Table of contents

The paper deals with various classes of functions that have zero integrals over all balls of a fixed radius in hyperbolic spaces. We describe these classes in terms of series in special functions and prove a uniqueness theorem. These results enabled us to obtain a definitive version of the local two-radii theorem.

In this paper we introduce and study the notion of an entropy solution of the Dirichlet problem for a class of non-linear elliptic fourth-order equations whose right-hand sides admit arbitrary growth with respect to the variable corresponding to the unknown function and belong to the space  for each fixed value of this variable. We prove the existence and uniqueness of an entropy solution. We establish the existence of so-called -solutions and -solutions of the problem and prove that the entropy solutions belong to certain Sobolev spaces.

This paper deals with the solutions of th-order differential equations of the form

where  belongs to the Carathéodory class , .

We describe homogeneous canonical transformations of the cotangent bundle of a manifold with conical singular points and compute the index of an elliptic Fourier integral operator obtained by the quantization of such a transformation. The answer involves the index of an elliptic Fourier integral operator on a smooth manifold and the residues of the conormal symbol.

For an Enriques surface  over a number field  with a -rational point we prove that the -component of is finite if and only if . For a regular projective smooth variety satisfying the Tate conjecture for divisors over a number field, we find a simple criterion for the finiteness of the -component of . Moreover, for an arithmetic model  of  we prove a variant of Artin's conjecture on the finiteness of the Brauer group of . Applications to the finiteness of the -components of Shafarevich-Tate groups are given.

We solve Ul'yanov's problem [1] of representing elements of class  as series with respect to arbitrary function systems.

We study entire functions on infinite-dimensional spaces. The basis is the study of spaces of Gateaux holomorphic functions that are bounded on certain subsets (bounded entire functions). The main goal is to characterize the Fourier image of the corresponding spaces of generalized entire functions (ultra-distributions) by an infinite-dimensional Paley-Wiener theorem. We introduce entire functions of exponential type and prove a generalization of the classical Paley-Wiener theorem. The crucial point of our theory is the dimension-invariant estimate given by Lemma 4.12.