Limiting weak-type behaviors for singular integrals with rough $L\log L(\mathbb{S}^n)$ kernels (2106.14051v1)
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 $\lim_{\lambda\to0+}\lambda|{x\in\mathbb{R}n:|T_\Omega(f)(x)|>\lambda}| = n{-1}|\Omega|_{L1(\mathbb {S}n)}|f|_{L1(\mathbb{R}n)},\quad0\le f\in L1(\mathbb{R}n).$ Moreover,$(n{-1}|\Omega|_{L1(\mathbb{S}{n-1})}$ 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.