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.

We construct the upper and lower Bellman functions for the L p (quasi)-norms of BMO functions. These appear as solutions to a series of Monge-Ampère boundary value problems on a non-convex plane domain. The knowledge of the Bellman functions leads to sharp constants in inequalities relating average oscillations of BMO functions and various BMO norms.the smallest such C being the corresponding norm (quasi-norm for 0 < p < 1),It is known that all p-based norms defined by (1.3) are equivalent and so (1.2) defines the same space for all p > 0. This fact is usually seen as a consequence of the John-Nirenberg inequality, although using that inequality to prove it will produce suboptimal constants of norm equivalence. One of the primary motivations of this work is to quantify this equivalence precisely, in dimension 1. To this end, we relate all BMO p norms to the BMO 2 norm. The reason BMO 2 norm plays a central role here is that it allows us to take advantage of the self-duality of L 2 (Q). From now on, we reserve the name BMO for BMO 2 :Thus, we would like to find the best constants c p , C p in the double inequalitiesProof. First, let us note that it is sufficient to prove this lemma for a one-sided cut, for example, for c = −∞. We then get the full statement by applying this argument twice. Indeed, if we denote by C d ϕ the cut-off of ϕ from above at height d, i.e.. Take a cube J and let J 1 = {s ∈ J : ϕ(s) ≤ d} and J 2 = {s ∈ J : ϕ(s) > d}. If either J 1 = ∅ or J 2 = ∅, the statement is trivial. Thus, we may assume that J k = ∅. Let β k = |J k |/|J|, k = 1, 2. We have the following identity:which proves the lemma, because ϕ

Abstract. We find explicitly the Bellman function for the dyadic maximal operator on L p as the solution of a Bellman PDE of Monge-Ampère type. This function has been previously found by A. Melas [M] in a different way, but it is our PDE-based approach that is of principal interest here. Clear and replicable, it holds promise as a unifying template for past and current Bellman function investigations.

Abstract. We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if Q is a cube in R n and h :Under some additional assumptions on h we obtain estimates on ϕ BMO in terms of the supremum above. We also show that even though the condition h(t) −→ t→∞ ∞ is not necessary for this implication to hold, it becomes necessary if one considers the dyadic BMO.

We compute the exact John-Nirenberg constant of BMO p (0, 1) for 1 ≤ p ≤ 2, which has been known only for p = 1 and p = 2. We also show that this constant is attained in the weak-type John-Nirenberg inequality and obtain a sharp lower estimate for the distance in BMO p to L ∞ . These results rely on sharp L p -and weak-type estimates for logarithms of A∞ weights, which in turn use the exact expressions for the corresponding Bellman functions.