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A multiparameter integral inequality for the dyadic maximal operator and applications

Eleftherios N. Nikolidakis

TL;DR

This work addresses sharp multiparameter integral inequalities for the dyadic maximal operator and the corresponding Bellman functions with three integral variables. It develops a tree-based, linearization-inspired approach and a symmetrization principle to prove a sharp three-parameter inequality for $M_{\mathcal{T}}$, then determines the domain and sharpness of the Bellman function $B_{p,q}^{\mathcal{T}}(f,A,F)$ and derives lower bounds in critical subdomains. By connecting these inequalities to Hardy-type transforms and extremizer constructions, the paper extends Melas' one-parameter results to a richer multiparameter setting and provides detailed structure for the associated Bellman function, including exact range, sharpness, and limiting behavior. The results enhance understanding of sharp bounds for dyadic martingales and offer precise multidimensional norm estimates with potential applications in harmonic analysis and related areas.

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.

A multiparameter integral inequality for the dyadic maximal operator and applications

TL;DR

This work addresses sharp multiparameter integral inequalities for the dyadic maximal operator and the corresponding Bellman functions with three integral variables. It develops a tree-based, linearization-inspired approach and a symmetrization principle to prove a sharp three-parameter inequality for , then determines the domain and sharpness of the Bellman function and derives lower bounds in critical subdomains. By connecting these inequalities to Hardy-type transforms and extremizer constructions, the paper extends Melas' one-parameter results to a richer multiparameter setting and provides detailed structure for the associated Bellman function, including exact range, sharpness, and limiting behavior. The results enhance understanding of sharp bounds for dyadic martingales and offer precise multidimensional norm estimates with potential applications in harmonic analysis and related areas.

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.

Paper Structure

This paper contains 7 sections, 9 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A multiparameter inequality for $M_{\mathcal{T}}$
  4. A first application of Theorem 1.1
  5. The range of $(f,A,F)$
  6. Sharpness of Theorem 1.1 - connections with $B_{p,q}^{\mathcal{T}}(f,A,F)$
  7. A lower bound for $B_{p,q}(f,A,F)$ when $(f,A,F)$ satisfies $\omega_q \left ( \frac{f^q}{A} \right ) < \omega_p \left ( \frac{f^p}{F} \right )$

Key Result

Theorem 1.1

Let $\beta\geq\gamma\geq 0$, and $K$ an arbitrary measurable subset of $X$, with measure $k\in (0,1]$. Then for every $\phi\in L^p(X,\mu)$ such that $\int_X \phi \; d\mu = f$ and $\int_X \phi^p \; d\mu = F$ the following inequality is true

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 5.1
  • proof
  • Remark 5.1
  • Theorem 5.1
  • ...and 7 more