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Regular version of the site

Article

Bellman function for extremal problems in BMO

Transactions of the American Mathematical Society. 2016. Vol. 368. P. 3415-3468.
Nikolay N. Osipov, Ivanisvili P., Stolyarov D. M., Vasyunin V. I., Zatitskiy P. B.

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 L^p-norms of BMO functions.