Vector-Valued Inequalities in the Morrey Type Spaces

We will obtain the strong type and weak type estimates for vector-valued analogues of classical Hardy-Littlewood maximal function, weighted maximal function, and singular integral operators in the weighted Morrey spaces when and , and in the generalized Morrey spaces for , where is a growth function on satisfying the doubling condition. 1. Introduction The classical Hardy-Littlewood maximal function is defined for a locally integrable function on by where the supremum is taken over all balls containing . It is well known that the maximal operator maps into for all and into . Let be a sequence of locally integrable functions on . For any , we define and This nonlinear operator was introduced by Fefferman and Stein in [1], and since then it has played an important role in the development of modern harmonic analysis. In this remarkable paper [1], Fefferman and Stein extended the classical maximal theorem to the case of vector-valued functions. Theorem 1 (see [1]). Let . Then, for every , there exists a constant independent of such that Theorem 2 (see [1]). Let . Then, for every , there exists a constant independent of such that A weight is a nonnegative, locally integrable function on ; denotes the ball with the center and radius . For , a weight function is said to belong to , if there is a constant such that, for every ball (see [2, 3]), For the case ,？？ , if there is a constant such that, for every ball , A weight function if it satisfies the condition for some . We say that , if for any ball there exists an absolute constant such that It is well known that if with , then . Moreover, if , then, for all balls and all measurable subsets of , there exists such that Given a ball and , denotes the ball with the same center as whose radius is times that of . For a given weight function and a measurable set , we also denote the Lebesgue measure of by and the weighted measure of by , where . Given a weight function on , for , the weighted Lebesgue space is defined as the set of all functions such that We also denote by the weighted weak space consisting of all measurable functions such that In particular, when equals to a constant function, we will denote and simply by and . In [4], Andersen and John considered the weighted version of Fefferman-Stein maximal inequality and showed the following. Theorem 3 (see [4]). Let and . Then, for every , there exists a constant independent of such that Theorem 4 (see [4]). Let and . Then, for every , there exists a constant independent of such that Given a weight , the weighted maximal function is defined as where the