A sharp integral inequality for the dyadic maximal operator and a related stability result

TL;DR

This paper establishes a sharp integral inequality for the dyadic maximal operator, simplifies its proof, and explores a stability condition for sequences of functions where equality is approached.

Contribution

It introduces a more straightforward proof of a known sharp inequality and provides a necessary and sufficient stability condition for near-equality in the limit.

Findings

Proved a simplified, sharp integral inequality for the dyadic maximal operator.

Derived a stability criterion for sequences approaching equality in the inequality.

Connected the inequality to the evaluation of the Bellman function for the operator.

Abstract

We prove a sharp integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables is possible, as can be seen in [3]. Our inequality of interest is proved in this article by a simpler and more immediate way. We also study a stability result in connection with this inequality, that is we provide a necessary and sufficient condition, for a sequence of functions,under which we obtain equality in the limit. The proof of this result is based on the proof of the related inequality which we present in this article.