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

TL;DR

This paper establishes weak type bounds for rough maximal singular integrals near $L^1$, extending known results to functions with minimal regularity and incorporating weighted inequalities.

Contribution

It proves new weak type bounds for rough maximal singular integrals under minimal regularity conditions and weighted settings.

Findings

Weak type $L ext{log} ext{log}L$ bounds for $T_ abla^*$ with $ abla o 0$

Weighted weak type bounds with $A_1$ weights and explicit dependence

Extension of bounds to rough kernels in $L ext{log}L$ space

Abstract

In this paper it is shown that for $\Omega\in L\log L(\mathbb{S}^{d-1})$, the rough maximal singular integral operator $T_\Omega^*$ 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 $T_\Omega^*$ 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 $T_\Omega$.