In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John-Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge-Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.

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

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