An endpoint estimate for the commutators of singular integral operators with rough kernels

Abstract

Let $\Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere $S^{d-1}$, $T_{\Omega }$ be the homogeneous singular integral operator with kernel $\frac{\Omega (x)}{|x{|}^d}$ and $T_{\Omega ,b}$ be the commutator of $T_{\Omega }$ with symbol $b$. In this paper, we prove that if $\Omega \in L(\log L{)}^2(S^{d-1})$, then for $b\in \mathrm{BMO}({\mathbb{R}}^d)$, $T_{\Omega ,b}$ satisfies an endpoint estimate of $L\log L$ type.

arXiv commentWe reorganized the manuscript