In this paper we develop the method of finding 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-Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.

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.

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Abstract. Under some assumptions on the vectors a1, .., an ∈ R k and the function B : R n → R we find the sharp estimate of the expression´R k B(u1(a1 · x), . . . , un(an · x))dx in terms of´R uj (y)dy, j = 1, . . . , n. In some particular case we will show that these assumptions on B imply that there is only one Brascamp-Lieb inequality.

We unify several Bellman function problems treated in [1,2,4,5,6,9,10,11,12,14,15,16,18,19,20,21,22,23,24] into one setting. For that purpose we define a class of functions that have, in a sense, small mean oscillation (this class depends on two convex sets in R 2 ). We show how the unit ball in the BMO space, or a Muckenhoupt class, or a Gehring class can be described in such a fashion. Finally, we consider a Bellman function problem on these classes, discuss its solution and related questions.Since Slavin [12] and Vasyunin [18] proved the sharp form of the John-Nirenberg inequality (see [15]), there have been many papers where similar principles are used to prove sharp estimates of this kind. However, there is no theory or even a unifying approach; moreover, the class of problems to which the method can be applied has not been described yet. There is a portion of heuristics in the folklore that is each time applied to a new problem in a very similar manner. The first attempt to build a theory (at least for BMO) was made in [16], then the theory was developed in the paper [4] (see the short report [5] also). We would also like to draw the reader's attention to the forthcoming paper [6], which can be considered as a description of the theory for the BMO space in a sufficient generality. Problems of this kind were considered not only in BMO, but in Muckenhoupt classes, Gehring classes, etc (see [1,2,11,13,19,20]). In this short note, we define a class of functions and an extremal problem on it that includes all the problems discussed above. We believe that the unification we offer gives a strong basis for a theory that will distinguish a certain class of problems to which the method is applicable in a direct way. In Section 1 we state the problem and discuss related questions. Section 2 contains a detailed explanation how our classes of functions include the unit ball in BMO as well as the "unit balls" in Muckenhoupt classes and Gehring classes. Finally, in Section 3 we give hints to the solution of the problem (as the reader may expect looking at previous papers, it is rather lengthy and technical, so we omit a description of the solution, but concentrate on an analogy with the case of BMO considered in [4,6,22]).

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In 2006 Carbery raised a question about an improvement on the naïve norm inequality f + g p p ≤ 2 p−1 (f p p + g p p) for two functions f and g in L p of any measure space. When f = g this is an equality, but when the supports of f and g are disjoint the factor 2 p−1 is not needed. Carbery's question concerns a proposed interpolation between the two situations for p > 2 with the interpolation parameter measuring the overlap being f g p/2. Carbery proved that his proposed inequality holds in a special case. Here, we prove the inequality for all functions and, in fact, we prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all real p = 0.