The Bellman function of two variables for the dyadic maximal operator

TL;DR

This paper determines the Bellman function of two variables for the dyadic maximal operator in a general tree-like setting, offering a new simple proof that advances understanding of maximal operators in harmonic analysis.

Contribution

It introduces a new, elementary proof for the Bellman function of the dyadic maximal operator in a general setting, extending previous results.

Findings

Explicit Bellman function for the dyadic maximal operator

Elementary proof method introduced

Generalization to tree-like maximal operators

Abstract

For p>1 we find the Bellman function of two variables associated with the dyadic maximal operator on Rn.Actually we do that in the more general setting of tree-like maximal operators.We provide a simple and elementary proof,different from those in [4] and [9].