# Bellman function for extremal problems in BMO

@article{Ivanisvili2012BellmanFF, title={Bellman function for extremal problems in BMO}, author={Paata Ivanisvili and Nikolay N. Osipov and Dmitriy M. Stolyarov and Vasily I. Vasyunin and Pavel Zatitskiy}, journal={Transactions of the American Mathematical Society}, year={2012}, volume={368}, pages={3415-3468} }

In this paper we develop the method of nding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John{Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge{Amp ere equation in a parabolic strip for suciently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a…

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## 52 Citations

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## References

SHOWING 1-10 OF 26 REFERENCES

Sharp results in the integral-form John-Nirenberg inequality

- Mathematics
- 2007

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 L^p estimates on BMO

- Mathematics
- 2011

We construct the upper and lower Bellman functions for the $L^p$ (quasi)-norms of BMO functions. These appear as solutions to a series of Monge--Ampere boundary value problems on a non-convex plane…

Mutual estimates of Lp-norms and the Bellman function

- Mathematics
- 2009

In this paper, we describe the range of the Lp-norm of a function under fixed Lp-norms with two other different exponents p and under a natural multiplicative restriction of the type of the…

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

- Mathematics, Computer Science
- 2008

This work explores the way of solving Monge--Amp\`ere equation by a sort of method of characteristics to find the Bellman function of certain classical Harmonic Analysis problems, and, therefore, of finding full structure of sharp constants and extremal sequences for those problems.

The sharp $A_p$ constant for weights in a reverse-H\

- Mathematics
- 2007

In a recent paper V. Vasyunin presented a proof of the reverse H\"older inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the…

Introduction to H[p] spaces

- Mathematics
- 1998

Preface Preface to the first edition 1. Rudiments 2. Theorem of the brothers Reisz. Introduction to the space H1 3. Elementary boundary behaviour theory for analytic functions 4. Application of…

Introduction to Hp Spaces

- Mathematics
- 1999

Preface Preface to the first edition 1. Rudiments 2. Theorem of the brothers Reisz. Introduction to the space H1 3. Elementary boundary behaviour theory for analytic functions 4. Application of…

Bellman function in stochastic control and harmonic analysis

- Mathematics, Computer Science
- 2001

This work shows how the homonym function in harmonic analysis is (and how it is not) the same stochastic optimal control Bellman function, and presents several creatures from Bellman’s Zoo.