Weak type bounds for rough maximal singular integrals near $L^1$

Abstract

In this paper it is shown that for $\Omega \in L\log L({\mathbb{S}}^{d-1})$, the rough maximal singular integral operator $T_{\Omega }^{\ast }$ is of weak type $L\log \log L({\mathbb{R}}^d)$. Furthermore, for $w\in A_1$ and $\Omega \in L^{\infty }({\mathbb{S}}^{d-1})$, it is shown that $T_{\Omega }^{\ast }$ is of weak type $L\log \log L(w)$ with weight dependence $[w{]}_{A_1}[w{]}_{A_{\infty }}\log ([w{]}_{A_{\infty }}+1),$ which is same as the best known constant for the singular integral $T_{\Omega }$.

23 pages. Comments are welcome