Research Article | Open Access
Multilinear Singular Integrals and Commutators on Herz Space with Variable Exponent
The paper establishes some sufficient conditions for the boundedness of singular integral operators and their commutators from products of variable exponent Herz spaces to variable exponent Herz spaces.
In recent years, the interest in multilinear analysis for studying boundedness properties of multilinear integral operators has grown rapidly. The subject was founded by Coifman and Meyer  in their seminal work on singular integral operators like Calderón commutators and pseudosdifferential operators having multiparameter function input. Subsequently, many authors including Christ and Journé , Kenig and Stein , and Grafakos and Torres  have substantially added to the exiting theory.
Let be a locally integrable function defined away from the diagonal in , which for satisfies the estimates and for , whenever , and whenever for all . Then is called -linear Calderón-Zygmund kernel. In this paper, we consider an -linear singular integral operator associated with the kernel , which is initially defined on product of the Schwartz space and takes its values in the space of tempered distribution such that for , where , the space of compactly supported bounded functions. If is bounded from to with and , then we say that is an -linear Calderón-Zygmund operator. It has been proved in  that is a bounded operator on product of Lebesgue spaces and endpoint weak estimates hold. For the boundedness of and its commutators on the product of Herz-type spaces we refer the reader to see [5, 6] and , respectively.
In the last few decades, however, a number of research papers have appeared in the literature which study the boundedness of integral operators, including the maximal function, singular operators, and fractional integral and commutators on function spaces with nonstandard growth conditions. Such kind of spaces is named as variable exponent function spaces which include variable exponent Lebesgue, Sobolev, Lorentz, Orlicz, and Herz-type function spaces. Among them the most fundamental and widely explored space is the Lebesgue space with the exponent depending on the point of the space. We will describe it briefly in the next section; however, we refer to the book  and the survey paper  for historical background and recent developments in the theory of spaces. Despite the progress made, the problems of boundedness of multilinear singular integral operators and their commutators on spaces remain open. Recently, Huang and Xu  proved the boundedness of such integral operators on the product of variable exponent Lebesgue space. Motivated by their results, in this paper we will study the multilinear singular integral on Herz space with variable exponent. Similar results for the boundedness of commutators generated by these operators and BMO functions are also provided.
Herz-type spaces are an important class of function spaces in harmonic analysis. In [11, 12], Izuki independently introduced Herz space with variable exponent , by keeping the remaining two exponents and as constants. Variability of alpha was recently considered by Almeida and Drihem  in proving the boundedness results for some classical operators on such spaces. More recently, Samko  introduced the generalized Herz-type spaces where all the three exponents were allowed to vary. In this paper, we will study the multilinear singular operators on variable exponent Herz space introduced in .
Throughout this paper, denotes a positive constant which may change from one occurrence to another. The next section contains some basic definitions and the main results of this study. Finally, the last section includes the proofs of main results along with some supporting lemmas.
2. Main Results
Let be a measurable subset of with Lebesgue measure . Given a measurable function , then for some we define the variable exponent Lebesgue space as and the space as The Lebesgue space becomes a Banach function space when equipped with the norm
Given a locally integrable function on , the Hardy-Littlewood maximal operator is defined by where . We denote We also define
Proposition 1. Let be an open set. If satisfies then one has .
Define to be the set of measurable functions such that Given , one can define the space as above. Since we will not use it in the this paper, we omit the details here and refer the reader to see [10, 17].
Let , and be the characteristic function of the set for .
Definition 2. For , and , the homogeneous Herz space with variable exponent is defined by where
In the sequel, unless stated otherwise, we will work on the whole space and will not mention it. Taking and in , Huang and Xu in  define the three commutator operators for suitable functions and . One of them is the operator As corollaries of their main results they give the following estimates.
Theorem A. Let be 2-linear Calderón-Zygmund operator and . If such that , then there exists a constant independent of the functions , and such that hold, where for and .
Theorem B. Let be 2-linear Calderón-Zygmund operator, , and . If such that , then there exists a constant independent of the functions , and such that hold, where for and .
Motivated by these results, here we give the following two theorems.
Theorem 3. Let be 2-linear Calderón-Zygmund operator and . Furthermore, let , , , , where are constants defined in the next section such that , , and . If is bounded from to , then hold for all , , where for and .
Theorem 4. Let be 2-linear Calderón-Zygmund operator, , and . Furthermore, let , , , , where are constants defined in the next section, such that , , . If is bounded from to , then hold for all , , where for and .
3. Proofs of the Main Results
In this section, we will prove main results stated in the last section. The ideas of these proofs mainly come from [5, 7]. We use the notation to denote the conjugate index of . Here we give some lemmas which will be helpful in proving Theorems 3 and 4.
Lemma 6 (see ). If , then there exist a constant such that for all balls in ,
Lemma 7 (see ). If , then there exists constants such that for all balls in and all measurable subsets ,
Recently, Izuki  established a relationship between Lebesgue space with variable exponent and space which can be stated in the form of the following lemma.
Lemma 9. One has that for all and all with ,
The next lemma is the generalized Minkowski's inequality and is useful in proving vector valued inequalities.
Lemma 10 (see ). If , then there exists a constant such that for all sequences of functions satisfying ,
Proof of Theorem 3. In order to make computations easy, first we have to prove the following inequality:
where for , ,
If , , then we have ; hence by the boundedness of , we obtain
In the other cases, we see that , for , , . Thus by the generalized Hölder's inequality,
Applying Lemma 5, we have
Now, for , , by Lemmas 6 and 8, it is easy to show that
By a similar argument for , , we get
In view of (29)–(33), (27) is obvious. Now by Minkowski’s inequality and (27), we get Since , then by definition
It remains to show that and . By symmetry, we only approximate . If , then by the well-known inequality and the inequality we have
If , Hölder's inequality and inequality (36) yield Therefore, for By symmetry, for we have Finally, we obtain By virtue of Lemma 10 and the fact that , it is easy to show that holds for all , , where for .
Thus the proof of Theorem 3 is complete.
Proof of Theorem 4. Similar to the proof of Theorem 3, for the case , , we use boundedness of to obtain
For other possibilities we have for , , . Thus, we consider the following two cases.
Case I (, ). We denote by , where and consider the following decomposition: Thus, Now, we will estimate each , separately. Applying Lemma 5, we have Therefore, by virtue of generalized Hölder's inequality and Lemmas 6, 8, and 9, we get Similarly, using Lemma 9, we approximate as Therefore, in view of Lemmas 6, 9, and 8, we have By symmetry, the estimate for is similar to that for ; therefore, Finally, it remains to estimate . For that, we use Lemma 9 to write Thus, by Hölder's inequality and Lemmas 6 and 8, we obtain Combining the estimates for , , , and , we have
Case II(, ). We denote by , where . In this case, we consider the following decomposition: Thus, Let us first compute . As in the proof of Theorem 3, in this case we estimate as By virtue of generalized Hölder's inequality and Lemmas 6–9, we obtain Next, we approximate . Using Lemma 9, it is easy to see that Thus, in view of Lemmas 6, 9, and 8, we get