Sharp L^p estimates on BMO Leonid SlavinV. Vasyunin 42A0542B3549K20BMOnorm equivalenceexplicit Bellman functionMonge-Ampere equation 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. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4651 10.1512/iumj.2012.61.4651 en Indiana Univ. Math. J. 61 (2012) 1051 - 1110 state-of-the-art mathematics http://iumj.org/access/