Estimates for Bellman functions related to dyadic-like maximal operators on weighted spaces

TL;DR

This paper introduces new estimates for Bellman functions associated with dyadic maximal operators on weighted spaces, utilizing symmetrization principles and double maximal operator estimates to improve understanding of weighted inequalities.

Contribution

It provides novel bounds for Bellman functions related to dyadic and martingale maximal operators on weighted spaces, expanding the theoretical framework.

Findings

New estimates for Bellman functions are established.

Conditions on weights imply bounds for maximal operators.

Utilizes symmetrization and double maximal operator techniques.

Abstract

We provide some new estimates for Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to weighted spaces. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we introduce certain conditions on the weight that imply estimate for the maximal operator on the corresponding weighted space. Also using a well known estimate for the maximal operator by a double maximal operators on different m easures related to the weight we give new estimates for the above Bellman type functions.