Bilinear Calderón-Zygmund operators on Sobolev, BMO and Lipschitz spaces

摘要

In this paper, the authors establish the necessary and sufficient condition such that the bilinear Calderón-Zygmund operators are bounded from L i p α ( R n ) × L n / α ( R n ) $Lip_{lpha}(mathbb{R}^{n})imes L^{n/lpha} (mathbb{R}^{n})$ to B M O ( R n ) $BMO(mathbb{R}^{n})$ space and from L i p α ( R n ) × L p ( R n ) $Lip_{lpha}(mathbb{R}^{n})imes L^{p}(mathbb{R}^{n})$ to L i p α − n / p ( R n ) $Lip_{lpha-n/p}(mathbb{R}^{n})$ space. As an application, the bilinear Riesz transform is a good example which meets the related conditions. Furthermore, the authors also establish another necessary and sufficient condition for the bilinear Calderón-Zygmund operators to be bounded from L i p α ( R n ) × B M O ( R n ) $Lip_{lpha}(mathbb{R}^{n})imes BMO(mathbb{