The $L(\log L)^{\epsilon}$ endpoint estimate for maximal singular integral operators

TL;DR

This paper establishes endpoint estimates for maximal singular integral operators involving logarithmic Orlicz spaces, extending previous results and providing new quantitative bounds in weighted settings.

Contribution

It introduces sharp endpoint estimates for maximal singular integrals using $L( ext{log}L)^{ ext{epsilon}}$ spaces, extending prior endpoint results to these operators.

Findings

Proves a weak-type endpoint estimate involving $L( ext{log}L)^{ ext{epsilon}}$ spaces.

Derives sharp $L^p$ estimates with explicit dependence on weights.

Provides a quantitative two-weight bump estimate.

Abstract

We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim \frac{1}{\epsilon} \int_{\mathbb{R}^n} |f(x)| M_{L(\log L)^{\epsilon}} (w)(x)dx$, for $w\geq 0, 0<\epsilon \leq 1.$ This follows from the sharp $L^p$ estimate $ \|T^*f \|_{ L^{p}(w) } \lesssim p' (\frac{1}{\delta})^{1/p'} \|f \|_{L^{p}(M_{ L(\log L)^{p-1+\delta}} (w))}$, for $1<p<\infty, w\geq 0, 0<\delta \leq 1. $ As as a consequence we deduce that $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim [w]_{A_{1}} \log(e+ [w]_{A_{\infty}}) \int_{\mathbb{R}^n} |f| w dx, $ extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.