The Boundedness Of Riesz Potential Operators In Generalized Morrey Type Spaces

In this dissertation, we are mainly devoted to the following three prob-lems on Riesz potentials in various generalized Morrey spaces:The first one is to es-tablish the Trudingcr type inequality involving the Ricsz potential operators on Morrey spaces with two variable exponents. The next is to attain the Trudinger type inequality on the generalized grand-Morrey spaces. The last one is to realize the boundedness of the Riesz potential operators on Morrey-Lorentz spaces. The paper is made up of four parts for details as follows:In the first chapter, we briefly introduce the background and the recent development related to this topic. Here we recall the Lebesgue spacees with variable exponents to de-fine generalized Morrey spaces with two variable exponents and so-called grand-Morrey spaces. In addition, the Riesz potential operator with variable exponents is introduced.In chapter 2, we focus on the Trudinger inequality on Morrey spaces with two variable exponents Lp(x),λ(x)(1≤p(x)<∞,0<λ(x)≤n), and obtain the following inequality when ‖f‖Lp(x),λ(x)≤1. This is an extension of the well-known Trudinger inequality: with f belonging to Morrey spaces Lp,λ, which is a quite different argument from the usual Morrey spaces with constant exponent. We estimate the Riesz potentials Morrey spaces with variable exponents, and obtain the similar conclusion.The third chapter is concerned with the Trudinger inequality in generalized grand-Morrey spaces Lp),v,θ. Actually, the grand-Lebesgue spaes Lp) is a weak form of the usual Lebesgue spaces LP. As a consequence, we conclude the inequality as follows: while 0< v< n and while v=n. Here, to this end we employ a similar approach in the second chapter.In the forth chapter, we focus on the boundedness of Riesz potential operators in a local Morrey-Lorentz spaces Mp,q;λloc. We conclude that the Riesz potential is bounded from Mp,q;λloc to Mλp/λ-αp,q;λ loc.