The Estimates Of Singular Integrals With Variable Kernel And Fractional Differentiations

In this dissertation,we shall mainly study the boundedness of singular integrals with variable kernel and fractional differentiations in some important spaces.The main results are as follows.In Section 1,the boundedness of convolution operators Tm,j in homogeneous Morrey-Herz spaces MKp,qα.λ(Rn)is established at first.Secondly,with the aid of the methods of the decompositions of spherical harmonical on the unit sphere,the boundedness is established for a class of singular integral operators T with variale kernel and its adjoint T*and pseudo-adjoint T#and the fractional differentiation operator Dγ on the weighted Morrey-Herz spaces.In Section 2,the boundedness of a class of singular integral operators T with variale kernel and its adjoint T*and pseudo-adjoint T#and the fractional diffcrenti-ation operator Dγ is discussed on the Weighted Morrey-Herz spaces MKp,qα.λ(ω1,ω2).At the same time,it is obtained that convolution operatorsTm,j is a bounded oper-ator on MKp,qα.λ(ω1,ω2).In Section 3,We proved the convolution operators Tm,j is a bounded operator from weak Morrey-Herz spaces MKp,1α.λ(Rn)into Morrey-Herz spaces MKp,1α.λ(Rn)by the disscussing it’s weak(1,1)inequality as q = 1.Furthermore,we proved the singular integrals with variable kernel and fractional differentiations are bounded operators from MKp,1α.λ(Rn)into the space MKp,1α.λ(Rn).