%0 Journal Article
%T On the Riesz Potential and Its Commutators on Generalized Orlicz-Morrey Spaces
%A Vagif S. Guliyev
%A Fatih Deringoz
%J Journal of Function Spaces
%D 2014
%R 10.1155/2014/617414
%X We consider generalized Orlicz-Morrey spaces including their weak versions . In these spaces we prove the boundedness of the Riesz potential from to and from to . As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of , in . 1. Introduction The theory of boundedness of classical operators of the real analysis, such as the maximal operator, fractional maximal operator, Riesz potential, and the singular integral operators, and so forth, has been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [1¨C3]. This boundedness extended to several function spaces which are generalizations of -spaces, for example, Orlicz spaces, Morrey spaces, Lorentz spaces, Herz spaces, and so forth. Orlicz spaces, introduced in [4, 5], are generalizations of Lebesgue spaces . 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 [6¨C8]). It is well known that the Riesz potential of order ( ) plays an important role in harmonic analysis, PDE, and potential theory (see [2]). Recall that is defined by The classical result by Hardy-Littlewood-Sobolev states that, if , then the operator is bounded from to if and only if and, for , the operator is bounded from to if and only if . For boundedness of on Morrey spaces , see Peetre (Spanne) [9] and Adams [10]. The boundedness of from Orlicz space to was studied by OˇŻNeil [11] and Torchinsky [12] under some restrictions involving the growths and certain monotonicity properties of and . Moreover Cianchi [6] gave a necessary and sufficient condition for the boundedness of from to and from to weak Orlicz space , which contain results above. In [13] the authors study the boundedness of the maximal operator and the Calder¨®n-Zygmund operator from one generalized Orlicz-Morrey space to and from to the weak space . Our definition of Orlicz-Morrey spaces (see Section 3) is different from that of Sawano et al. [14] and Nakai [15, 16]. The main purpose of this paper is to find sufficient conditions on general Young functions and functions , which ensure the
%U http://www.hindawi.com/journals/jfs/2014/617414/