Properties Of Hypersingular Integral Operator On Some Function Spaces

In recent years,there are many results about the hypersingular integral operator D_α on Euclidean space Rn.Meanwhile,some properties of hypersingular integral operator D_α have been obtained with non-doubling measures.Inspired by these results,the paper discusses the boundedness of hypersingular integral operator D_α on some function spaces and nonhomogeneous space（X,d,μ）.Our results will enrich the theory of hypersingular integral operator D_α.At first,we discuss the boundedness of hypersingular integral operator D_α on Rn.Hypersingular integral operator D_α is a bounded operator not only from Sobolev space Bs（Rn）to Bs-α（Rn）,but also from Lipschitz space Lipβ（Rn）to a subspace C*β-α,p（Rn）of Lipschitz space Lipβ-α（Rn）.Then,the definition of hypersingular integral D_α on nonhomogeneous space（X,d,μ）is introduced,and we get the boundedness of hypersingular integral operator D_α on Lips-chitz space Lipβ（μ）.Finally,we discuss the composition T = D_α1I_α2 of a hypersingular integral operator D_α1 and a standard fractional integral operator I_α2.When α1=α2,T = D_α1 Iα1 is a Calderon-Zygmund operator;When α2>α1,α= α2-α1,Tα = D_α1I_α2 is a fractional integral operator.