The Bellman function of the dyadic maximal operator related to Kolmogorov's inequality

TL;DR

This paper computes the Bellman function for the dyadic maximal operator related to Kolmogorov's inequality, providing an alternative proof and characterizing extremal sequences of functions.

Contribution

It offers a precise calculation of the Bellman function and characterizes extremal sequences, enhancing understanding of the dyadic maximal operator's behavior.

Findings

Explicit Bellman function formula derived

Alternative proof of Kolmogorov inequality results

Characterization of extremal function sequences

Abstract

We precisely compute the Bellman function of two variables of the dyadic maximal operator in relation to Kolmogorov inequality. In this way we give an alternative proof of the results in [5].Additionally, we characterize the sequences of functions that are extremal for this Bellman function.The proof of this is based on that is given in this paper for the Bellman function we are interested in.