Table of contents

The modulus of continuity of a function (, ) 1-periodic in each variable is defined by

The following estimate is established for the nonincreasing rearrangement of a function (, ; ):

(1)

Also, analytic functions of Hardy class in the unit disk are considered. It is proved that the inequality (1) () holds for the rearrangements of their boundary values also when (this is false for real functions of class ). Inequality (1) is used to find necessary and sufficient conditions for the space () of functions with a given majorant of the -modulus of continuity to be imbedded in the Orlicz classes , where satisfies the -condition and on . For the solution of this problem follows from estimates obtained earlier by the author (Zbl. 314#46031). An analogous result is established for classes of functions in the Hardy space (). The imbeddings with limiting exponent (Sobolev and Hardy-Littlewood theorems) are limiting cases of the results in this article. Bibliography: 27 titles.

Let be a simply connected bounded domain on the complex plane , let , and assume that is a closed rectifiable Jordan curve. Denote by the Lebesgue linear measure on . For a function analytic on and for let

where is the Euclidean distance from to . It is proved that if for some

(1)

then has a finite nontangential boundary value for almost all , and

where the integral on the left-hand side is understood as an -integral. It is also proved that under condition (1) the function is representable in by the Cauchy -integral of its nontangential boundary values on . Further, if is regular (i.e., for all and , where the constant is independent of and ), then these assertions are valid if condition (1) holds for some . The question of representability of integrals of Cauchy type by Cauchy -integrals is studied. In particular, well-known results of Ul'yanov on this question are carried over to the case of domains with a regular boundary. It is proved that the condition of regularity of the boundary cannot be weakened here. Bibliography: 18 titles.

Theorems are proved on the selection of convergence subsystems and -subsystems with lacunary index density. Bibliography: 10 titles.

Asymptotic formulas for polynomials determined by orthogonality relations are obtained. The weight functions are concentrated on disjoint intervals. Bibliography: 24 titles.

It is proved that the distribution function for the maximum of the modulus of a set of jointly Gaussian random variables with given variance and zero mean is minimal if these variables are independent. For let

As a corollary of the result mentioned, the precise orders of the constants are computed (), and various improvements of these inequalities are obtained. The estimates are used in particular to construct lacunary analogues of the Rudin-Shapiro trigonometric polynomials. Bibliography: 24 titles.

For functions holomorphic in tube domains over a cone, the author studies the connection between their asymptotic behavior at the origin along orbits of one-parameter groups of linear transformations and asymptotic properties of the real parts of the boundary values of these functions. It is shown that, as a rule, the presence of certain asymptotics of the real part of a boundary value of a holomorphic function guarantees similar asymptotics for the whole function within the domain. Bibliography: 6 titles.

By a method different from known ones, the author obtains rather precise easily verifiable algebraic conditions for imbedding Sobolev spaces of infinite order. Bibliography: 18 titles.

It is shown that, if a generating manifold does not contain proper submanifolds of the same CR dimension as , then all CR functions can be extended from into some wedge with edge . In particular, extension of all CR functions into a wedge necessarily obtains for manifolds of finite type. Bibliography: 21 titles.

This is a continuation of Mat. Sb. 131(173) (1986), no. 1, 3-26=Math. USSR Sb. 59 (1988), no. 1, 1-23. The author proves sufficient conditions that must be satisfied by the spectral data of any two similar Sturm-Liouville boundary value problems with nonseparated boundary conditions. Characteristic properties are obtained for conformal mappings of domains connected with such problems onto the upper half-plane. Bibliography: 4 titles.

The purpose of this article is to describe the formulation of a selfadjoint spectral problem with boundary conditions on a sufficiently thin manifold. Namely, let be a selfadjoint operator in , let be a smooth manifold, let be the restriction of to the lineal in consisting of all functions which vanish in a neighborhood of . It is shown that the deficiency elements of this restriction can be represented as "tensor layers" with densities of a definite class of smoothness, concentrated on the "boundary" of . If is sufficiently thin, there is only one family of deficiency elements, and it is analogous to the single-layer potentials. In this case, calculation of the boundary form and the description of the selfadjoint extensions appears to be quite simple. This case is studied in detail because the investigation of the simplest model of the three-particle problem of quantum mechanics reduces to it. Bibliography: 16 titles.

This article is concerned with the problem of the removal of singularities of CR functions , where is a smooth generating manifold in a domain in . Under appropriate conditions on the Levi form of this manifold an analog of the Riemann removable singularities theorem for holomorphic functions is established. Bibliography: 12 titles.

We study integrable distributions over the -algebra of truncated polynomials, where is a field of characteristic . We obtain an analogue of the theorem of Frobenius; we describe the equivalence classes of TI-distributions, i.e., of those distributions with respect to which the algebra has no nontrivial -invariant ideals; we show that over a perfect field any TI-distribution is equivalent to a general Lie algebra of Cartan type ; and we find all the forms of the Zassenhaus algebra, in the process making essential use of the theory of representations of the chromatic quiver of Kronecker. Bibliography: 13 titles.

A theorem is proved on the uniform convergence of diagonal Padé approximants for Stieltjes type meromorphic functions (with a finite number of poles). Bibliography: 13 titles.

A sharp or optimal interpolation theorem is proved for the Lorentz spaces , generalizing the Marcinkiewicz theorem and refining the Riesz-Thorin theorem and the Stein-Weiss theorem. This theorem extends to the spaces of the real method constructed from any Banach pair; thus it extends also to Besov spaces. Bibliography: 12 titles.

Necessary and sufficient conditions are established for the existence, on the domain's boundary, of a limit in , , of solutions of second order elliptic equations having Tricomi type degeneracy on the boundary. Solvability of the Dirichlet problem for such an equation in spaces of type is proved. Bibliography: 9 titles.

The main result is Theorem 1.Let be a bounded convex domain in , , with . Let . Let be an entire function of exponential type whose zero set is the union of planes , , . Suppose the following conditions hold: (a) there exist constants , , , , such that the estimate

holds if the point satisfies , , ; (b) for every the restriction of the entire function to the plane is not identically zero; (c) there exist constants and such that for either or .Then every analytic function in the domain can be represented by a series

converging in the topology of . Bibliography: 11 titles.

In the framework of a geometric approach to nonlinear systems of partial differential equations, a Bäcklund correspondence is defined according to a representation of it as a differential connection between two such systems, and a construction of it is presented. Examples of the practical realization of this construction are given. Bibliography: 7 titles.

Let be a Banach space with norm , and let be a compact topological space with -algebra of measurable sets on which a nonnegative regular Borel measure  is given. Further, let be the Banach space of Bochner-integrable functions , with the norm , and let be a multivalued mapping and a single-valued mapping, where is a compact topological space. Under certain assumptions it is proved that for any there exists a continuous mapping such that the following conditions hold for any : and

where is the distance in from a point to a set. Bibliography: 11 titles.