Compactness of commutators of bilinear maximal Calderón-Zygmund singular integral operators

Abstract

Let $T$ be a bilinear Calderón-Zygmund singular integral operator and $T_{\ast }$ be its corresponding truncated maximal operator. The commutators in the $i$-$th$ entry and the iterated commutators of $T_{\ast }$ are defined by $$T_{\ast ,b,1}(f,g)(x)=\underset{\delta >0}{\sup }|{\iint }_{|x-y|+|x-z|>\delta }K(x,y,z)(b(y)-b(x))f(y)g(z)dydz|,$$ $$T_{\ast ,b,2}(f,g)(x)=\underset{\delta >0}{\sup }|{\iint }_{|x-y|+|x-z|>\delta }K(x,y,z)(b(z)-b(x))f(y)g(z)dydz|,$$ \begin{align*} T_{\ast,(b_1,b_2)}(f,g)(x)=\sup\limits_{δ>0}\bigg|\iint_{|x-y|+|x-z|>δ} K(x,y,z)(b_1(y)-b_1(x))(b_2(z)-b_2(x))f(y)g(z)dydz\bigg|. \end{align*} In this paper, the compactness of the commutators $T_{\ast ,b,1}$, $T_{\ast ,b,2}$ and $T_{\ast ,(b_1,b_2)}$ on $L^r({\mathbb{R}}^n))$ is established.

arXiv commentNew version, corrected the fomer mistakes

References in corpus (1)

• Compactness properties of commutators of bilinear fractional integrals

Cited by in corpus (1)

• On Some properties of dyadic operators