Table of contents

The following analogue of the well-known theorem of Rudin for the polydisk is proved for the ball. Given any positive lower semicontinuous integrable function on the sphere , there is a positive singular measure on such that , and the difference between the Poisson integrals of the function and the measure is a pluriharmonic function (in the unit ball , with ). This implies immediately the existence of an inner function in . A certain weakened version of the Pick-Nevanlinna theorem on interpolation of inner functions is also obtained for . The results obtained are applied to the Hardy classes () in the ball and in the polydisk. Bibliography: 17 titles.

A criterion is obtained for the absolute continuity or singularity of probability measures corresponding to certain classes of two-parameter discrete random fields. The proof is based on some lemmas which describe the sets of convergence of two-parameter semimartingales. Bibliography: 5 titles.

We consider a unitary representation of a Lie group given by a positive polarization. Suppose the group contains an abelian normal subgroup of a suitable sort. It is then shown that if we replace the polarization by a certain new one that contains the corresponding abelian ideal, the equivalence class of the representation is left unchanged. Bibliography: 5 titles.

An asymptotic expansion of the solution of the equation , where the function tends to zero as faster than any power of , is constructed for the second boundary value problem in the exterior of a cylinder and a half-cylinder in three-dimensional space. The asymptotic expression as is constructed with accuracy to any power of uniformly with respect to all directions. The Fourier transform in the coordinate along the axis of the cylinder is used for the cylinder, and the asymptotic behavior of the Green function is determined along the way. The results obtained for the cylinder with subsequent application of the method of asymptotic successive approximations are used in the case of the boundary value problem for the half-cylinder. Bibliography: 7 titles.

A new solution of the restricted Burnside problem for sufficiently large odd exponents is presented. The proof is considerably shorter than the original one given by P.S.  Novikov and S.I. Adyan in 1968 (although the bound for the exponent is worse: ). It is based on a geometrical interpretation of deducibility of relations in a group from its defining relations. Bibliography: 7 titles. Figures: 16.

An a priori energy estimate analogous to the inequalities expressing St. Venant's principle in elasticity theory is obtained for the solution of a polyharmonic equation with the conditions of the first boundary-value problem in an n-dimensional domain. These estimates are used to study the behavior of the solution and its derivatives near irregular boundary points and at infinity as a consequence of the geometric properties of the boundary in a neighborhood of these points. Moreover, the estimates obtained are used to prove a uniqueness theorem for the solution of the Dirichlet problem in unbounded domains. Bibliography: 13 titles.

As is well known, the mixed problem for the entire class of Petrovskiĭ well-posed partial differential equations has not been studied. In this paper, a certain subclass of Petrovskiĭ well-posed equations for which it is possible to state and study mixed problems, is isolated. In the rectangle , consider the equation

with boundary conditions

for , where , , , , , is a continuous linear functional in , , and for

, , and with initial conditions and . Well-posedness conditions are found for this problem. Bibliography: 9 titles.

The Schrödinger equation with a time-dependent Hamiltonian is considered in the space . It is assumed that , , , and ; . It is shown that each solution of the Schrödinger equation which exits any compact subset of configuration space must have free asymptotics. More precisely, if for any there is a sequence such that , then, for some , , . This provides an effective description of the ranges of the wave operators relating the problems with the free Hamiltonian and the complete Hamiltonian . Examples show that the conditions imposed are best possible. The case of functions periodic in is treated separately; in this case the description of the ranges of the wave operators can be given in spectral terms for and any . More general differential operators are also considered. Bibliography: 14 titles.