**Part 1. Bellman function method and geometry of Lp spaces. **

We will talk about several problems concerning the geometry of Lp spaces and explain how to solve them using Bellman function machinery. First, following paper [3] we will concentrate on the uniform convexity property of Lp. We will discuss the classical Clarkson’s and Hanner’s inequalities (see [1] and [2] correspondingly), prove both of them using the Bellman function method.

Second, following paper [4] we will discuss some improvements of the classical H ̈older’s inequality. Namely, we will obtain an inequality which might be considered as some analog of the Pythagorean theorem for Lp spaces. The Bellman function method provides us to find sharp versions of the inequality.

**Part 2. Locally concave functions and sharp estimates of integral functionals.**

In the second part of the course, we will discuss several problems about integral functionals. We will see how the Bellman function method might be applied to these problems and how corresponding geometric problems about the so-called locally concave functions naturally arise. The basic example here is the classical John-Nirenberg inequality on the BMO space (the space of functions of bounded mean oscillation) about the exponential decay of the distributions of the functions from BMO. We will demonstrate the main principles of the method and construct the Bellman function corresponding to the problem. This method allows one to calculate sharp constants in this and other integral inequalities on the BMO space, on the Muckenhoupt weights classes, and on other classes of functions of the similar nature. Further, we will talk about the probabilistic nature of these problems, and also discuss the mysterious connection of these problems and some questions of estimating the distributions of martingale transformations. This part of the talk is based on the series of papers, see for example [5], [6] and [7].

**References**

[1] J. A. Clarkson, Uniformly convex spaces, Transactions of the American Mathematical Society, 40(3), 396-414, (1936).

[2] O. Hanner, On the uniform convexity of Lp and lp, Arkiv for Matematik, 3(3), 239-244, (1956).

[3] P. Ivanisvili, D. M. Stolyarov, P. B. Zatitskiy, Bellman VS Beurling: sharp estimates of uniform convexity for Lp spaces, St. Petersburg Mathematical Journal, 27:2 (2016), 333- 343 (originally in Russian). http://arxiv.org/abs/1405.6229.

[4] H. Hedenmalm, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Sharpening H ̈older’s inequality, Journal of Functional Analysis, 275, 5, 1280–1319, (2018). https://arxiv.org/abs/1708.08846

[5] P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems on BMO, Transactions of the AMS, 368:5 (2016), 3415– 3468. http://arxiv.org/abs/1205.7018

[6] P. Ivanisvili, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems in BMO II: evolution, Memoirs of the AMS, 255:1220 (2018). http://arxiv.org/abs/1510.01010[7] D. M. Stolyarov, P. B. Zatitskiy, Theory of locally concave functions and its applications to sharp estimates of integral functionals, Advances in Mathematics 291 (2016), 228-273. https://arxiv.org/abs/1412.5350

**Monday, 6 May 2019, 11:00 - 13:00 **

**Tuesnday, 7 May 2019, 11:00 - 13:00 **

**Wednesday, 8 May 2019, 11:00 - 13:00 **

**Thursday, 9 May 2019, 11:00 - 13:00**