Properties Of Hypersingular Integral Operator On Some Function Spaces

In recent years,there are many results about the hypersingular integral operator D_? on Euclidean space Rn.Meanwhile,some properties of hypersingular integral operator D_? have been obtained with non-doubling measures.Inspired by these results,the paper discusses the boundedness of hypersingular integral operator D_? on some function spaces and nonhomogeneous space?X,d,??.Our results will enrich the theory of hypersingular integral operator D_?.At first,we discuss the boundedness of hypersingular integral operator D_? on Rn.Hypersingular integral operator D_? is a bounded operator not only from Sobolev space Bs?Rn?to Bs-??Rn?,but also from Lipschitz space Lip??Rn?to a subspace C*?-?,p?Rn?of Lipschitz space Lip?-??Rn?.Then,the definition of hypersingular integral D_? on nonhomogeneous space?X,d,??is introduced,and we get the boundedness of hypersingular integral operator D_? on Lips-chitz space Lip????.Finally,we discuss the composition T = D_?1I_?2 of a hypersingular integral operator D_?1 and a standard fractional integral operator I_?2.When ?1=?2,T = D_?1 I?1 is a Calderon-Zygmund operator;When ?2>?1,?= ?2-?1,T? = D_?1I_?2 is a fractional integral operator.