Sharp Lorentz estimates for dyadic-like maximal operators and related Bellman functions

TL;DR

This paper precisely evaluates Bellman functions for dyadic maximal operators and related martingale maximal operators, providing sharp Lorentz estimates and symmetrization techniques to determine supremums under fixed integral and Lorentz norm constraints.

Contribution

The authors introduce a symmetrization principle to exactly evaluate Bellman functions for dyadic maximal operators and martingale maximal operators with Lorentz norm constraints.

Findings

Exact Bellman function evaluations for dyadic maximal operators.

Sharp Lorentz estimates for maximal operators on martingales.

Determination of supremums under fixed integral and Lorentz norm conditions.

Abstract

We precisely evaluate Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to local Lorentz type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function $\phi$ when the integral of $\phi$ is fixed and also the same Lorentz quasinorm of $\phi$ is fixed. Also we find the corresponding supremum when the integral of $\phi$ is fixed and several weak type conditions are given.