On a sharp upper bound related to the Bellman function of three integral variables of the dyadic maximal operator

TL;DR

The paper investigates sharp upper bounds for integral expressions tied to the Bellman function of three integral variables for the dyadic maximal operator in a tree-like Bellman-function setting. Building on Nikol's characterization of the constant $t(s_1,s_2)$, the authors derive precise conditions that determine when $t$ lies on either branch of its definition, and establish monotonicity properties and boundary curves governing the sign of $t'(0)$ across the $(s_1,s_2)$-domain. They show a dichotomy: for small $s_1$, $t'(0)<0$ for all admissible $s_2$, while for larger $s_1$ the region splits into subregions with $t'(0)<0$ or $t'(0)>0$, and they relate these to the solvability of $F_{s_1,s_2}(t)=0$ or $t(0)= heta_p(s_1)$, where $\theta_p$ is a fixed branch of the $\omega_p$-related inverse. These results refine the three-variable Bellman-function problem and set the stage for a full treatment of the three-variable Bellman problem $B_{p,q}^{\mathcal{T}}(f,A,F)$ in the given tree-like framework, with implications for sharp inequalities for the dyadic maximal operator.

Abstract

We study properties for the sharp upper bound for integral quantities related to the Bellman function of three integral variables of the dyadic maximal operator, that is determined in [11].

Theorems & Definitions (15)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• Lemma 2.5
• proof
• Lemma 2.6