Limiting weak-type behaviors for singular integrals with rough $L\log L(\mathbb{S}^n)$ kernels

Abstract

Let $\Omega$ be a function of homogeneous of degree zero and vanish on the unit sphere $\mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_\Omega$ associated with rough kernel $\Omega$. We show that, if $\Omega\in L\log L(\mathbb S^{n})$, then $\lim_{\lambda\to0^+}\lambda|\{x\in\mathbb{R}^n:|T_\Omega(f)(x)|>\lambda\}| = n^{-1}\|\Omega\|_{L^1(\mathbb {S}^n)}\|f\|_{L^1(\mathbb{R}^n)},\quad0\le f\in L^1(\mathbb{R}^n).$ Moreover,$(n^{-1}\|\Omega\|_{L^1(\mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_\Omega$ when $\Omega\in L\log L(\mathbb{S}^{n-1})$. Corresponding results for rough bilinear singular integral operators defined in the form $T_{\vec\Omega}(f_1,f_2) = T_{\Omega_1}(f_1)\cdot T_{\Omega_2}(f_2)$ have also been established.