A bilinear sparse domination for the maximal singular integral operators with rough kernels

Abstract

Let $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have mean value zero, $T_{\Omega }$ be the homogeneous singular integral operator with kernel $\frac{\Omega (x)}{|x{|}^d}$ and $T_{\Omega }^{\ast }$ be the maximal operator associated to $T_{\Omega }$. In this paper, the authors prove that if $\Omega \in L^{\infty }(S^{d-1})$, then for all $r\in (1,\infty )$, $T_{\Omega }^{\ast }$ enjoys a $(L^{\Phi },L^r)$ bilinear sparse domination with bound $C{r}^{\prime }\Vert \Omega {\Vert }_{L^{\infty }(S^{d-1})}$, where $\Phi (t)=t\log \log ({\mathrm{e}}^2+t)$.

arXiv commentrewrite the proof of main theorem