The endpoint Fefferman-Stein inequality for the strong maximal function

TL;DR

This paper establishes an endpoint Fefferman-Stein inequality for the strong maximal function in higher dimensions, extending previous two-dimensional results to all dimensions greater than one, with implications for weighted inequalities.

Contribution

The authors prove a new endpoint Fefferman-Stein inequality for the strong maximal function in all dimensions greater than one, generalizing prior two-dimensional findings.

Findings

Proved the inequality for all n > 1

Extended Mitsis' 2D result to higher dimensions

Provided bounds involving strong Muckenhoupt weights

Abstract

Let Mf denote the strong maximal function of f on R^n, that is the maximal average of f with respect to n-dimensional rectangles with sides parallel to the coordinate axes. For any dimension n>1 we prove the natural endpoint Fefferman-Stein inequality for M and any strong Muckenhoupt weight w: w({x \in R^n: M f (x) > t}) \lesssim_{w,n} \int_{R^n} |f|/t [1 + (log^+ |f|/t)^{n-1}] Mw. This extends the corresponding two-dimensional result of T. Mitsis.