Geodesic
On locally concave functions on simplest nonconvex domain
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 50, Tome 512 (2022), pp. 40-87 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that certain Bellman functions of several variables are minimal locally concave functions. This generalizes earlier results about Bellman functions of two variables. 
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P. B. Zatitskiy; D. M. Stolyarov. On locally concave functions on simplest nonconvex domain. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 50, Tome 512 (2022), pp. 40-87. http://geodesic.mathdoc.fr/item/ZNSL_2022_512_a4/

[1] V. I. Vasyunin, “Tochnaya konstanta v obratnom neravenstve Geldera dlya makenkhauptovskikh vesov”, Algebra i analiz, 15:1 (2003), 73–117

[2] V. I. Vasyunin, “Vzaimnye otsenki $l^p$-norm i funktsiya Bellmana”, Zap. nauchn. semin. POMI, 355, 2008, 81–138

[3] A. N. Shiryaev, Veroyatnost 2, MTsNMO, 2021

[4] D. Azagra, “Global and fine approximation of convex functions”, Proc. Lond. Math. Soc. (3), 107:4 (2013), 799–824 | DOI | MR

[5] D. Azagra, D. Stolyarov, Inner and outer smooth approximation of convex hypersurfaces. When is it possible?, arXiv: 2204.07498

[6] H. Brezis, L. Nirenberg, “Degree theory and BMO. I. Compact manifolds without boundaries”, Selecta Math., 1:2 (1995), 197–263 | DOI | MR

[7] E. Dobronravov, On a minimal locally concave function on a ring, in preparation

[8] B. Guan, “The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature”, Trans. Amer. Math. Soc., 350:12 (1998), 4955–4971 | DOI | MR

[9] T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach spaces, v. I, Martingales and Littlewood–Paley theory, Springer, 2016 | MR

[10] P. Ivanishvili, 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”, C. R. Math. Acad. Sci. Paris, 350:11 (2012), 561–564 | DOI | MR

[11] P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, “On Bellman function for extremal problems in $\mathrm{BMO}$”, C. R. Math. Acad. Sci. Paris, 353:12 (2015), 1081–1085 | DOI | MR

[12] P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, “Bellman function for extremal problems in $\mathrm{BMO}$”, Trans. Amer. Math. Soc., 368 (2016), 3415–3468 | DOI | MR

[13] P. Ivanishvili, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems on $\mathrm{BMO}$ II: evolution, Mem. Amer. Math. Soc., 255, no. 1220, 2018 | MR

[14] F. John, L. Nirenberg, “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426 | DOI | MR

[15] B. Kirchheim, S. Müller, V. Šverák, “Studying nonlinear pde by geometry in matrix space”, Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003, 347–395 | DOI | MR

[16] A. Reznikov, “Sharp weak type estimates for weights in the class $A_{p_1,p_2}$”, Rev. Mat. Iberoam., 29:2 (2013), 433–478 | DOI | MR

[17] R. T. Rockafellar, Convex analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, N.J., 1970 | MR

[18] L. Slavin, V. Vasyunin, “Sharp results in the integral form John–Nirenberg inequality”, Trans. Amer. Math. Soc., 363:8 (2011), 4135–4169 | DOI | MR

[19] L. Slavin, V. Vasyunin, “Sharp $L^p$ estimates on $\mathrm{BMO}$”, Indiana Univ. Math. J., 61:3 (2012), 1051–1110 | DOI | MR

[20] E. M. Stein, Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993 | MR

[21] D. Stolyarov, V. Vasyunin, P. Zatitskiy, “Sharp multiplicative inequalities with BMO I”, J. Math. Anal. Appl., 492:2 (2020), 124479, 17 pp. | DOI | MR

[22] D. Stolyarov, P. Zatitskiy, “Sharp transference principle for BMO and $A_p$”, J. Funct. Anal., 281:6, 109085 | DOI | MR

[23] D. M. Stolyarov, P. B. Zatitskiy, “Theory of locally concave functions and its applications to sharp estimates of integral functionals”, Adv. Math., 291 (2016), 228–273 | DOI | MR

[24] V. Vasyunin, A. Volberg, “Sharp constants in the classical weak form of the John–Nirenberg inequality”, Proc. Lond. Math. Soc., 108:6 (2014), 1417–1434 | DOI | MR

[25] V. Vasyunin, P. Zatitskiy, I. Zlotnikov, “Sharp multiplicative inequalities with BMO II”, J. Math. Anal. Appl., 515:2, 126430, arXiv: 2111.05565 | DOI | MR

[26] V. I. Vasyunin, The sharp constant in the John–Nirenberg inequality, PDMI preprints 20/2003 http://www.pdmi.ras.ru/preprint/2003/03-20.html