Boundedness Of Several Integral Operators

In this thesis, we study the boundedness of several kinds of integral operators on a-modulation spaces, Campanato spaces and generalized Morrey spaces. The operators which we consider include fractional integral operators, multilinear operators, Hausdorff operators and sublinear operators with rough kernel, etc.This thesis is divided into six chapters.The first chapter of this thesis is introduction. First, we introduce some of the relevant background and known results of the operators we consider in this thesis. Then, we introduce the structure of this thesis.In the chapter two, we obtain the boundedness of the fractional integral oper-ators, the bilinear fractional integral operators and the bilinear Hilbert transform on a-modulation spaces.In the chapter three, we study several kinds of Hausdorff operators on Cam-panato spaces and obtain their sharp bounds. Furthermore, we also discuss the boundedness of multilinear Hausdorff operators on Morrey spaces.In the chapter four, we consider a new type of Hausdorff operators with power weights on n-dimensional product spaces. The sharp bounds on Lebesgue spaces with power weights are obtained. Our theorems extend some known results on Hardy’s inequality.In the chapter five, we consider the Hausdorff operators and their commu-tators with CBMO functions on generalized central Morrey spaces.In the last chapter, we establish the boundedness of commutators generated by sublinear operators and BMO functions on generalized Morrey spaces. We also obtain the boundedness of commutators generated by sublinear operators and Lipschitz functions on these spaces. As an application, we get the bounded-ness of commutators generated by Marcinkiewicz operators or rough fractional integral operators with BMO functions on generalized Morrey spaces. Further-more, we establish the regularity on generalized Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients.