Uchiyama's lemma and the John-Nirenberg inequality

Using integral formulas based on Green's theorem and in particular, a lemma of Uchiyama, we give simple proofs of comparisons of different bounded mean oscillation 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 BMOA ∪ (H¹)* and BMO ∪ Re(H¹) *.