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

TL;DR

The paper extends the Christ–Fefferman sparse domination paradigm to the one-sided Hardy–Littlewood maximal operator $M^{+}$, aiming to obtain sharp weighted and two-weight estimates without relying on dyadic grids. It establishes a tight one-sided bound $ orm{M^{+}f}_{L^{p}(w)} \le c_p [w]_{A_p^{+}}^{1/(p-1)} \norm{f}_{L^{p}(w)}$ and a mixed-type bound $\norm{M^{+}f}_{L^{p}(w)} \le c_p ([w]_{A_p^{+}} [\sigma]_{A_{\infty}^{-}})^{1/p} \norm{f}_{L^{p}(w)}$ with $\sigma = w^{-1/(p-1)}$, together with a one-sided two-weight bump framework for sufficiency and necessity of corresponding mixed weak-type inequalities. A fractional/weighted variant and a two-weight quantitative estimate are developed, using Lorentz and Luxemburg norms with Young functions to produce explicit constants. The results provide a dyadic-free, sparse-type approach to one-sided weighted theory, yielding new proofs of sharp one-sided estimates and a flexible framework for future one-sided weighted inequalities with explicit constants. Overall, the work broadens the scope of sparse domination techniques to one-sided operators and clarifies the role of one-sided bump conditions in quantitative weighted theory.

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 Pérez and Rela, and also one sided counterparts of some very recent mixed weak type results due to Sweeting.

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7