Boundedness Of Fractional Integral With Variable Kernel On Some Spaces

In this dissertation,we shall mainly study the boundedness of fractional integral operators with variable kernel in some function spaces.The main results are as follows.In section 1,we recall some definitions and lemmas associated with the paper.And also give some proofs of the partial lemmas.In section 2,we mainly prove that the fractional integral operator with variable kernel T?,? is bounded from Morrey spaces L?/?,?(Rn)to the spaces BMO(Rn),and also from Lp,?(Rn)to a class of the Campanato spaces Ll,n(?/n-?/np)(Rn).In section 3,we research the boundedness for the fractional integrals with vari-able kernel on weighted Hardy space.By applying the atomic decomposition and molecular decomposition of weighted Hardy space,we prove that the fractional in-tegrals with variable kernel is bounded from the weighted Hardy space to weighted Lebesgue space Lp?(Rn).