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On a constant related to the Bellman function of three integral variables of the dyadic maximal operator: Part A

Eleftherios N. Nikolidakis

TL;DR

The paper addresses a Bellman-function problem for the dyadic maximal operator with three integral variables, focusing on the monotonicity of the associated constant $t(s_1,s_2)$ in the second variable $s_2$. Building on prior work, it analyzes the implicit defining relation for $t$ and conducts a detailed derivative analysis using auxiliary functions $H_p$, $H_q$, $\omega_p$, $\omega_q$ and the parameter $\alpha(s_2)$, establishing that $t(s_1,s_2)$ increases with $s_2$ for small values and decreases for larger $s_2$, with a critical threshold $s_2'=H_q(\omega_p(s_1))$. This unimodal behavior sharpens the understanding of the Bellman function in the three-variable setting and may aid in determining the optimal constant in the corresponding inequality. The results contribute to the refinement of Bellman-function techniques for dyadic maximal operators and have potential implications for sharper $L^p$-inequalities via three-variable extremal problems.

Abstract

We study the behaviour of the constant that is provided in the articles [12] and [13], which is connected with the determination of the Bellman function of three integral variables of the dyadic maximal operator. More precisely we study the monotonicity properties of this constant with respect to the second variable from which it depends.

On a constant related to the Bellman function of three integral variables of the dyadic maximal operator: Part A

TL;DR

The paper addresses a Bellman-function problem for the dyadic maximal operator with three integral variables, focusing on the monotonicity of the associated constant in the second variable . Building on prior work, it analyzes the implicit defining relation for and conducts a detailed derivative analysis using auxiliary functions , , , and the parameter , establishing that increases with for small values and decreases for larger , with a critical threshold . This unimodal behavior sharpens the understanding of the Bellman function in the three-variable setting and may aid in determining the optimal constant in the corresponding inequality. The results contribute to the refinement of Bellman-function techniques for dyadic maximal operators and have potential implications for sharper -inequalities via three-variable extremal problems.

Abstract

We study the behaviour of the constant that is provided in the articles [12] and [13], which is connected with the determination of the Bellman function of three integral variables of the dyadic maximal operator. More precisely we study the monotonicity properties of this constant with respect to the second variable from which it depends.
Paper Structure (2 sections, 4 theorems, 58 equations)

This paper contains 2 sections, 4 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. The monotonicity properties of $t(s_1,s_2)$ with respect to the variable $s_2$

Key Result

Lemma 2.1

Let $(s_1,s_2)$ satisfying $s_2=s'_2=H_q({\omega_p(s_1)})$. Then if we define $\tau'=\tau({s_1,s'_2})=\frac{p-q}{p}\,\frac{t^p-s_1}{t^{p-q}-\frac{s_1}{s'_2}}$, then we have that $\tau'=H_q({\omega_p(s_1)})$.

Theorems & Definitions (8)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof