Boundedness of Multilinear Calderón-Zygmund Operators on Grand Variable Herz Spaces

Abstract

In this paper, we prove the boundedness of multilinear Calderón-Zygmund operators on product of grand variable Herz spaces. These results generalize the boundedness of multilinear Calderón-Zygmund operators on product of variable exponent Lebesgue spaces and variable Herz spaces.

1. Introduction

There has been increased interest in the study of multilinear singular integral operators in recent years. The class of multilinear singular integrals with standard Calderón-Zygmund kernels provides the foundation and starting point of research of the theory. Such a class of multilinear Calderón-Zygmund operators was introduced and first studied by Coifman and Meyer [1–3] and later by Grafakos and Torres [4]. For the boundedness and other properties of multilinear fractional integrals, we refer to, e.g., [5–8].

Variable Lebesgue spaces were introduced in [9], but stayed under the radar for a considerable amount of time. Apart from some previous sporadic episodes, the research boom on such spaces can be traced back to the foundational paper [10]. Since then, these spaces have attracted much attention of mathematicians, not only because of their connection with harmonic analysis but also due to their usefulness in application to a wide range of problems, see, e.g., [11]. The standard references to the general theory of variable Lebesgue spaces are [12, 13].

The classical definition of Herz spaces was introduced in [14]. Many studies can be found related to these spaces and its variations, which include variable Herz spaces, continual Herz spaces, and Herz spaces with variable smoothness and integrability. For details, see [15–21] and references therein.

Grand Lebesgue spaces on bounded sets, which proved to be useful in application to partial differential equations, were introduced in [22, 23]. In the last years, various operators of harmonic analysis have been intensively studied on grand spaces, see for instance [20, 24–32]. Grand Lebesgue sequence spaces were introduced recently in [33], where several operators of harmonic analysis were studied, e.g., maximal, convolutions, Hardy, Hilbert, and fractional operators.

In this paper, we prove the boundedness of multilinear Calderón-Zygmund operators on grand variable Herz spaces which were introduced in [34]. The present paper is organized in the following way. Apart from the introduction, in Section 2, we recall some definitions and results related to variable exponent spaces. Section 3 contains some details about multilinear Calderón-Zygmund kernels and the proof of the main result.

Notations. (i) is the set of natural numbers and (ii) is the set of integers(iii) is the set of negative integers(iv) for and (v) is the ball of radius center at the point (vi) for all (vii) is a spherical layer(viii)(ix)(x) means that and means that (xi)constants (often different constant in the same chain of inequalities) will mainly be denoted by or

2. Function Spaces with Variable Exponent

In this section, we recall definitions and results related to variable exponent Lebesgue spaces, variable Herz spaces, and grand variable Herz spaces.

2.1. Lebesgue Space with Variable Exponent

For the current section, we refer to [10–13, 35] unless and until stated otherwise. Let be a real-valued measurable function on with values in . For we suppose that

where . By , we denote the space of measurable function on such that

It is a Banach space, see [13, 35], endowed with norm:

By , we denote the conjugate exponent of , defined by . In the sequel, we use

log condition:where does not depend on ;

decay condition at 0:holds for some ; and

decay condition at: there exists a number , such that where does not depend on .

Given a function , the Hardy-Littlewood maximal operator is defined by

We adopt the following notations: (i)(ii) consists of all measurable functions satisfying and (iii) and denote the classes of which satisfy (5) and (6), respectively(iv) is the set of all for which is bounded on

For the following lemma, we refer to, e.g., [12].

Lemma 1 (Generalized Hölder’s Inequality). Given , define by . Then, there exists a constant such that for all and , and

The following lemma appears in [21].

Lemma 2. Let and Then where and depend on , but do not depend on .

2.2. Herz Spaces with Variable Exponent

The classical Herz spaces were first introduced in [14]. We recall the definition of variable exponent Herz spaces.

Definition 3. Let , , and . The homogeneous Herz space is defined by where The nonhomogeneous Herz space is defined by where For the boundedness of integral operators on Herz type spaces, we refer to, e.g., [18, 19, 36].

2.3. Grand Lebesgue Sequence Space

In this section, we recall the definition of grand Lebesgue sequence space. For the following definition and statements, see [33]. In what follows, stands for one of the sets , and .

Definition 4. Let and . The grand Lebesgue sequence space is defined by the norm

2.4. Grand Variable Herz Space

Following [34], we now introduce the grand variable Herz spaces.

Definition 5. Let , , , and The homogeneous grand variable Herz space is defined by where

The following lemma, see [15], is helpful to estimate the norm of characteristics functions.

Lemma 6. Let and let If , then with the implicit constants independent of and
The left-hand side equivalence remains true for every if we assume, additionally, that

3. Boundedness of Multilinear Calderón-Zygmund Operators

Consider the multilinear operator of the form where and , the space of compactly supported functions. Let be a locally integrable function defined away from the diagonal , which satisfies the size estimate for some and all with for some For smoothness, assume that for some , provided that and whenever for all

Such kernels are called Calderón-Zygmund kernels and the class of all functions satisfying (20), (21), and (22) with parameters , and will be denoted by -CZK, compare [4]. We say that be as in (19) is an -linear Calderón-Zygmund operator, if (i)The related kernel belongs to -CZK class(ii) is bounded from to for some , and

Grafakos and Torres [4] proved the boundedness of from to for some , and , and from to .

The boundedness of the multilinear Calderón-Zygmund operator on variable exponent Lebesgue spaces was proved in [37], as stated below.

Lemma 7. Let , , with , and for some . Then, the -linear Calderón-Zygmund operator is bounded on the product of variable exponent Lebesgue spaces. Moreover, with the constant independent of .

We now state and prove the boundedness of multilinear Calderón-Zygmund operator on grand variable Herz spaces.

Theorem 8. Let , , such that , , and for some . Let , , and be log-Hölder continuous both at the origin and at infinity for with Suppose that and Then, the -linear Calderón-Zygmund operator is bounded on the product of grand variable Herz spaces. Moreover, with the constant independent of .

Proof. We restrict ourselves to , the general case following in a similar manner. Defining and , we decompose the component functions of as For future usage, we divide into the following sets and, for and arbitrary subsets of , we define From Definition 5 and (26), we have where It is necessary to estimate , and , since , , and can be obtained in a similar manner as , , and , respectively. Estimation for : splitting and by the asymptotic ( and ) and ( and ), we get For , , , and , we have yielding From estimate (35), Hölder’s inequality, , and Lemma 2, we obtain Taking into account the previous estimate for , the equality and Hölder’s inequality, we have By Hölder’s inequality, Fubini’s theorem for series, , and defining , we obtain The estimate is obtained, mutatis mutandis, via the estimation for . With this estimates at hand, we obtain To estimate , we split as follow The estimate follows in similar manner as in with simply replaced by and used the fact
For estimate , by Hölder’s inequality, Lemma 2, and the inequality , we obtain From the estimate of , the equality , and Hölder’s inequality, we have Invoking the Hölder inequality and defining , we have Similar estimate, with the corresponding changes, is obtained for , from which we obtain Hence Estimation for : as in the case of , we obtain the following estimate Notice that, for , , , , and , we have from which, taking Lemma 2 into consideration and elementary computations, we obtain From the estimate for and Hölder’s inequality, we get Notice that . For the estimate , we reason as follows The term is estimated by To estimate , by Hölder’s inequality, Lemma 2, and the inequality , we have Thus using (51) and by Hölder’s inequality, we have The term is equal to and for we use similar arguments as for , replacing with .
For the term , by Hölder’s inequality and Lemma 2, we have Taking into consideration (53) and applying Hölder’s inequality, we get Noting that and estimating in a similar fashion as , we obtain , from which we get Estimation for : we have For , , , , and , we get from which, taking into account the elementary inequality , it follows By Hölder’s inequality, Lemma 2, , and , we obtain To estimate the term , using (59) and the Hölder inequality, we obtain We have , since .
As for , we split as follow By Hölder’s inequality, Fubini’s theorem for series, the inequality and , we obtain For , using the same argument as above and the inequality , we obtain So
The term can be estimated as For , , , , and Lemma 2, we obtain By (65) and Hölder’s inequality, we get Note that The estimate for can be obtained in a similar way as for , by replacing with
For estimate , by Hölder’s inequality and Lemma 2, we have Using (67) and Hölder’s inequality, we obtain We have and the estimate for the term is similar to Taking all the estimates into account yields Estimation for : We have By the -boundedness of , see Lemma 7, we obtain that By Hölder’s inequality, we have Similarly, we can obtain similar estimate for , replacing by with Therefore Estimation for : we have For , , , and , we have yielding By Hölder’s inequality, Lemma 2, , and inequality , we obtain By (77) and invoking Hölder’s inequality, we get Note that the estimate is equal to that of and is obtained by similar argument used in
For , we have Using (79) and Hölder’s inequality, we get The estimate is similar to and Hence Estimation for : we estimate as usual For , , and , we have , whereas for , , and , we have . Under the assumptions for the obtained inequalities, we have When , by Hölder’s inequality, Lemma 2, and , we obtain Thus The estimates for the terms and are obtained in the same way as for . Therefore Finally, for the term , we have The terms and can be estimated in a similar manner as for that Thus Taking into account the estimates (44), (55), (69), (73), (81), and (88), we get which completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest.

Authors’ Contributions

All the authors have contributed equally in preparation and finalization of the manuscript.

Acknowledgments

We gratefully acknowledge the constructive comments of the anonymous referees.