Extremal sequences for the Bellman function of the dyadic maximal operator and applications to the Hardy operator

TL;DR

This paper investigates extremal sequences related to the Bellman function of the dyadic maximal operator, showing they behave like eigenfunctions, and applies this understanding to the Hardy operator.

Contribution

It introduces a novel characterization of extremal sequences as approximate eigenfunctions for the dyadic maximal operator and extends this to the Hardy operator.

Findings

Extremal sequences behave as eigenfunctions of the dyadic maximal operator.

The eigenfunction behavior is used to analyze the Hardy operator.

Results provide new insights into the structure of extremal functions for these operators.

Abstract

We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator.