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

TL;DR

This paper investigates the monotonicity of the constant $t(s_1,s_2)$ arising in the Bellman-function framework for the dyadic maximal operator with three integral variables. It analyzes the implicit relation connecting $t$, $s_1$, $s_2$, and auxiliary functions $\omega_p$, $\omega_q$, $\tau$, and $\alpha(s_2)$, performing a sign-analysis of $\partial t/\partial s_1$ to establish monotonicity. The main result is that $t(s_1,s_2)$ is strictly decreasing in the first variable $s_1$ for admissible $(s_1,s_2)$, with no flat intervals in the domain. This advances the understanding toward a complete characterization of the three-variable Bellman function $B_{p,q}^{\mathcal{T}}(f,A,F)$ and its sharp bounds for the dyadic maximal operator, contributing to refined $L^p$-inequality analyses in this martingale-dyadic setting.

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 first variable from which it depends.

Theorems & Definitions (1)

• Theorem 2.1