A bilinear sparse domination for the maximal singular integral operators with rough kernels

TL;DR

This paper establishes a bilinear sparse domination for the maximal singular integral operator with rough kernels, extending understanding of their boundedness properties in harmonic analysis.

Contribution

It proves a new bilinear sparse domination result for maximal singular integrals with rough kernels under certain conditions, advancing the theoretical framework.

Findings

Maximal singular integral operator with rough kernels admits bilinear sparse domination.

The domination bound depends on the kernel's $L^{ty}$ norm and a specific Orlicz function.

Results hold for all $r$ in (1, inite).

Abstract

Let $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have mean value zero, $T_{\Omega}$ be the homogeneous singular integral operator with kernel $\frac{\Omega(x)}{|x|^d}$ and $T_{\Omega}^*$ be the maximal operator associated to $T_{\Omega}$. In this paper, the authors prove that if $\Omega\in L^{\infty}(S^{d-1})$, then for all $r\in (1,\,\infty)$, $T_{\Omega}^*$ enjoys a $(L^\Phi,\,L^r)$ bilinear sparse domination with bound $Cr'\|\Omega\|_{L^{\infty}(S^{d-1})}$, where $\Phi(t)=t\log\log ({\rm e}^2+t)$.