The best constant for the centered maximal operator on radial decreasing functions

TL;DR

This paper proves that the optimal constant in the weak type (1,1) inequality for the centered Hardy-Littlewood maximal operator on radial decreasing functions is exactly 1.

Contribution

It establishes the precise value of the best constant for the centered maximal operator on a specific class of functions, filling a gap in harmonic analysis.

Findings

The optimal constant is exactly 1 for radial decreasing functions.

The result applies to the weak type (1,1) inequality.

It advances understanding of maximal operators on symmetric functions.

Abstract

We show that the lowest constant appearing in the weak type (1,1) inequality satisfied by the centered Hardy-Littlewood maximal operator on radial integrable functions is 1.