Boundedness Of Singular Integral Under An Extended Kernel Condition

When studying singular integrals,a special condition proposed by Grafakos and Ste-fanov is often referred as the condition on the kernels.In the first chapter of this paper,we briefly introduce the development of singular integrals,mainly focus on its bound-edness properties,as well as the Grafakos-Stefanov condition and its related theorems.In the second chapter,Grafakos-Stefanov s condition is generalized to a wide variety of function space and thus some new boundedness theorem for both the singular integral and maximum singular integral are obtained.An example is constructed to show that our newly extended condition on the kernel does provide some new staff comparing the Grafakos-Stefanov's one.In the last chapter,the conditions considered in this paper are applied to the Marcinkiewicz integral and some new boundedness theorems are proved.