# Brief research summary:

My research interests lie in the intersection of Complex Analysis, Harmonic analysis and Geometry.

Many problems I am working on originate from applications, such as Control Theory, Stationary Random Processes and Dynamical Systems.

My earlier research deals with approximations (Menshov's type theorem); BMO-regular lattices and Hardy-type subspaces; Corona Problem and related topics.

Recent research projects involve classical harmonic analysis: Hardy--Littlewood maximal function, sharp weighted estimates for square functions, $A_{p}$ weights and $BMO$ classes. Some of the most recent and my favorite projects in these directions deal with developing a modern Bellman function technique.

Another direction of research concerns about isoperimetric problems: an attempt to find a unified approach to isoperimetric inequalities by reducing them to a Laplacian eigenvalue problem, $\bar{\partial}$-equation, backwards heat equation and Monge--Amp`ere equation with a drift.

# Publications and preprints:

1. P. Ivanishvili, S. V. Kislyakov, Correction up to a function with sparse spectrum and uniformly convergent Fourier series, Zap. Nauchn. Sem. POMI, Vol. 376, 25-47, (2010)

2. P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems on $\text{BMO}$, Comptes Rendus Mathematique, Volume 350, Issues 11-12, Pages 561-564, June (2012)

3. P. Ivanishvili, $J$-closed finite collections of Hardy-type subspaces, Zap. Nauchn. Sem. POMI, Vol. 401, 82-92 (2012)

4. P. Ivanisvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems in BMO, Trans. Amer. Math. Soc (2015) doi: dx.doi.org/10.1090/tran/6460

5. P. Ivanisvili   Inequality for Burkholder's martingale transform. Analysis & PDE 8-4 (2015), 765--806. DOI 10.2140/apde.2015.8.765

6. P. Ivanisvili,   Bellman function approach to the sharp constants in uniform convexity. Advances in Calculus of Variations DOI: 10.1515/acv-2016-0008, August 2016

7. P. Ivanisvili, D. M. Stolyarov, P. B. Zatitskiy   Bellman VS Beurling: sharp estimates of uniform convexity for $L^p$ spaces (2014) To appear in St. Petersburg Mathematical Journal

8. P. Ivanisvili, A. Volberg   Hessian of Bellman functions and uniqueness of Brascamp--Lieb inequality, J. London Math. Soc. (2015), doi:10.1112/jlms/jdv040

9. P. Ivanisvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy,   Sharp estimates of integral functionals on classes of functions with small mean oscillation. To appear in Comptes Rendus Mathematique

10. P. Ivanisvili, A. Volberg   Bellman partial differential equation and the hill property for classical isoperimetric problems (2015)

11. P. Ivanisvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy,   Bellman function for extremal problems in BMO II: evolution., 148 pages, to appear in Memoirs of the AMS

12. P. Ivanisvili, A. Volberg     Isoperimetric functional inequalities via the maximum principle: the exterior differential systems approach. (2015)

13. P. Ivanisvili,     A bundling problem revisited. preprint, (2016)

14. P. Ivanisvili, B. Jaye F. Nazarov   Lower bounds for uncentered maximal functions in any dimension, Int. Math. Res. Notices Vol. 2016, Issue 00, 2016

15. P. Ivanisvili,   Boundary value problem and the Ehrhard inequality (2016)

16. P. Ivanisvili, A. Volberg,   Improving Beckner's bound via Hermite functions (2016)

17. P. Ivanisvili, A. Volberg, F. Nazarov   On weak and strong sharp weighted estimates of square function

18. P. Ivanisvili, A. Volberg,   Poincaré inequality 3/2 on the Hamming cube

# In progress:

19. P. Ivanisvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy,  Bellman functions on general non-convex planar domains.