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 $T_{\Omega }^*$ 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 $r\in (1,\,\infty )$, $T_{\Omega }^*$ enjoys a $(L^\Phi ,\,L^r)$ bilinear sparse domination with $\Phi (t)=t\log \log (\textrm{e}^2+t)$. Some applications of this bilinear sparse domination are also given.