An alternative proof of a sharp generalization of an integral inequality for the dyadic maximal operator and applications

TL;DR

This paper presents an alternative proof of a sharp integral inequality for the dyadic maximal operator, enabling the evaluation of its Bellman function and extending previous results in the field.

Contribution

The authors provide a new proof technique for a key inequality, facilitating the analysis of the dyadic maximal operator's Bellman function and generalizing earlier findings.

Findings

Established a new proof method for the integral inequality

Enabled evaluation of the Bellman function for the dyadic maximal operator

Extended previous inequalities to broader contexts

Abstract

We give an alternative proof of a sharp generalization of an 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. This last mentioned inequality, which was first noticed in [3],also generalizes in a certain direction the results of [7].