A Christ-Fefferman type approach to the one sided maximal operator

TL;DR

This paper introduces a Christ-Fefferman inspired approach to analyze the one sided maximal operator, providing new proofs and extensions of weighted and two-weight estimates in harmonic analysis.

Contribution

It offers a novel Christ-Fefferman type method for the one sided maximal function, yielding alternative proofs and new results in weighted inequalities.

Findings

Provided an alternative proof for the sharp weighted estimate of the one sided maximal function.

Extended two weight bump conditions to the one sided setting.

Established one sided counterparts of recent mixed weak type results.

Abstract

In this paper, an approach to the one sided maximal function in the spirit of the Christ-Fefferman proof for the strong type weighted estimates of the maximal function is provided. As applications of that approach, we provide an alternative proof of the sharp weighted estimate for the one sided maximal function that was settled by one of us and de la Torre, a one sided two weight bumps counterpart of a result of P\'erez and Rela, and also one sided counterparts of some very recent mixed weak type results due to Sweeting.