Sharp L^p estimates on BMO

TL;DR

This paper develops Bellman functions for BMO norms to derive sharp constants in inequalities relating BMO oscillations and norms, using solutions to Monge-Ampère boundary value problems.

Contribution

It constructs explicit Bellman functions for BMO norms via Monge-Ampère equations, providing sharp constants in related inequalities.

Findings

Explicit Bellman functions for BMO norms are obtained.

Sharp constants in BMO inequalities are derived.

Solutions involve Monge-Ampère boundary value problems.

Abstract

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--Amp\`ere 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.