A Characterization of the boundedness of the median maximal function on weighted L^p spaces

TL;DR

This paper introduces a median-based maximal function and characterizes its boundedness on weighted L^p spaces, showing it is bounded for all p > 0 if and only if the weight belongs to A_infinity, with a sharp bound for p=1.

Contribution

It provides a qualitative characterization of the boundedness of the median maximal function on weighted L^p spaces, contrasting with the classical Hardy-Littlewood maximal operator.

Findings

Median maximal function is bounded on L^p(w) for all 0 < p < ∞ if and only if w ∈ A_∞.

The paper establishes a sharp bound for the median maximal function on L^1(w), proportional to [w]_{A_1}.

The characterization does not specify the dependence on [w]_{A_∞}.

Abstract

We introduce and study the median maximal function \mathcal{M} f, defined in the same manner as the classical Hardy-Littlewood maximal function, only replacing integral averages of f by medians throughout the definition. This change has a qualitative impact on the mapping properties of the maximal operator: in contrast with the Hardy-Littlewood operator, which is not bounded on L^1, we prove that \mathcal{M} is bounded on L^p(w) for all 0 < p < \infty, if and only if w \in A_{\infty}. The characterization is purely qualitative and does not give the dependence on [w]_{A_{\infty}}. However, the sharp bound \|\mathcal{M}\|_{L^1(w) \to L^1(w)} \lesssim [w]_{A_1} is established.