Endpoint estimates for the commutators of multilinear CalderónZygmund operators with Dini type kernels
Abstract
Let $T_{\vec{b}}$ and $T_{\Pi b}$ be the commutators in the $j$th entry and iterated commutators of the multilinear CalderónZygmund operators, respectively. It was wellknown 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1601.02173
 Bibcode:
 2016arXiv160102173L
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 42B25;
 47G10
 EPrint:
 23 pages