The John-Nirenberg inequality with sharp constants

TL;DR

This paper determines the exact sharp constants in the one-dimensional John-Nirenberg inequality, establishing the optimal values for both the exponential decay rate and the coefficient, which enhances the understanding of the inequality's precise bounds.

Contribution

The paper proves the exact sharp constants for the John-Nirenberg inequality in one dimension, specifically identifying the optimal values of C_1 and C_2.

Findings

C_2=2/e is the sharp constant for the exponential decay.

The best possible C_1 corresponding to C_2=2/e is (1/2)e^{4/e}.

The results provide the precise bounds for the inequality.

Abstract

We consider the one-dimensional John-Nirenberg inequality: $$ |\{x\in I_0:|f(x)-f_{I_0}|>\a\}|\le C_1|I_0|\exp\Big(-\frac{C_2}{\|f\|_{*}}\a\Big). $$ A. Korenovskii found that the sharp $C_2$ here is $C_2=2/e$. It is shown in this paper that if $C_2=2/e$, then the best possible $C_1$ is $C_1= \frac{1}{2}e^{4/e}$.