Boundedness Of Hausdorff Operators On Some Function Spaces

The study of Hausdorff operator can be traced back to 1921. It was introduced to solve the convergence of the series by Hausdorff. Hausdorff operator contains Hardy op-erator, Cesaro operator etc as its special case. It is widerly used in other mathematical branches, such as probabilily theory, multicomplex function theory and geometrical anal-ysis etc. This paper mainly studies the boundedness of the quasi-Hausdorff operator and its commutator on some function spaces. These spaces contain Triebel-Lizorkin space,Campanato space and Bp,? space. In addition, this paper discusses the boundedness of the multilinear Hausdorff operator and the anisotropic Hausdorff operators on the Morrey space.This thesis contains for the following four chapters:In the first chapter, we mainly introduce the background and their recent develop-ments of Hausdorff operators .In the second chapter, we mainly study the boundedness of the quasi-Hausdorff op-eratorH?f(x)=?Rf(tx)d?(t),x?Rn,and its formal adjoint operatorH?*f(x)=?R|t|-nf(x/t)d?(t),x?Rnon the Triebel-Lizorkin space and Campanato space. In the meantime, we also obtain the boundedness for its commutator on the Bp,? space.In the third chapter, we get the sharp bound for the multilinear Hausdorff operator(?)on the Morrey space, and the sufficient and necessary conditions of the boundedness of the anisotropic Hausdorff operator(?)on the Morrey space. Besides, we discuss the boundedness of the anisotropic multilinear Hausdorff operator on the Morrey space.In the fourth chapter, we summarize this paper and put forward the further studies in the future.