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

: 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 general - ization of the classical sharp maximal operator


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: 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 n be an n-dimensional Euclidean space and ( ) = ×⋯× n m n n be an m-fold product space ( ∈ m ). We denote by S ( ) n the space of all Schwartz functions on n and by S ( ) ′ n its dual space, the set of all tempered distributions on n . Let ( ) ∞ C c n denote the set of smooth functions with compact support in n .
The following notations can be found from [22,42,43].  , and < ≤ < ∞ p q 1 , thenT α is called as a standard m-linear fractional Calderón-Zygmund operator. Let be a collection of locally integrable functions, then the m-linear commutator of T α with → b is defined as follows: , α when = m 1. To clarify the notation, the commutators can be formally written as follows: the Lipschitz spaces and the variable Lipschitz spaces (see Definition 2.10), respectively. Let 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 m 1, 2, , ) belonging to ( ) δ or ( ( )) ⋅ 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 ( ) n log , and P ( ) n .
Theorem 1.1. Let T α be an m-linear fractional Calderón-Zygmund operator given in (1.4). Suppose that be a collection of locally integrable functions. Assume that the associated kernel K α of T α is a homogeneous of degree ( ) − − mn α and the Fourier series of (i) The above result gives a characterization of the Lipschitz spaces ( ) δ in terms of the boundedness of → T α b ,Σ between variable Lebesgue spaces. (ii) 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. (iii) When = α 0, the above result coincides with Theorem 1.1 in [46].
be a collection of locally integrable functions. Assume that the associated kernel K α of T α is a homogeneous of degree ( ) − − mn α and the Fourier series of 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 belonging to the products of variable Lipschitz spaces.
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 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 tQ stands for the cube with the same center as Q and with side length ( ) ( ) = l tQ tl Q . Denote by | | S the Lebesgue measure and by χ S the characteristic function for a measurable set ⊂ S n . ( ) B x r , means the ball is centered at x and of radius r, and , we denote by ( ) ′ q x its conjugate index, namely, We will occasionally use the notational , , , m 1 for convenience. For a set E and a positive integer m, we will use the notation sometimes.

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.
L q n is defined as follows: is defined as follows: for all compact subsets . q n q n loc Next, we define some classes of variable exponent functions. Given a function ( ) ∈ f L n loc 1 , the Hardy-Littlewood maximal operator M is defined as follows: Now, we introduce the log-Hölder continuity.
x y x y ln , 12 , , , n where C denotes a universal positive constant that may differ from line to line, and C does not depend on x y , .
(ii) The set C ( ) ∞ n log consists of all log-Hölder continuous functions ( ) ⋅ q at infinity satisfies condition was originally defined in this form in [9].

Auxiliary propositions and lemmas
Next, we give some auxiliary propositions and lemmas we need.
The following conditions are equivalent: 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]).
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][28][29][30]. The generalized Hölder's inequality in Orlicz space (see [28,35,36] for details and the more general cases).
and Q be a cube in n . Then, , , m n 1 satisfy the condition Then, for any ( ) The following results are also needed.
Then, there exists a positive constant C such that the inequality holds for every cube ⊂ Q n and a.e. ∈ x Q (see Lemma 4.4 in [39] or p. 126, Corollary 4.5.9 in [16]).
is the harmonic mean of ( ) ⋅ q . Then, the following conditions are equivalent (see Theorem 4.5.7 in [16] Proposition 4.66 in [12]): uniformly for all cubes ⊂ Q n .
, 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]).
where a is given in equation (2.4), and let = / η k 1 0 0 (which will be used in the following sections). Note that, if P ( ) ( ) ⋅ ∈ p n log , the estimates (2.4) and (2.5) imply the doubling condition for the functional ( ) if there exists a positive constant C such that for every cube Q, the inequality holds. When = γ 0, the inequality above is the ( ) ( ) ⋅ p n class given by Cruz-Uribe et al. in [11], which characterizes the boundedness of the Hardy-Littlewood maximal operator on ( ) The first part of the following results was proved in [3] (see Lemma 4.1) and gives a relation between the ( ) classes (see also Lemma 4.14 in [39]). On the basis of the first part, note that if Some notations of Lipschitz-type function spaces are stated as follows.
The space Λ δ of the Lipschitz continuous functions with order δ is defined as follows: where f is the locally integrable function on n , and the smallest constant > C 0 will be denoted as the Lipschitz norm by ‖ ‖ f Λδ .
1. The space ( ) δ is defined to be the set of all locally integrable functions f , i.e., there exists a positive constant > C 0 such that where the supremum is taken over every cube ⊂ Q n and ( ) The least constant C will be denoted by and an exponent function . We say that a locally integrable function f belongs to ( ) The least constant C will be denoted by defined by the set of the measurable functions f such that (see [40] for more details) , see [37] or [38]) Let w be a weight and ≤ < δ 0 1, we say that a locally integrable function f belongs to ( ) δ w if there exists a positive constant C such that the inequality holds for every ball ⊂ B n . The least constant C will be denoted by Remark 2.11. (i) In (1) of Definition 2.10, it is well known that the space Λ δ coincides with the space ( ) δ (see [23,39]). (ii) In (2) of Definition 2.10, it is not difficult to see that, for = δ 0, the space ( ) δ coincides with the space of bounded mean oscillation functions BMO (see [25]). (iii) 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]): (iv) In (4) of Definition 2.10, denote { } z max , 0 by + z (see [4]). In addition, when ( ) = r x r is constant, ( ( )) ⋅ δ coincides with the space ( ) / − / n β n r .
(v) In (5) of Definition 2.10, it is not difficult to see that, for = δ 0, the space ( ) δ w coincides with one of the versions of weighted bounded mean oscillation spaces (see [32] and ∈ ∞ a T . Then, The following inequalities are also necessary (see (2.16) . (i) Then, there is a positive constant = C C p q , such that for any measurable function f, we have , then there is a positive constant = C C p q , such that for any measurable function f, we have The following results can be obtained from [26] or [45].
The following definition of the multilinear fractional integral operator was considered by several authors (see [19,26,31,44]). . The multilinear fractional integral is defined as follows: If we take = m 1, then 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].
. Define the variable exponent ( ) ⋅ q as follows: Then, there exists a positive constant C such that (2) (See Lemma 3.8 in [39]) Then, there is a positive constant C such that for every ∈ j , we have The following pointwise results can be founded in [41].
, then, for every > s 0 and ∈ x n , there exists a positive constant C such that

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 n .
(1) Set ≤ < δ 0 1. The δ-sharp maximal operator is defined as follows: where the supremum is taken over all cube ⊂ Q n containing x, and | | ( ) denotes the average of f over the cube ⊂ Q n .
(ii) For any > γ 0, the generalization of the operator ( ) ⋅ ♯ f δ is defined as follows: and > ε 0, define the following operators via is the Luxemburg-type average defined as follows:

Definition 2.21. (Multilinear fractional maximal functions) For all locally integrable functions
and ∈ x n , ≤ < α mn 0 , (1) the multilinear maximal fractional functions α and r α , are defined as follows: where the supremum is taken over all the cubes Q containing x.
The following result is a generalization to the variable context of a pointwise estimate of commutators.
Then, there exists a positive constant C such that 1, it is enough to show that, for some constant C Q , there exists a positive constant C such that where : there is at least one 0 m j 1 . Let λ be some positive constant to be chosen. It is easy to see that By taking , we obtain that Let us first estimate I . By taking < < / r ε γ 1 and using Hölder's inequality, we obtain that To estimate III, we first consider the case when . Note that, for ∈ w z Q , and any ( , .    From equations (2.9)-(2.11), there holds the following inequality, which will be used later,   , similar to the above, we have that  (1) We first prove 1 2 . Assume that Note that the homogeneity of K α , set , we may assume, without loss of do not play any significant role in the proof.  , i.e., ( , 2 x y l x y l 1 2 , which means that ( ) − − x y x y , 1 2 is bounded away from the singularity of K α . Without loss of generality, let ( ) ( ) ( ) (1) We first prove 1 2 . Assume that (2) Now, we prove the second part. Assume that