Endpoint estimates for the commutators of multilinear Calder\'{o}n-Zygmund operators with Dini type kernels

TL;DR

This paper establishes sharp endpoint boundedness results for commutators of multilinear Calderón-Zygmund operators with Dini type kernels, extending understanding of their behavior on Hardy spaces and weak Lebesgue spaces.

Contribution

It provides natural sharp endpoint estimates for these commutators under Dini type conditions, which was previously not well-understood.

Findings

Bounded from product Hardy space to weak L^{1/m, } space

Utilizes a key lemma by M. Christ for the analysis

Employs complex decomposition and case analysis

Abstract

Let $T_{\vec{b}}$ and $T_{\Pi b}$ be the commutators in the $j$-th entry and iterated commutators of the multilinear Calder\'{o}n-Zygmund operators, respectively. It was well-known that $T_{\vec{b}}$ and $T_{\Pi b}$ were not of weak type $(1,1)$ and $(H^1, L^1)$, but they did satisfy certain endpoint $L\log L$ type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that $T_{\vec{b}}$ and $T_{\Pi b}$ are bounded from product Hardy space $H^1\times\cdot\cdot\cdot\times H^1$ to weak $L^{\frac{1}{m},\infty}$ space, whenever the kernel satisfies a class of Dini type condition. This was done by using a key lemma given by M. Christ, a very complex decomposition of the integrand domains and splitting and estimating the commutators very carefully into several terms and cases.