The John--Nirenberg constant of ${\rm BMO}^p$, $p>2$

TL;DR

This paper extends the calculation of the John--Nirenberg constant for ${ m BMO}^p$ spaces to the case where p>2, revealing more complex Bellman functions and advancing understanding of BMO space properties.

Contribution

It computes the John--Nirenberg constant for ${ m BMO}^p$ when p>2, using Bellman functions with more intricate structures than in the previous p≤2 case.

Findings

John--Nirenberg constant determined for p>2

Bellman functions exhibit more complex structures for p>2

Advances understanding of ${ m BMO}^p$ space properties

Abstract

This paper is a continuation of earlier work by the first author who determined the John--Nirenberg constant of ${\rm BMO}^p\big((0,1)\big)$ for the range $1\le p\le 2.$ Here, we compute that constant for $p>2.$ As before, the main results rely on Bellman functions for the $L^p$ norms of logarithms of $A_\infty$ weights, but for $p>2$ these functions turn out to have a significantly more complicated structure than for $1\le p\le 2.$