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. 1. Introduction 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 [1] 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 [2], Kenig and Stein [3], and Grafakos and Torres [4] 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 [4] 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 [7], 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 [8] and the survey paper [9] 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 [10] proved the boundedness of such integral operators on the product of variable exponent Lebesgue space.