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 $b\in { \rm BMO}(\mathbb{ R}^d)$, $T_{ \Omega,\,b}$ satisfies an endpoint estimate of $L\log L$ type.
Categories: math.CA