TY - JOUR

T1 - BELLMAN FUNCTION FOR EXTREMAL PROBLEMS IN BMO

AU - Ivanishvili, P.

AU - Osipov, N.

AU - Stolyarov, D.

AU - Vasyunin, V.

AU - Zatitskiy, P.

PY - 2016/5

Y1 - 2016/5

N2 - 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-Ampere 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.

AB - 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-Ampere 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.

U2 - 10.1090/tran/6460

DO - 10.1090/tran/6460

M3 - статья

VL - 368

SP - 3415

EP - 3468

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -