Bellman Function for Extremal Problems in BMO II: Evolution

Abstract

This volume contains the proceedings of the conference Local and Global Methods in Algebraic Geometry, held from May 12-15, 2016, at the University of Illinois at Chicago, in honor of Lawrence Ein's 60th birthday. The articles cover a broad range of topics in algebraic geometry and related fields, including birational geometry and moduli theory, analytic and positive characteristic methods, geometry of surfaces, singularity theory, hyper-Kähler geometry, rational points, and rational curves. Contents: R. Lazarsfeld, Some remarks on the work of Lawrence Ein; A. Calabri and C. Ciliberto, Contractible curves on a rational surface; F. Catanese, On the canonical map of some surfaces isogenous to a product; C. Ciliberto, F. Flamini, C. Galati, and A. L. Knutsen, Degeneration of differentials and moduli of nodal curves on 3 surfaces; I. Coskun and J. Huizenga, Weak Brill-Noether for rational surfaces; R. Datta and K. E. Smith, Excellence in prime characteristic; M. González Villa, A. Libgober, and L. Maxim, Motivic zeta functions and infinite cyclic covers; C. Hacon, M. Popa, and C. Schnell, Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun; S. Ishii and W. Niu, A strongly geometric general residual intersection; J. Kollár, Quadratic solutions of quadratic forms; S. J Kovács, Non-Cohen-Macaulay canonical singularities; N. Mok, Full cones swept out by minimal rational curves on irreducible Hermitian symmetric spaces as examples of varieties underlying geometric substructures; M. Mustaţă and Y. Nakamura, A boundedness conjecture for minimal log discrepancies on a fixed germ; E. Sernesi, The Wahl map of one-nodal curves on K3 surfaces; Y.-T. Siu, Skoda's ideal generation from vanishing theorem for semipositive Nakano curvature and Cauchy-Schwarz inequality for tensors; C. Voisin, Hyper-Kähler compactification of the intermediate Jacobian fibration of a cubic fourfold: The twisted case.

Lower Bounds for Uncentered Maximal Functions in Any Dimension

“…It can be derived by using the methods recently developed in works of N. Osipov, D. Stolyarov, V. Vasyunin, P. Zatitskiy and P. Ivanisvili (see e.g. [7,8,6]). Lemma 3.…”

Theory of locally concave functions and its applications to sharp estimates of integral functionals

“…The paper [33] was the first to appear. Following this route, many authors have managed to prove various inequalities, see the papers [1,4,8,7,15,26,22,24,31,28,29,32,33,34,35,36,37,38]. Though the method turned out to be very useful, there was absolutely no theory that allowed to calculate the Bellman functions "mechanically".…”

“…In [8], most of them were unified into a single theory (see also the short report [9]). However, for the treatment in the full generality, see the forthcoming paper [7] (still on the space BMO! ).…”

“…All the mentioned papers (including the latest papers [8,7]), in some sense, employ a miracle. The main theorem of the present paper was usually proved for a specific case in such a fashion: it was assumed to be valid, from these assumptions, the author guessed the "formula" for the Bellman function, after that he guessed the optimizers, and finally, he verified the concavity, thus proving that his guess for the Bellman function was right (and also proving the main theorem for his particular case).…”

Sharpening Hölder's inequality

“…The closest one is the Bellman function in [13]. See [11] for a more geometric point of view and [9], [10], and [12] for a study of a related problem. We also refer the reader to [15], [16], [19], and [20] for history and basics of the Bellman function theory.…”