ) ) Q ∣ = C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ ∫ Q ( I 21 + I 22 ) d z ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ‖ χ Q ‖ q ′ ( ⋅ ) ∣ Q ∣ 1 / β + α / n − 1 ∑ k = 1 ∞ ( a k 0 k ( δ ˜ ( x ) − η ) + a k 0 k ( α − n / p ( ⋅ ) ) ) ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n − 1 ‖ χ Q ‖ q ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

Therefore,

I 2 ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n ‖ χ Q ‖ q ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

For the term I 3 , we first estimate the difference ∣ T α , b 1 ( f 11 , f 22 ) ( x ) − ( T α , b 1 ( f 11 , f 22 ) ) Q ∣ . If x , z ∈ Q , then

∣ T α , b 1 ( f 11 , f 22 ) ( x ) − ( T α , b 1 ( f 11 , f 22 ) ) Q ∣ ≤ 1 ∣ Q ∣ ∫ Q ∣ T α , b 1 ( f 11 , f 22 ) ( x ) − T α , b 1 ( f 11 , f 22 ) ( z ) ∣ d z ≤ 1 ∣ Q ∣ ∫ Q ∣ ∫ a k 0 Q × ( R n \ a k 0 Q ) ( b 1 ( x ) − b 1 ( y 1 ) ) K α ( x , y → ) ∏ i = 1 2 f i ( y i ) d y → − ∫ a k 0 Q × ( R n \ a k 0 Q ) ( b 1 ( z ) − b 1 ( y 1 ) ) K α ( z , y → ) ∏ i = 1 2 f i ( y i ) d y → ∣ d z ≤ 1 ∣ Q ∣ ∫ Q ∫ a k 0 Q × ( R n \ a k 0 Q ) ( ∣ b 1 ( x ) − b 1 ( y 1 ) ∣ ∣ K α ( x , y → ) − K α ( z , y → ) ∣ + ∣ b 1 ( x ) − b 1 ( z ) ∣ ∣ K α ( z , y → ) ∣ ) ∣ ∏ i = 1 2 f i ( y i ) ∣ d y → d z .

Proceeding as in equation (3.3). Since 1 < β < r − such that 0 < δ ( ⋅ ) n = 1 β − 1 r ( ⋅ ) , then for every x and y , by applying Lemma 2.19 and the hypothesis, we obtain

∣ b 1 ( x ) − b 1 ( y ) ∣ ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ x − y ∣ δ ( x ) .

Therefore, using equations (1.1) and (1.2) to deduce

∣ T α , b 1 ( f 11 , f 22 ) ( x ) − ( T α , b 1 ( f 11 , f 22 ) ) Q ∣ ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ ∫ Q ∑ k = 1 ∞ ∫ ( a k 0 ( k + 1 ) Q ) 2 ∣ x − y 1 ∣ δ ( x ) ∣ x − z ∣ η ( ∣ x − y 1 ∣ + ∣ x − y 2 ∣ ) 2 n + η − α + ∣ x − z ∣ δ ( x ) ( ∣ z − y 1 ∣ + ∣ z − y 2 ∣ ) 2 n − α ∣ ∏ i = 1 2 f i ( y i ) ∣ d y → d z ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ ∫ Q ∑ k = 1 ∞ ∫ ( a k 0 ( k + 1 ) Q ) 2 ∣ a k 0 k Q ∣ δ ( x ) / n ∣ Q ∣ η / n ∣ a k 0 k Q ∣ 2 + η / n − α / n + ∣ Q ∣ δ ( x ) / n ∣ a k 0 k Q ∣ 2 − α / n ∣ ∏ i = 1 2 f i ( y i ) ∣ d y → d z ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ∑ k = 1 ∞ ( a k 0 k ( δ ( x ) − 2 n − η + α ) + a − k 0 k ( 2 n − α ) ) ∣ Q ∣ δ ( x ) / n + α / n − 2 ∏ i = 1 2 ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) .

Since 1 q ( ⋅ ) = 1 r ( ⋅ ) + 1 p ( ⋅ ) = 1 r ( ⋅ ) + 1 p 1 ( x ) + 1 p 2 ( x ) , then α n + δ ( ⋅ ) n − 1 p ( ⋅ ) = δ ˜ ( ⋅ ) n = α n + 1 β − ( 1 r ( ⋅ ) + 1 p ( ⋅ ) ) = α n + 1 β − 1 q ( ⋅ ) . From the fact that p i ( ⋅ ) ≥ ( p i ) ∞ implies that p i ′ ( ⋅ ) ≤ ( p i ′ ) ∞ ( i = 1 , 2 ) , by using (1) in Lemma 2.8 and the doubling condition to obtain

∣ Q ∣ δ ( ⋅ ) / n + α / n − 2 ∏ i = 1 2 ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) ≤ C a k 0 k ( n / p 1 ′ ( ⋅ ) + n / p 2 ′ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n − 1 ∣ Q ∣ 1 / q ′ ( ⋅ ) .

Proceeding as in equation (3.4), using (3) and (4) in Lemma 2.8, we obtain

∣ Q ∣ δ ( ⋅ ) / n + α / n − 2 ∏ i = 1 2 ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) ≤ C a k 0 k ( n / p 1 ′ ( ⋅ ) + n / p 2 ′ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n − 1 ‖ χ Q ‖ q ′ ( ⋅ ) .

Since ( p j ) ∞ ≤ p j ( x ) ≤ ( p j ) + < 2 n / α ( j = 1 , 2 ) implies that p ( ⋅ ) < n / α , and max { η 0 , δ ˜ + } < η ≤ 1 . Consequently, it results that

∣ T α , b 1 ( f 11 , f 22 ) ( x ) − ( T α , b 1 ( f 11 , f 22 ) ) Q ∣ ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ∑ k = 1 ∞ ( a k 0 k ( δ ( x ) − 2 n − η + α ) + a − k 0 k ( 2 n − α ) ) ∣ Q ∣ δ ( x ) / n + α / n − 2 ∏ i = 1 2 ‖ χ a k 0 ( k + 1 ) Q ‖ p i ′ ( ⋅ ) ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n − 1 ‖ χ Q ‖ q ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ∑ k = 1 ∞ ( a k 0 k ( δ ˜ ( x ) − η ) + a k 0 k ( α − n / p ( ⋅ ) ) ) ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n − 1 ‖ χ Q ‖ q ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

Therefore,

I 3 ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n ‖ χ Q ‖ q ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

For I 4 , similar to the estimate of I 3 , we can obtain

I 4 ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∣ Q ∣ 1 / β + α / n ‖ χ Q ‖ q ′ ( ⋅ ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

Combining these estimates, we obtain

1 ∣ Q ∣ 1 / β + α / n ‖ χ Q ‖ q ′ ( ⋅ ) ∫ Q ∣ T α , b 1 ( f → ) ( x ) − C Q ∣ d x ≤ C ‖ b 1 ‖ L ( δ ( ⋅ ) ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) .

Furthermore, we have

1 ∣ Q ∣ 1 / β + α / n ‖ χ Q ‖ q ′ ( ⋅ ) ∫ Q ∣ T α , b j ( f → ) ( x ) − C Q ∣ d x ≤ C ‖ b j ‖ L ( δ ( ⋅ ) ) ∏ i = 1 2 ‖ f i ‖ p i ( ⋅ ) ,

which implies that T α , b j ( f → ) ∈ L ( δ ˜ ( ⋅ ) ) ( j = 1 , 2 ) .

The proof is completed.□

Acknowledgments

The authors cordially thank the anonymous referees who gave valuable suggestions and useful comments, which have led to the improvement of this article.

  1. Funding information: This work was partly supported by the National Natural Science Foundation of China (Grant No. 11571160), Scientific Project-HLJ (No. 2019-KYYWF-0909, 1355ZD010), and the Reform and Development Foundation for Local Colleges and Universities of the Central Government (No. 2020YQ07).

  2. Author contributions: All authors contributed equally to the writing of this article. All authors read the final manuscript and approved its submission.

  3. Conflict of interest: The authors state that there is no conflict of interest.

  4. Data availability statement: All data generated or analysed during this study are included in this published article.

References

[1] A. L. Bernardis, S. Hartzstein, and G. G. Pradolini, Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type, J. Math. Anal. Appl. 322 (2006), 825–846. 10.1016/j.jmaa.2005.09.051Search in Google Scholar

[2] A. L. Bernardis, G. G. Pradolini, M. Lorente, and M. S. Riveros, Composition of fractional Orlicz maximal operators and A1-weights on spaces of homogeneous type, Acta Math. Sinica, English Series 26 (2010), 1509–1518. 10.1007/s10114-010-8445-4Search in Google Scholar

[3] A. L. Bernardis, E. D. Dalmasso, and G. G. Pradolini, Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces, Annales Academiæ Scientiarum Fennicæ Mathematica 39 (2014), 23–50. 10.5186/aasfm.2014.3904Search in Google Scholar

[4] A. Cabral, G. G. Pradolini, and W. A. Ramos, Extrapolation and weighted norm inequalities between Lebesgue and Lipschitz spaces in the variable exponent context, J. Math. Anal. Appl. 436 (2016), 620–636. 10.1016/j.jmaa.2015.12.020Search in Google Scholar

[5] S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), 7–16. 10.1512/iumj.1982.31.31002Search in Google Scholar

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

[7] R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635. 10.2307/1970954Search in Google Scholar

[8] R. R. Coifman and Y. Meyer, Commutateurs daintégrales singulières et opérateurs multilinéaires, in: Annales de lainstitut Fourier, vol. 28, Imprimerie Louis-Jean, Gap, 1978, pp. 177–202. 10.5802/aif.708Search in Google Scholar

[9] D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, The maximal function on variable Lp spaces, Annales Academiæ Scientiarum Fennicæ Mathematica. 28 (2003), 223–238. Search in Google Scholar

[10] D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, The boundedness of classical operators on variable Lp spaces, Annales Academiæ Scientiarum Fennicæ Mathematica. 31 (2006), 239–264. Search in Google Scholar

[11] D. Cruz-Uribe, L. Diening, and P. Hästö, The maximal operator on weighted variable Lebesgue spaces, Fract. Calculus Appl. Anal. 14 (2011), 361–374. 10.2478/s13540-011-0023-7Search in Google Scholar

[12] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer Science & Business Media, Heidelberg, 2013. 10.1007/978-3-0348-0548-3Search in Google Scholar

[13] D. Cruz-Uribe and L.-A. D. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), 447–493. 10.1512/iumj.2014.63.5232Search in Google Scholar

[14] L. Diening and M. Růzicka, Calderón-Zygmund operators on generalized Lebesgue spaces Lp(⋅) and problems related to fluid dynamics, J. für die Reine und Angewandte Mathematik 563 (2003), 197–220. 10.1515/crll.2003.081Search in Google Scholar

[15] L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bulletin des Sciences Mathématiques 129 (2005), 657–700. 10.1016/j.bulsci.2003.10.003Search in Google Scholar

[16] L. Diening, P. Harjulehto, P. Hästö, and M. Růzicka, Lebesgue and Sobolev spaces with variable exponents, vol. 2017 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2011. 10.1007/978-3-642-18363-8Search in Google Scholar

[17] X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446. 10.1006/jmaa.2000.7617Search in Google Scholar

[18] J. García-Cuerva and J. L. R. De Francia, Weighted norm inequalities and related topics, vol 116 of North-Holland Mathematics Studies, Elsevier Science Publishers B.V., Netherlands, 1985. Search in Google Scholar

[19] L. Grafakos, On multilinear fractional integrals, Studia Math. 102 (1992), 49–56. 10.4064/sm-102-1-49-56Search in Google Scholar

[20] L. Grafakos and R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), 124–164. 10.1006/aima.2001.2028Search in Google Scholar

[21] L. Grafakos and D. He, Multilinear Calderón-Zygmund operators on Hardy spaces, II, J. Math. Anal. Appl. 416 (2014), 511–521. 10.1016/j.jmaa.2014.02.051Search in Google Scholar

[22] V. Guliyev and M. Omarova, Multilinear singular and fractional integral operators on generalized weighted Morrey spaces, Azerbaijan J. Math. 5 (2015), 104–132. 10.1155/2015/594323Search in Google Scholar

[23] E. Harboure, O. Salinas, and B. Viviani, Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces, Trans. Amer. Math. Soc. 349 (1997), 235–255. 10.1090/S0002-9947-97-01644-9Search in Google Scholar

[24] S. Janson, Mean oscillation and commutators of singular integral operators, Arkiv för Matematik 16 (1978), 263–270. 10.1007/BF02386000Search in Google Scholar

[25] F. John and L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math. 14 (1961), 415–426. 10.1002/cpa.3160140317Search in Google Scholar

[26] C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), 1–15. 10.4310/MRL.1999.v6.n1.a1Search in Google Scholar

[27] O. Kováčik and J. Rákosník, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41 (1991), 592–618. 10.21136/CMJ.1991.102493Search in Google Scholar

[28] A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres, and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009), 1222–1264. 10.1016/j.aim.2008.10.014Search in Google Scholar

[29] W. Li, John-Nirenberg inequality and self-improving properties, J. Math. Res. Exposition 25 (2005), 42–46. Search in Google Scholar

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

[31] K. Moen, Weighted inequalities for multilinear fractional integral operators, Collectanea Math. 60 (2009), 213–238. 10.1007/BF03191210Search in Google Scholar

[32] B. Muckenhoupt and R. Wheeden, Weighted bounded mean oscillation and the Hilbert transform, Studia Math. 3 (1976), 221–237. 10.4064/sm-54-3-221-237Search in Google Scholar

[33] A. Nekvinda, Hardy-Littlewood maximal operator on Lp(x)(Rn), Math. Inequal. Appl. 7 (2004), 255–266. 10.7153/mia-07-28Search in Google Scholar

[34] M. Paluszyński, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg, and Weiss, Indiana Univ. Math. J. 44 (1995), 1–17. 10.1512/iumj.1995.44.1976Search in Google Scholar

[35] C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), 163–185. 10.1006/jfan.1995.1027Search in Google Scholar

[36] C. Pérez and R. Trujillo-González, Sharp weighted estimates for multilinear commutators, J. London Math. Soc. 65 (2002), 672–692. 10.1112/S0024610702003174Search in Google Scholar

[37] G. G. Pradolini, A class of pairs of weights related to the boundedness of the fractional integral operator between Lp and Lipschitz spaces, Commentationes Mathematicae Universitatis Carolinae 42 (2001), 133–152. Search in Google Scholar

[38] G. G. Pradolini, Two-weighted norm inequalities for the fractional integral operator between Lp and Lipschitz spaces, Commentationes Mathematicae 41 (2001), 147–169. Search in Google Scholar

[39] G. G. Pradolini and W. A. Ramos, Characterization of Lipschitz functions via the commutators of singular and fractional integral operators in variable Lebesgue spaces, Potential Anal. 46 (2017), 499–525. 10.1007/s11118-016-9592-8Search in Google Scholar

[40] M. J. Ramseyer, O. Salinas, and B. Viviani, Lipschitz type smoothness of the fractional integral on variable exponent spaces, J. Math. Anal. Appl. 403 (2013), 95–106. 10.1016/j.jmaa.2012.12.074Search in Google Scholar

[41] M. J. Ramseyer, Operators in generalized Lebesgue spaces (Operadores en espacios de Lebesgue generalizados), Ph.D. thesis, Facultad de Ingeniería Química, Universidad Nacional del Litoral, 2013. Search in Google Scholar

[42] E. T. Sawyer, C.-Y. Shen, and I. Uriarte-Tuero, A Two Weight Theorem for α-Fractional Singular Integrals in Higher Dimension, 2013, arXiv: http://arXiv.org/abs/arXiv:1305.5104v7. Search in Google Scholar

[43] E. T. Sawyer, C.-Y. Shen, and I. Uriarte-Tuero, A two weight theorem for α-fractional singular integrals with an energy side condition, Revista Matemática Iberoamericana 32 (2016), 79–174. 10.4171/RMI/882Search in Google Scholar

[44] J. Tan, Z. Liu, and J. Zhao, On some multilinear commutators in variable Lebesgue spaces, J. Math. Inequal. 11 (2017), 715–734. 10.7153/jmi-11-57Search in Google Scholar

[45] H. Wang and J. Xu, Multilinear fractional integral operators on central Morrey spaces with variable exponent, J. Inequal. Appl. 2019 (2019), 1–23. 10.1186/s13660-019-2264-7Search in Google Scholar

[46] J. Wu and P. Zhang, Characterization of Lipschitz functions via commutators of multilinear singular integral operators in variable Lebesgue spaces, Acta Mathematica Sinica, English Series (2023), In Press. 10.1007/s10114-023-2164-0Search in Google Scholar

[47] Q. Xue, Weighted estimates for the iterated commutators of multilinear maximal and fractional-type operators, Studia Math. 217 (2013), 97–122. 10.4064/sm217-2-1Search in Google Scholar
Received: 2023-01-03
Accepted: 2023-04-10
Published Online: 2023-07-05

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

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

Articles in the same Issue

  1. Research Articles
  2. Separation functions and mild topologies
  3. A non-geodesic analogue of Reshetnyak’s majorization theorem
  4. Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard spaces
  5. On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow
  6. Exceptional families of measures on Carnot groups
  7. Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces
  8. Identifying 1-rectifiable measures in Carnot groups
  9. Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces
  10. Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities
  11. Extremal subsets in geodesically complete spaces with curvature bounded above
https://doi.org/10.1515/agms-2022-0153
Keywords for this article
multilinear commutator; multilinear fractional integral operator; Lipschitz function space; variable exponent
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Separation functions and mild topologies
  3. A non-geodesic analogue of Reshetnyak’s majorization theorem
  4. Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard spaces
  5. On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow
  6. Exceptional families of measures on Carnot groups
  7. Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces
  8. Identifying 1-rectifiable measures in Carnot groups
  9. Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces
  10. Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities
  11. Extremal subsets in geodesically complete spaces with curvature bounded above
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 19.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/agms-2022-0153/html
Scroll to top button