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 ${\lim }_{\lambda \to 0^{+}}\lambda |\{x\in {\mathbb{R}}^n:|T_{\Omega }(f)(x)|>\lambda \}|=n^{-1}\Vert \Omega {\Vert }_{L^1({\mathbb{S}}^n)}\Vert f{\Vert }_{L^1({\mathbb{R}}^n)},0\le f\in L^1({\mathbb{R}}^n).$ Moreover,$(n^{-1}\Vert \Omega {\Vert }_{L^1({\mathbb{S}}^{n-1})}$ 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.

26 pages. arXiv admin note: text overlap with arXiv:2011.11512