奇异积分在Herz型Sobolev空间上的有界性

Boundedness of sigular integral operator on Herz type space.

采用Hardy空间的原子分解理论、Riesz-Thorin内插定理以及调和分析中的一些基本方法讨论了粗糙核奇异积分算子TΩ,βf(x)=p.v.∫Rnb(|y|)Ω(y′)|y|-n-βf(x-y)dy,当Ω∈Hr(Sn-1)r=n-1n-1+β时,是从Herz型Sobolev空间到Herz型空间有界的.其中b(.)是一个有界函数,β≥0,Ω是Sn-1上满足某些消失性条件的分布.

The sigular integral operator T_(Ω,β)f(x)=p.v.∫_(R^n)b(|y|)Ω(y′)|y|^(-n-β)f(x-y)dy defined on all-test function f is studied, where b is a bounded function, β≥0,Ω(y′) is an integrable function on unit sphere S^(n-1) satisfying certain cancellation conditions. It is proved that, for 0<α<n1-1q,1<q<∞,0<p<∞,T_(Ω,β)(extends) to be a bounded operator from the Herz type Sobolev space to Herz type space with Ω being a distribution in the Hardy space H^r(S^(n-1)) with r=n-1n-1+β.