Extremal sequences for the Bellman function of three variables of the dyadic maximal operator in relation to Kolmogorov's inequality

TL;DR

This paper characterizes extremal sequences for the Bellman function of a three-variable dyadic maximal operator, showing they behave like eigenfunctions, advancing understanding of inequalities related to the operator.

Contribution

It introduces a new characterization of extremal sequences for the Bellman function in relation to Kolmogorov's inequality, using an approach based on eigenfunction behavior.

Findings

Extremal sequences behave approximately like eigenfunctions.

The Bellman function for the three-variable case has been precisely evaluated.

The approach links extremal sequences to eigenvalues of the operator.

Abstract

We give a characterization of the extremal sequences for the Bellman function of three variables of the dyadic maximal operator in relation to Kolmogorov's inequality. In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue. For this approach we use the one introduced in [4], where the respective Bellman function has been precisely evaluated.