Bellman function for extremal problems in BMO

Ivanisvili, Paata; Osipov, Nikolay; Stolyarov, Dmitriy; Vasyunin, Vasily; Zatitskiy, Pavel

doi:10.1090/tran/6460

Bellman function for extremal problems in BMO
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by Paata Ivanisvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin and Pavel B. Zatitskiy PDF

Trans. Amer. Math. Soc. 368 (2016), 3415-3468 Request permission

Abstract:

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.

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