一个奇异积分算子

本文研究了奇异积分算子Tf(x)=p.v.H*f(x)的L~2有界性,其中H(x)=b(x)K(x),K(x)满足经典条件,b(x)是有界经向函数.新的结果改进了Fefferman,R.以及笔者本人以前昕建立的定理.

A Singula Integral Operator

Let K(x)=(x)/|x|n be a local integrable function in the region and be a homogeneous function of degree zero, satisfying the cancella- tion ion condition where Sn-1 denotes the unit sphere in Rnand d(x) is the Lebesgue measure on it. Suppose that b(x) is a bounded radial function and H(x) =b(x)K(x). The aim of this paper is to discuss the boundedness of the operator Tf(x)=p.v.H*f(x). Let F be a measurable subset of Sn-1 and {Bk} a sequence of balls on Sn-1 covering F and satisfying dk=diam(Bk) 1/2. Then the quantity called the entropy of F. For any the quantity is is called the entropy of the function We prove the following Theorem 1. If n 2 and the entropy , then the operator T is bounded in L2(Rn). In order to study Lp-boundedness of the operator T we introduce the concept of entropy. Let , F be a measurable subset of Sn-1 and {Bk} a sequence of balls on Sn-1 covering F. Then the quantity is called the entropy of F, For any the quantity is called the entropy of Theorem 2. If and for some then the operator T is bounded in Lp(Rn), where