Vasily Vasyunin

Sharp results in the integral-form John-Nirenberg inequality

We consider the strong form of the John-Nirenberg inequality for the L 2 -based BMO. We construct explicit Bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant, as well as the precise bound on the inequalitys range of validity, both previously unknown. The results for the two cases are substantially different. The paper not only gives another instance in the short list of such explicit calculations, but also presents the Bellman function method as a sequence of clear steps, adaptable to a wide variety of applications.

Sharp constants in the classical weak form of the John–Nirenberg inequality

The sharp constants in the classical John--Nirenberg inequality are found by using Bellman function approach.

Weak integral conditions for BMO

We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if

On weak weighted estimates of the martingale transform and a dyadic shift

Q

On Weak Weighted Estimates of Martingale Transform

is a cube in

Quasi-similarity of contractions having a 2 × 1 characteristic function

\mathbb{R}^n

Monge--Amp\`ere equation and Bellman optimization of Carleson Embedding Theorems

and

Monge-Ampère equations and Bellman functions : The dyadic maximal operator

h:[0,\infty)\to[0,\infty)

Burkholder’s function via Monge–Ampère equation

is such that

On Bellman function for extremal problems in BMO

h(t)\underset{t\to\infty}{\longrightarrow}\infty,