Weighted Lorentz spaces: sharp mixed $A_p-A_{\infty}$ estimate for maximal functions

TL;DR

This paper establishes sharp weighted estimates for the Hardy-Littlewood maximal function within Lorentz spaces, providing new bounds that improve understanding of weighted inequalities in harmonic analysis.

Contribution

It introduces a rearrangement-free method to obtain sharp mixed $A_p-A_{ abla}$ estimates for maximal functions in Lorentz spaces, applicable to two-weight scenarios.

Findings

Proved sharp mixed $A_p-A_{ abla}$ estimate for maximal functions in Lorentz spaces.

Developed a rearrangement-free approach for weighted inequalities.

Derived new bounds for the strong maximal operator and dual settings.

Abstract

We prove the sharp mixed $A_{p}-A_{\infty}$ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely \[ \|M\|_{L^{p,q}(w)} \lesssim_{p,q,n} [w]^{\frac1p}_{A_p}[\sigma]^{\frac1{\min(p,q)}}_{A_{\infty}}, \] where $\sigma=w^{\frac{1}{1-p}}$. Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for $M$ in a dual setting.