Sharp results in the integral-form John--Nirenberg inequality

TL;DR

This paper derives sharp constants and bounds for the strong form of the John-Nirenberg inequality in BMO using explicit Bellman functions, highlighting differences between continuous and dyadic cases.

Contribution

It constructs explicit Bellman functions for the inequality, providing the first sharp constants and bounds for both continuous and dyadic BMO settings.

Findings

Sharp constants obtained for the inequality

Explicit Bellman functions constructed for both cases

Differences between continuous and dyadic results highlighted

Abstract

We consider the strong form of the John-Nirenberg inequality for the $L^2$-based BMO. We construct explicit Bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant as well as the precise bound on the inequality's range of validity, both previously unknown. The results for the two cases are substantially different. The paper not only gives another instance in the short list of such explicit calculations, but also presents the Bellman function method as a sequence of clear steps, adaptable to a wide variety of applications.