Spherical convolution operators in spaces of variable Hölder order

In this paper, we study the images of operators of the type of spherical potential of complex order and of spherical convolutions with kernels depending on the inner product and having a spherical harmonic multiplier with a given asymptotics at infinity. Using theorems on the action of these operators in Hölder-variable spaces, we construct isomorphisms of these spaces. In Hölder spaces of variable order, we study the action of spherical potentials with singularities at the poles of the sphere. Using stereographic projection, we obtain similar isomorphisms of Hö lder-variable spaces with respect to n-dimensional Euclidean space (in the case of its one-point compactification) with some power weights.     

Linear Elliptic Boundary Value Problems with Non‐Smooth Data: Campanato Spaces of Functionals

In this paper linear elliptic boundary value problems of second order with non‐smooth data L^∞‐coefficients, sets with Lipschitz boundary, regular sets, non‐homogeneous mixed boundary conditions) are considered. It will be shown that such boundary value problems generate isomorphisms between certain Sobolev‐Campanato spaces of functions and functionals, respectively.     

Twisted cocycles of lie algebras and corresponding invariant functions

We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation - so called (α,β,γ)-derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants - we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction.     

Variational principle, characteristic electric multipoles, and higher polarizing moments in field theory

Based on the variational principle, we introduce a new notion: the characteristic electric multipoles constituting a system of basic distributions of charge on the boundary of a spatial domain. Inside the domain, potentials of the characteristic multipoles are harmonic polynomials whose orders determine the minimum orders of nonzero spherical multipole moments of the characteristic multipoles. Using the characteristic multipole formalism, we solve the moment problem in electrostatics and construct the superconductor Lagrangian in an electrostatic field. We express the empty-space Green's function for the Laplace equation using the characteristic multipole potentials. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 441–454, June, 1999     

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

ABSTRACT

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered.     

紧致齐性空间上的调和分析（Ⅰ）：富里埃级数的Poisson求和

首先介绍了紧致齐性空间上调和分析的若干基础性结果,并给出这些结果的较简洁的证明。接着,我们定义了紧致齐性空间上函数的卷积(熟知n维球面是一个紧致齐性空间),这一定义看来对研究紧致齐性空间上的调和分析向题是相当有用的。最后,用定义的卷积,研究了紧致齐性空间上Fourier级数的Poisson求和。     

Multirectangular invariants for power Köthe spaces

Using some new linear topological invariants, isomorphisms and quasidiagonal isomorphisms are investigated on the class of first type power Köthe spaces [Proceedings of 7th Winter School in Drogobych, 1976, pp. 101-126; Turkish J. Math. 20 (1996) 237-289; Linear Topol. Spaces Complex Anal. 2 (1995) 35-44]. This is the smallest class of Köthe spaces containing all Cartesian and projective tensor products of power series spaces and closed with respect to taking of basic subspaces (closed linear hulls of subsets of the canonical basis). As an application, it is shown that isomorphic spaces from this class have, up to quasidiagonal isomorphisms, the same basic subspaces of finite (infinite) type.     

On some properties of Gel’fond–Leont’ev generalized integration operators

In the class of linear continuous operators that act in the spaces of functions analytic in domains, we describe, in various forms, isomorphisms that commute with a power of the Gel’fond–Leont’ev generalized integration operator. We also obtain representations of all closed subspaces of the space of analytic functions that are invariant with respect to a power of the Gel’fond–Leont’ev generalized integration operator.     

Hyperbolic Harmonic Functions: Weak Approach with Applications in Function Spaces

Harmonic functions with respect to the Poincare metric on the unit ball are called hyperbolic harmonic functions. We establish the weak formulation of hyperbolic harmonic functions and use it in the study of hyperbolic harmonic function spaces. In particular, we give the Carleson measure characterization for the whole spectrum of spaces, whose analytic counterparts include among else Bloch spaces, Bergman-spaces, Besov-spaces, and Q_p-spaces. The second author was supported by the Finnish Cultural Foundation.     

Jordan Structures in Harmonic Functions and Fourier Algebras on Homogeneous Spaces

We study the Jordan structures and geometry of bounded matrix-valued harmonic functions on a homogeneous space and their analogue, the harmonic functionals, in the setting of Fourier algebras of homogeneous spaces.Supported by EPSRC grant GR/G91182 and NSERC grant 7679.     

Stochastic Navier–Stokes Equations Driven by Lévy Noise in Unbounded 3D Domains

Martingale solutions of the stochastic Navier–Stokes equations in 2D and 3D possibly unbounded domains, driven by the Lévy noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo–Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tightness criteria in a certain space contained in some spaces of càdlàg functions, weakly càdlàg functions and some Fréchet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces.     

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator \({\mathcal{L} = -{\rm d}/{\rm d}t + A}\) in homogeneous function spaces. We provide sufficient conditions for invertibility of such operators in terms of the spectral properties of the operator A and the semigroup generated by A. We introduce a homogeneous space of functions with absolutely summable spectrum and prove a generalization of the Gearhart–Prüss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-linear equations.     

Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities

The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.     

Multiply Hyperharmonic Functions

We consider multiply hyperharmonic functions on the product space of two harmonic spaces in the sense of Constantinescu and Cornea. Earlier multiply superharmonic and harmonic functions have been studied in Brelot spaces notably by GowriSankaran. Important examples of Brelot spaces are solutions of elliptic differential equations. The theory of general harmonic spaces covers in addition to Brelot spaces also solution of parabolic differential equations. A locally lower bounded function is multiply hyperharmonic on the product space of two harmonic spaces if it is a hyperharmonic function in each variable for every fixed value of the other. We prove similar results as in Brelot spaces, but our approach is different. We study sheaf properties of multiply hyperharmonic functions. Our main theorem states that multiply hyperharmonic functions are lower semicontinuous and satisfy the axiom of completeness with respect to products of relatively compact sets. We also study nearly multiply hyperharmonic functions.     

Statistical Convergence in Probability for a Sequence of Random Functions

In this paper, we introduce a new type of convergence for a sequence of random functions, namely, statistical convergence in probability, which is a natural generalization of convergence in probability. In this approach, we allow such a sequence to go far away from the limit point infinitely many times by presenting random deviations, provided that these deviations are negligible in some sense of measure. In this context, the set of values of a random function is considered as a probabilistic metric (PM) space of random variables, and some basic results are obtained using the tools of PM spaces.     

Correctness of general boundary-value problems in a half-space for differential equations with constant coefficients in the classes of functions of power growth and decay

We present the necessary and sufficient conditions of correctness of general boundary-value problems in a half-space for homogeneous differential equations with constant coefficients in the spaces of functions that have the exponential behavior at infinity in a separated variable and are rapidly decaying in the remaining variables. On their basis, we obtained the sufficient conditions of solvability of the indicated problems in the spaces of distributions of slow growth.     

On Riesz systems of harmonic conjugates in

In continuation of recent studies, we discuss two constructive approaches for the generation of harmonic conjugates to find null solutions to the Riesz system in . This class of solutions coincides with the subclass of monogenic functions with values in the reduced quaternions. Our first algorithm for harmonic conjugates is based on special systems of homogeneous harmonic and monogenic polynomials, whereas the second one is presented by means of an integral representation. Some examples of function spaces illustrating the techniques involved are given. More specifically, we discuss the (monogenic) Hardy and weighted Bergman spaces on the unit ball in consisting of functions with values in the reduced quaternions. We end up proving the boundedness of the underlying harmonic conjugation operators in certain weighted spaces. Copyright © 2013 John Wiley & Sons, Ltd.     

On quasiaffine spaces and coherent configurations

In [2] André deduced a (1?1) correspondence between the class of homogeneous coherent configurations and the class of certain noncommutative spaces which he called quasiaffine. In this note we establish a (1?1) correspondence between (not necessarily homogeneous) coherent configurations and weakly quasiaffine spaces which generalizes André's. Furthermore we consider some applications of this correspondence to quasiaffine spaces; especially we characterize such spaces with maximal diameter with respect to one direction (compare with [5]).