%0 Journal Article
%T -Admissible Sublinear Singular Operators and Generalized Orlicz-Morrey Spaces
%A Javanshir J. Hasanov
%J Journal of Function Spaces
%D 2014
%R 10.1155/2014/505237
%X We study the boundedness of -admissible sublinear singular operators on Orlicz-Morrey spaces . These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator and Calder車n-Zygmund singular integral operator. 1. Introduction As it is well known that Morrey [1] introduced the classical Morrey spaces to investigate the local behavior of solutions to second-order elliptic partial differential equations (PDE), we recall its definition as where ,ˋˋ . Here and everywhere in the sequel stands for the ball in of radius centered at . Let be the Lebesgue measure of the ball and , where . was an expansion of in the sense that . We also denote by the weak Morrey space of all functions for which where denotes the weak -space (for see Definition 4). Morrey found that many properties of solutions to PDE can be attributed to the boundedness of some operators on Morrey spaces. Maximal functions and singular integrals play a key role in harmonic analysis since maximal functions could control crucial quantitative information concerning the given functions, despite their larger size, while singular integrals, Hilbert transform as its prototype, nowadays intimately connected with PDE, operator theory and other fields. Let . The Hardy-Littlewood (H-L) maximal function of is defined by The Calder車n-Zygmund (C-Z) singular integral operator is defined by and bounded on , where is a ※standard singular kernel,§ that is, a continuous function defined on and satisfying the estimates It is well known that the maximal and singular integral operators play an important role in harmonic analysis (see [2, 3]). Orlicz spaces, introduced in [4, 5], are generalizations of Lebesgue spaces (see also, [6每8]). They are useful tools in harmonic analysis and its applications. For example, the Hardy-Littlewood maximal operator is bounded on for , but not on . Using Orlicz spaces, we can investigate the boundedness of the maximal operator near more precisely (see [9每11]). We find it convenient to define the generalized Orlicz-Morrey spaces in the following form (see Definition 3 for the notion of Young functions). Definition 1. Let be a positive measurable function on and a Young function. We define the generalized Orlicz-Morrey space as the space of all functions with finite quasinorm Remark 2. The Calder車n-Zygmund (C-Z) singular integral operators are bounded and expressed as (4) for all , with standard kernel . Then, one can prove that is of weak type and type , , for , and then is uniquely extended to an -bounded operator by the density
%U http://www.hindawi.com/journals/jfs/2014/505237/