Multilinear Singular and Fractional Integral Operators on Weighted Morrey Spaces

We will study the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights. 1. Introduction and Main Results Multilinear Calderón-Zygmund theory is a natural generalization of the linear case. The initial work on the class of multilinear Calderón-Zygmund operators was done by Coifman and Meyer in [1] and was later systematically studied by Grafakos and Torres in [2–4]. Let be the -dimensional Euclidean space, and let be the -fold product space ( ). We denote by the space of all Schwartz functions on and by its dual space, the set of all tempered distributions on . Let and be an -linear operator initially defined on the -fold product of Schwartz spaces, and taking values into the space of tempered distributions, Following [2], for given , we say that is an -linear Calderón-Zygmund operator if, for some and with , it extends to a bounded multilinear operator from into and if there exists a kernel function in the class - , defined away from the diagonal in such that whenever and . We say that is a kernel in the class - if it satisfies the size condition for some and all with for some . Moreover, for some , it satisfies the regularity condition that whenever and also that, for each fixed with , whenever . In recent years, many authors have been interested in studying the boundedness of these operators on function spaces; see, for example, [5–8]. In 2009, the weighted strong and weak type estimates of multilinear Calderón-Zygmund singular integral operators were established in [9] by Lerner et al. New more refined multilinear maximal function was defined and used in [9] to characterize the class of multiple weights. Theorem A (see [9]). Let and be an -linear Calderón-Zygmund operator. If and with and satisfies the condition, then there exists a constant independent of such that where . Theorem B (see [9]). Let , and let be an -linear Calderón-Zygmund operator. If , and with , and satisfies the condition, then there exists a constant independent of such that where . Let , and let . For given , the -linear fractional integral operator is defined by For the boundedness properties of multilinear fractional integrals on various function spaces, we refer the reader to [10–16]. In 2009, Moen [17] considered the weighted norm inequalities for multilinear fractional integral operators and constructed the class of multiple weights (see also [18]). Theorem C (see [17, 18]). Let , , and let be an -linear fractional integral operator. If , and , and satisfies the