# A multiparameter integral inequality for the dyadic maximal operator and applications

## TL;DR

This paper establishes a sharp multiparameter integral inequality for the dyadic maximal operator, refines existing one-parameter bounds, and explores the Bellman function's domain and bounds for applications in harmonic analysis.

## Contribution

It introduces a new sharp multiparameter inequality for the dyadic maximal operator and determines the exact domain and bounds of the associated Bellman function.

## Key findings

- Proved a sharp multiparameter inequality for the dyadic maximal operator.
- Determined the exact domain of the Bellman function with three integral variables.
- Provided lower bounds for the Bellman function based on the inequality.

## Abstract

We prove a sharp multiparameter integral inequality for the dyadic maximal operator which refines the one-parameter inequality that is given by A.Melas in [4] which in turn is applied for the evaluation of the Bellman function of two integral variables for this maximal operator. Moreover we find the exact domain of definition of the related Bellman function of three integral variables and by using the results connected with the sharpness of this new multiparameter inequality we give lower bounds of this Bellman function.