The Commutators of Multilinear Maximal and Fractional-Type Operators on Central Morrey Spaces with Variable Exponent

We show that the maximal operator associated with multilinear Calderón-Zygmund singular integrals and its commutators are bounded on products of central Morrey spaces with variable exponent. Moreover, some bounded properties are obtained for the commutators of multilinear Calderón-Zygmund operators as well as for the corresponding fractional integrals.


Introduction
Let T be a multilinear operator initially defined on the m -fold product of Schwartz spaces and taking values into the space of tempered distributions, Following Grafakos and Torres [1], we say that T is an m -linear Calderón-Zygmund operator if, for some 1 ≤ p 1 , ⋯, p m < ∞, it extends to a bounded multilinear operator from L p 1 ðℝ n Þ × ⋯ × L p m ðℝ n Þ to L p ðℝ n Þ, where 1/p = 1/p 1 + ⋯+1 /p m , and if there exists a function K, defined away from the diagonal x = y 1 = ⋯ = y m in ðℝ n Þ m+1 , satisfying for all x ∉ T m j=1 supp f j and f 1 , ⋯, f m are C ∞ functions with compact support, for some A > 0 and all ðx, y 1 , ⋯, y m Þ ∈ ðℝ n Þ m+1 with x ≠ y k for some k, and for some ε > 0, whenever jx − x ′ j ≤ ð1/2Þ max fjx − y 1 j, ⋯, jx − y m jg, and also that whenever jy i − y i ′ j ≤ ð1/2Þ max fjx − y 1 j, ⋯, jx − y m jg for all 1 ≤ i ≤ m. Multilinear Calderón-Zygmund operators were introduced by Coifman and Meyer [2,3] in the 70s and were systematically studied by Grafakos and Torres [1]. They showed that multilinear Calderón-Zygmund operators map L p 1 × ⋯ × L p m into L p for some 1 < p 1 , ⋯, p m < ∞ and some 0 < p < ∞ with 1/p = 1/p 1 + ⋯+1/p m . Moreover, multilinear Calderón-Zygmund operators satisfy weak endpoint bounds when at least one p i is equal to one. Lerner et al. [4] established the multiple weights theory for multilinear Calderón-Zygmund operators and obtained some weighted estimates for these operators and their commutators. Grafakos and Kalton [5] studied the boundedness of these operators on products of Hardy spaces. Subsequently, under some additional conditions, Hu and Meng [6] proved the boundedness of these operators from product Hardy spaces into Hardy spaces.
A function b ∈ L loc ðℝ n Þ is said to belong to space BMO ðℝ n Þ, if where B is a ball of ℝ n with center at the origin and radius R, jBj is the Lebesgue measure of B, and b B ≔ ð1/jBjÞ Ð B f ðxÞdx.
where each term is the commutator of b j and T in the jth entry of T, that is, Also, the commutators of b ! and the maximal multilinear Calderón-Zygmund operator T * are defined by where each term is the commutator of b j and T * in the jth entry of T * , that is, This definition coincides with the commutator of Coifman et al. [17] when m = 1. The multilinear commutators T b ! were considered by Lerner et al. in [4]; they proved that if b ! ∈ ðBMOÞ m , 1 < p 1 , ⋯, p m < ∞, and p defined by 1/p = 1/p 1 + ⋯+1/p m , then where For further contributions in the study of the multilinear commutators T b ! and T * b !, we refer to [18,19] and the references therein.
On the other hand, as one of the most important operators, the multilinear fractional operator has also been intensively studied in the recent years. Kenig and Stein [20] considered the following multilinear fractional operator I α , 0 < α < mn, where m, n denote the nonnegative integers with m ≥ 1, n ≥ 2. They proved that I α is of strong type ðL p 1 × ⋯ × L p m , L q Þ and weak type ðL p 1 × ⋯ × L p m , L q,∞ Þ. Moen [21] developed a weighted theory that adapts to the multilinear fractional integral operators. He established the multiple weighted norm inequalities for the multilinear fractional integral operators and the corresponding multilinear fractional maximal operators. Following the work of [4], we define the commutators of b ! and the multilinear fractional operator I α as follows: where each term is the commutator of b j and I α in the jth entry of I α , that is, Journal of Function Spaces The commutators I α, b ! were first studied by Chen and Xue [22], in which some weighted strong bounds and Lðlog LÞ type endpoint estimates for I α, b ! are obtained. Lu and Yang [23,24] introduced the central bounded mean oscillation space C _ MO q ðℝ n Þ, 1 < q < ∞, which satisfies the following condition: This space can be regarded as a local version of BMOðℝ n Þ at the origin, but they have quite different properties. For example, the well-known John-Nirenberg inequality shows that the functions in BMOðℝ n Þ can be described by means of the condition However, the space C _ MO q ðℝ n Þ depends on q. More precisely, if q 1 < q 2 , then C _ MO q 2 ðℝ n Þ ⊂ C _ MO q 1 ðℝ n Þ. Therefore, there is no analogy of the classical John-Nirenberg inequality of BMOðℝ n Þ for the space C _ MO q ðℝ n Þ. One can imagine that the behavior of C _ MO q ðℝ n Þ may be quite different from that of BMOðℝ n: Þ. Definition 2. Let λ < 1/n and 1 < q < ∞. The λ-central bounded mean oscillation space C _ MO q,λ ðℝ n Þ is defined by Remark 3. When λ = 0, the space C _ MO q,λ ðℝ n Þ is just the space C _ MO q ðℝ n Þ defined above. If two functions which differ by a constant are regarded as a function in the space C _ MO q,λ ðℝ n Þ, then it becomes a Banach space. Obviously, (20) is equivalent to the following condition: Definition 4. Let λ ∈ ℝ and 1 < q < ∞. The central Morrey space B q,λ ðℝ n Þ consists of all f ∈ L q loc ðℝ n Þ such that The above λ-central BMO spaces and central Morrey spaces were introduced in Alvarez et al. [25]; since then, various operators were studied in these spaces. For some recent development, we mention that Fu et al. [26] established λ-central BMO estimates for a class of multisublinear operators on the product of central Morrey spaces. As its special cases, the corresponding results of multilinear Calderón-Zygmund operators can be deduced. Si and Xue [18] established λ-central BMO estimates for commutators of maximal multilinear Calderón-Zygmund operators and multilinear fractional operators on central Morrey spaces. In the last two decades, following the fundamental work of Kovácik and Rákosnk [27] on variable Lebesgue spaces L pð·Þ ðℝ n Þ and variable Sobolev spaces W k,pð·Þ ðℝ n Þ (here and below, the exponent pð·Þ is a function and not a constant), function spaces with variable exponent, such as variable Morrey spaces, variable Herz spaces, and variable Hardy spaces, have been widely studied by a significant number of authors; see [28][29][30][31][32][33][34][35][36][37] and the references therein. These spaces are of interest in their own right and also have applications to image restoration [38], fluid dynamics [39], and PDEs with nonstandard growth conditions [40].
In 2019, Fu et al. [41] introduced the λ-central BMO spaces and the central Morrey spaces with variable exponent and proved the boundedness of the fractional singular integrals and its commutator on those spaces. Subsequently, Wang and Xu [42] further obtained the boundedness of multilinear fractional integral operators and their commutators on central Morrey spaces with variable exponent. The λ -central BMO estimates for m-linear Calderón-Zygmund operators and their commutators on the product of central Morrey spaces with variable exponent are independently obtained by Wang [43] and Wang et al. [44].
Motivated by [18,41], the aim of this paper is to establish λ-central BMO estimates for the maximal multilinear Calderón-Zygmund operators T * and its commutators T * b ! on central Morrey spaces with variable exponent. Moreover, the similar boundedness properties for the multilinear commutators T b ! and I α, b ! are obtained. Throughout this paper, the symbol ℕ stands for the set of all natural numbers. We denote Bð0, RÞ ≔ fy ∈ ℝ n : jyj < rg simply by B, and Bð0, lRÞ by lB is a measurable set, then jEj means the Lebesgue measure of E and χ E denotes its characteristic function. p ′ ð·Þ is the conjugate exponent function defined by 1/pð·Þ + 1/p′ð·Þ = 1. The letter C stands for a positive constant whose value may change from appearance to appearance.

Preliminaries and Main Results
Let us first recall some basic properties of Lebesgue spaces with variable exponent; we refer to the surveys [45,46] and the monographs [47,48] for further details.
The variable Lebesgue spaces are a generalization of the classical L p spaces with the exponent p replaced by a measurable function pð·Þ: ℝ n ⟶ ð0,∞Þ. It consists of all measurable functions f on ℝ n such that 3 Journal of Function Spaces If ðxÞ ≥ 1 a.e., then this becomes a Banach space when equipped with the norm Given an open set Ω ⊂ ℝ n , the space L pð·Þ loc ðΩÞ is defined by In what follows, we define P ðℝ n Þ to be the set of measurable function pð·Þ: ℝ n ⟶ ½1,∞Þ such that where M is the Hardy-Littlewood maximal operator defined by A measurable function pð·Þ ∈ P ðℝ n Þ is called globally log-H€ older continuous if it satisfies The set of pð·Þ satisfying (28) and (29) is denoted by LHðℝ n Þ. In Cruz-Uribe et al. [49], Theorem 1.1 shows that if pð·Þ ∈ P ðℝ n Þ T LHðℝ n Þ, then pð·Þ ∈ Bðℝ n Þ. Suppose pð·Þ ∈ P ðℝ n Þ, then for all f ∈ L pð·Þ ðℝ n Þ and all g ∈ L p ′ ð·Þ ðℝ n Þ, the generalized H€ older inequality holds in the form ð with r p = 1 + 1/p − − 1/p + ; see [27], Theorem 14. The next Lemmas 5 and 6 are due to Izuki [33], Page 203.
(i) For all cubes (or balls) jQj ≤ 2 n and any x ∈ Q, we have (ii) For all cubes (or balls) jQj ≥ 1, we have where p ∞ = lim x⟶∞ pðxÞ: The proofs of Lemmas 7 and 8 can be found in [48]. Tan et al. [50] obtain the following result.
Our main results can be stated as follows.

Journal of Function Spaces
For E 1 , noting that T * is of type ðL p 1 ð·Þ × ⋯× L p m ð·Þ , L pð·Þ Þ (see [44], Theorem 1.3]), by Lemmas 6 and 8, we deduce that For E 2 , since jðx − y 1 , x − y 2 Þj 2n ≥ jx − y 2 j 2n~j 2 l Bj 2 for x ∈ B and y 2 ∈ ð2 l BÞ c , using H€ older's inequality and Lemma 5, we have Therefore, we have For E 3 , as in the estimation of E 2 , we get For E 4 , we note that jðx − y 1 , x − y 2 Þj 2n ≥ jx − y 1 j n · jx − y 2 j n~j 2 l 1 Bj · j2 l 2 Bj for x ∈ B, y 1 ∈ ð2 l BÞ c and y 2 ∈ ð2 l BÞ c , an application of H€ older's inequality and Lemma 5 gives Hence, we derive the estimate The proof of Theorem 2.1 is complete.

Proof of Theorem 2.2
By linearity, it is enough to consider the operator with only one symbol. Fix then b 1 ∈ C _ MO s 1 ð·Þ,u 1 ðℝ n Þ and consider the operator Without loss of generality, we may assume that m = 2.

Proof of Theorem 2.3
Fix b 1 ∈ C _ MO s 1 ð·Þ,β 1 ðℝ n Þ; as in the proof of Theorem 2.2, we consider the operator Without loss of generality, we may assume that m = 2.