Bellman function for extremal problems in BMO

We develop a general method for obtaining sharp integral estimates on BMO. Each such estimate gives rise to a Bellman function, and we show that for a large class of integral functionals, this function is a solution of a homogeneous Monge–Ampère boundary-value problem on a parabolic plane domain. Furthermore, we elaborate an essentially geometric algorithm for solving this boundary-value problem. This algorithm produces the exact Bellman function of the problem along with the optimizers in the inequalities being proved. The method presented subsumes several previous Bellman-function results for BMO, including the sharp John–Nirenberg inequality and sharp estimates of Lp -norms of BMO functions. 2010 Mathematics Subject Classification. Primary 42A05, 42B35, 49K20.