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

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 $Omegain Llog L(mathbb S^{n})$, then $lim_{lambdato0^+}lambda|{xinmathbb{R}^n:|T_Omega(f)(x)|>lambda}| = n^{-1}|Omega|_{L^1(mathbb {S}^n)}|f|_{L^1(mathbb{R}^n)},quad0le fin L^1(mathbb{R}^n).$ Moreover,$(n^{-1}|Omega|_{L^1(mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_Omega$ when $Omegain Llog L(mathbb{S}^{n-1})$. Corresponding results for rough bilinear singular integral operators defined in the form $T_{vecOmega}(f_1,f_2) = T_{Omega_1}(f_1)cdot T_{Omega_2}(f_2)$ have also been established.