Compactness of commutators of bilinear maximal Calder\'{o}n-Zygmund singular integral operators

TL;DR

This paper investigates the compactness properties of certain commutators associated with bilinear maximal Calderón-Zygmund singular integral operators, extending the understanding of their behavior on Lebesgue spaces.

Contribution

It establishes the compactness of commutators and iterated commutators of bilinear maximal Calderón-Zygmund operators on Lebesgue spaces, which was previously unexplored.

Findings

Proves compactness of $T_{ ext{*,b,1}}$, $T_{ ext{*,b,2}}$, and $T_{ ext{*,(b_1,b_2)}}$ on $L^r(R^n)$.

Extends the theory of commutators to bilinear maximal Calderón-Zygmund operators.

Provides new insights into the structure of these operators in harmonic analysis.

Abstract

Let $T$ be a bilinear Calder\'{o}n-Zygmund singular integral operator and $T_*$ be its corresponding truncated maximal operator. The commutators in the $i$-$th$ entry and the iterated commutators of $T_*$ are defined by $$ T_{\ast,b,1}(f,g)(x)=\sup_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta}K(x,y,z)(b(y)-b(x))f(y)g(z)dydz\bigg|, $$ $$T_{\ast,b,2}(f,g)(x)=\sup_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta}K(x,y,z)(b(z)-b(x))f(y)g(z)dydz\bigg|,$$ \begin{align*} T_{\ast,(b_1,b_2)}(f,g)(x)=\sup\limits_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta} 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.