Uchiyama's lemma and the John-Nirenberg inequality

TL;DR

This paper provides simplified proofs of key inequalities and norm comparisons in BMO spaces using integral formulas and Uchiyama's lemma, avoiding the traditional John-Nirenberg inequality.

Contribution

It introduces new, straightforward proofs of BMO norm comparisons and the strong John-Nirenberg inequality leveraging integral formulas and a lemma of Uchiyama.

Findings

Simplified proofs of BMO norm comparisons

Proof of the strong John-Nirenberg inequality

Inclusions of BMOA in dual of H^1 and BMO in dual of real H^1

Abstract

Using integral formulas based on Green's theorem and in particular a lemma of Uchiyama, we give simple proofs of comparisons of different BMO norms without using the John-Nirenberg inequality while we also give a simple proof of the strong John-Nirenberg inequality. Along the way we prove the inclusions of BMOA in the dual of H^1 and BMO in the dual of real H^1.