Skip to main content

Über dieses Buch

This proceedings volume gathers selected, peer-reviewed papers from the "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis VIII" (OTHA 2018) conference, which was held in Rostov-on-Don, Russia, in April 2018.

The book covers a diverse range of topics in advanced mathematics, including harmonic analysis, functional analysis, operator theory, function theory, differential equations and fractional analysis – all fields that have been intensively developed in recent decades. Direct and inverse problems arising in mathematical physics are studied and new methods for solving them are presented. Complex multiparameter objects that require the involvement of operators with variable parameters and functional spaces, with fractional and even variable exponents, make these approaches all the more relevant.

Given its scope, the book will especially benefit researchers with an interest in new trends in harmonic analysis and operator theory, though it will also appeal to graduate students seeking new and intriguing topics for further investigation.



Function Theory and Approximation Theory


Some General Properties of Operators in Morrey-Type Spaces

In the paper we consider general properties of operators acting from rearrangement invariant spaces into generalized Morrey-type spaces. We extract wide class of operators that preserve non-negativity and monotonicity of functions and prove two-sided estimates for their norms. As corollaries we obtain corresponding results for operators of embedding and for Hardy–Littlewood maximal operators. For operators commuting with a shift operator the results are extended to the case of global Morrey-type spaces. As an application of these approaches we establish a criterion of the embedding for a weighted Lorentz space into a Morrey-type space.
Mikhail L. Goldman, Elza Bakhtigareeva

Characterization of Parabolic Fractional Maximal Function and Its Commutators in Orlicz Spaces

In this paper, we give a necessary and sufficient condition for the boundedness of the parabolic fractional maximal operator and its commutators in Orlicz spaces.
Vagif S. Guliyev, Ahmet Eroglu, Gulnara A. Abasova

Finite Trees Inside Thin Subsets of

Bennett, Iosevich and Taylor proved that compact subsets of \({\mathbb R}^d\), \(d \ge 2\), of Hausdorff dimensions greater than \(\frac{d+1}{2}\) contain chains of arbitrary length with gaps in a non-trivial interval. In this paper we generalize this result to arbitrary tree configurations.
A. Iosevich, K. Taylor

Boundedness of Projection Operator in Generalized Holomorphic and Harmonic Spaces of Functions of Hölder Type

We prove boundedness of holomorphic projection in the generalized Hölder type space of holomorphic functions in the unit disc with prescribed modulus of continuity and in the variable exponent generalized Hölder spaces of holomorphic functions in the unit disc. We also prove boundedness of harmonic projection in analogous spaces of harmonic functions.
Alexey Karapetyants, Joel E. Restrepo

Generalized Fourier Series by the Double Trigonometric System

Necessary and sufficient conditions are obtained on the function M such that \(\{ M(x,y) e^{i kx}e^{i my}: (k,m)\in \varOmega \}\) is complete and minimal in \(L^{p}(\mathrm{I\!\!T}^{2})\) when \(\varOmega ^{c}=\{(0,0)\}\) and \(\varOmega ^{c} = 0\times {\!Z\!Z}\). If \(\varOmega ^{c} = 0\times {\!Z\!Z}_{0},\) \({\!Z\!Z}_{0} = {\!Z\!Z}\setminus \{0\}\) it is proved that the system \(\{ M(x,y) e^{i kx}e^{i my}: (k,m)\in \varOmega \}\) cannot be complete minimal in \(L^{p}(\mathrm{I\!\!T}^{2})\) for any \(M\in L^{p}(\mathrm{I\!\!T}^{2})\). In the case \(\varOmega ^{c}=\{(0,0)\}\) necessary and conditions are found in terms of the one dimensional case.
K. S. Kazarian

Hardy Type Inequalities in the Category of Hausdorff Operators

Classical Hardy’s inequalities are concerned with the Hardy operator and its adjoint, the Bellman operator. Hausdorff operators in their various forms are natural generalizations of these two operators. In this paper, we try to adjust the scheme used by Bradley for Hardy’s inequalities with general weights to the Hausdorff setting. It is not surprising that the obtained necessary conditions differ from the sufficient conditions as well as that both depend not only on weights but also on the kernel that generate the Hausdorff operator. For the Hardy and Bellman operators, the obtained necessary and sufficient conditions coincide and reduce to the classical ones.
Elijah Liflyand

Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over \({\mathbb {R}}^{n+1}\) are considered. Special attention is given to their origins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions are characterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on the other side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications, sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches.
Helmuth R. Malonek, Isabel Cação, M. Irene Falcão, Graça Tomaz

Paraproduct in Besov–Morrey Spaces

The paraproduct plays a key role in some highly singular partial differential equations. For example, that for Hölder–Zygmund spaces is used to solve stochastic differential equations, as demonstrated by Gubinelli, Imkeller, Perkowski, and Hairer. In this note, which is organized in a self-contained manner, the counterparts for Besov Morrey spaces are obtained.
Yoshihiro Sawano

Functional Analysis and Operator Theory


Analogs of the Khintchin—Kolmogorov Inequalities in Discrete Morrey Spaces

We prove certain estimates of the Rademacher sums in the discrete local \( LM_{l, L^p} \) and global \( GM_{l, L^p} \) Morrey function spaces. Using this estimates, we give necessary and sufficient conditions for the validity of an analogue of the Khinchin-Kolmogorov inequality in the spaces \( LM_{l, L^p} \) and \( GM_{l, L^p} \).
Evgenii I. Berezhnoĭ

Mellin Convolution Equations

In the present paper we collect results on Mellin convolution equations (MCEs), obtained recently. We start with the motivation and in the first section are exposed MCEs which encounter in applications. Further we expose the boundedness results of corresponding operators in the Lebesgue space with weight. Fourier convolution (Wiener-Hopf) equations are defined and their connection to MCEs is described. Solvability and Fredholm properties and the index formulae for MCEs are formulated in terms of the symbol functions assigned to them. Results on the Banach algebra generated by Mellin and Wiener-Hopf operators in the Lebesgue space are exposed: The symbol function is defined and the Fredholm criteria is formulated, the index formula is written. In conclusion we expose relatively new results on Fredholm property and index of MCEs in the Bessel potential spaces. These results are applied to the mixed boundary value problem (Mixed BVP) for the Laplace equation in an angle, which reduces to an equivalent MCE. Results on the Fredholm property, solvability and index of such equations are formulated and, in conclusion, applied to the above formulated mixed BVP for the Laplace equation.
Roland Duduchava

Integral Operators of the -Convolution Type in the Case of a Reflectionless Potential

The notion of the \(\mathcal L\)-convolution operator is introduced by changing the Fourier operator in the definition of the (regular) convolution operator to the operator intertwining the Sturm-Liouville operator \(\mathcal L\) with the multiplication operator. Along the same lines, the \(\mathcal L\)-Wiener-Hopf operator is introduced. For the latter, the invertibility is studied in the case of a reflectionless potential and piecewise continuous symbols.
Davresh Hasanyan, Armen Kamalyan, Martin Karakhanyan, Ilya M. Spitkovsky

Spectral Theory for Nonlinear Operators: Quadratic Case

In many cases, given a non-linear map, linearized systems near its fixed points do qualitatively capture its topological and algebraic properties. This suggests to extend the linear spectral theory to non-linear operators by considering spectra of linearizations in small neighborhoods of the fixed points. In the present paper, we develop this approach for quadratic maps. Several standard concepts such as asymptotic laws for splitting/gluing zeros of polynomial maps) are considered from new (and, possibly, unexpected) angles.
Yakov Krasnov

Factorization of Order Bounded Disjointness Preserving Multilinear Operators

Given a finite collection of disjointness preserving linear operators with values in an f-algebra, the mapping defined as their pointwise product is a disjointness preserving multilinear operator. The central result asserts that an arbitrary order bounded disjointness preserving multilinear operator from the Cartesian product of vector lattices into an arbitrary vector lattice has a similar structure. Using this fact, we establish a multilinear version of the Hart theorem on the associated lattice homomorphism, and also give some consequences on the structure of disjointness preserving homogeneous polynomials.
Anatoly G. Kusraev, Zalina A. Kusraeva

Robbins–Monro Conditions for Persistent Exploration Learning Strategies

We formulate simple assumptions, implying the Robbins–Monro conditions for the Q-learning algorithm with the local learning rate, depending on the number of visits of a particular state (local clock) and the number of iteration (global clock). It is assumed that the Markov decision process is communicating and the learning policy ensures the persistent exploration. The restrictions are imposed on the functional dependence of the learning rate on the local and global clocks. The result partially confirms the conjecture of Bradkte (1994).
Dmitry B. Rokhlin

On Widths of Invariant Sets

Many problems of approximation theory and operator theory can be reduced to the computation or estimation of n-widths. By definition, the n-width \(w_n(A)\) of a subset A in a Banach space X is the minimal distance of A from n-dimensional subspaces. If in X an isometric representation \(g\mapsto T_g\) of a group G acts and A is invariant under operators \(T_g\), then it is reasonable to estimate the minimal distances from A to invariant subspaces; in this case the invariant width \(w_n^G(A)\) is defined as the minimal distance to invariant subspaces of dimension at most n. This approach, initiated in the author’s work [5], allowed to study stability of Levi-Civita functional equations on amenable groups in classes of bounded functions. To catch more general classes of “approximate solutions”  one has to consider representations on linear G-spaces that contain invariant subspaces on which the representation acts isometrically—this is the subject of the present work. Some partial results are obtained for general (not necessarily amenable) groups if X is reflexive; for Hilbert spaces the result can be formulated in the form \(\ w_n^G(A)\le 7\sqrt{n+1}\ w_n(A)\).   New applications to the stability of functional equations are given.
Ekaterina Shulman

The Distance Function and Boundedness of Diameters of the Nearest Elements

The paper is concerned with approximative properties of sets versus the rate of change of the distance function. We solve a number of problems posed recently by W. B. Moor on sets with bounded diameters of sets of nearest points.
Igor’ G. Tsar’kov

Differential Equations and Mathematical Physics


The Influence of Oscillations on Energy Estimates for Damped Wave Models with Time-dependent Propagation Speed and Dissipation

The aim of this paper is to derive higher order energy estimates for solutions to the Cauchy problem for damped wave models with time-dependent propagation speed and dissipation. The model of interest is
$$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}-\lambda ^2(t)\omega ^2(t)\varDelta u +\rho (t)\omega (t)u_t=0, &{} (t,x)\in [0,\infty )\times \mathbb {R}^n, \\ u(0,x)=u_0(x), \,\,\,\, u_t(0,x)=u_1(x), &{} x\in \mathbb {R}^n, \end{array}\right. } \end{aligned}$$
The coefficients \(\lambda =\lambda (t)\) and \(\rho =\rho (t)\) are shape functions and \(\omega =\omega (t)\) is a bounded oscillating function. If \(\omega (t)\equiv 1\) and \(\rho (t)u_t\) is an effective dissipation term, then \(L^2-L^2\) energy estimates are proved in Bui and Reissig (Fourier analysis, trends in mathematics. Birkhäuser, Basel, [2]). In contrast, the main goal of the present paper is to generalize the previous results to coefficients including an oscillating function in the time-dependent coefficients. We will explain how the interplay between the shape functions and oscillating behavior of the coefficient \(\omega =\omega (t)\) will influence energy estimates.
Halit Sevki Aslan, Michael Reissig

On a Dirichlet Problem for One Improperly Elliptic Equation

The Dirichlet problem for sixth order improperly elliptic equation is considered. The functional class of boundary functions, where this problem is normally solvable is determined. If the roots of the characteristic equation satisfy some conditions, the number of linearly independent solutions of homogeneous problem and the number of linearly independent solvability conditions of in-homogeneous problem are obtained. Solutions of homogeneous problem and solvability conditions of in-homogeneous problem are obtained in explicit form.
Armenak H. Babayan, Seyran H. Abelyan

On the 1-dim Defocusing NLS Equation with Non-vanishing Initial Data at Infinity

We show global well-posedness of certain type of strong-in-time and weak-in-space solutions for the Cauchy problem of the 1-dimensional nonlinear Schrödinger equation, in various cases of open sets, bounded and unbounded. These solutions do not vanish at the boundary or at infinity.
Nikolaos Gialelis, Ioannis G. Stratis

On Time-Global Solutions of SDE Having Nowhere Vanishing Initial Densities

We deal with stochastic differential equations (SDE) on the non-compact finite-dimensional manifolds and consider their solutions such that the initial values have densities nowhere equal to zero. First, we obtain a sufficient condition under which all such solutions exist on the entire half-axis \([0,\infty )\). Further, we introduce the notion of the system, generated by the above equation, continuous at infinity. A sufficient condition the latter property to be satisfied is found. Then we find the necessary and sufficient condition for existence of the above solutions on \([0,\infty )\) for the case where the corresponding system is continuous at infinity.
Yuri E. Gliklikh

On Transmutation Operators and Neumann Series of Bessel Functions Representations for Solutions of Linear Higher Order Differential Equations

A new representation for solutions of linear higher order ordinary differential equations with a spectral parameter is obtained in terms of Neumann series of Bessel functions. The result is based on a Fourier-Legendre series representation for the Borel transform of the solution with respect to the spectral parameter. Estimates for the coefficients and for the convergence of the representation are derived. Numerical illustrations of the applicability of the obtained formulae are presented.
Flor A. Gómez, Vladislav V. Kravchenko

On a Boundary Value Problem with Infinite Index

The generalized Riemann boundary value problem for analytic functions is investigated in the weighted space. It is supposed that the weight function has infinitely many zeroes on the unit circumference. It is proved that the homogeneous problem has an infinite number of linearly independent solutions and under some additional conditions on the order of zeroes of the weight function these solutions determined in explicit form.
H. M. Hayrapetyan

A Numerical Realization of the Wiener–Hopf Method for the Kolmogorov Backward Equation

We propose a new numerical method for a certain type of boundary value problems for 3-dimensional partial differential equations, which are related to first passage time distributions of Itô diffusions. We consider the Kolmogorov backward equation, which arises in various applications including mathematical finance. The technique presented is based on a probabilistic interpretation of the problem, which involves a Markov chain approximation, and a Wiener–Hopf factorization. First, we use Carr’s time randomization and approximate the second component of the related diffusion process with a Markov chain. As the result, we reduce the original problem to a sequence of 1-dimensional differential equations with suitable boundary conditions, associated with Gaussian processes, whose constant coefficients are defined by the Markov chain constructed. We also suggest an improvement for the approximation procedure, which lowers the number of nodes used. Then we express an analytic solution to each problem in terms of a probabilistic form of Wiener–Hopf factorization. We implement explicit and approximate factorization formulae numerically using the Fast Fourier Transform algorithm and provide the results of numerical experiments to illustrate the performance of the method developed.
Oleg Kudryavtsev, Vasily Rodochenko

On Waves Processes in Transversally-Inhomogeneous Waveguides

Different types of problems on wave propagation in transversally inhomogeneous cylindrical waveguides are considered. General properties of the dispersion set of inhomogeneous cylindrical waveguides are presented. Asymptotic formulas for the dispersion set branches in the vicinity of radial resonance points are constructed. A condition of solvability of inhomogeneous problems is built. Different models of materials are used for numerical testing of the asymptotic formulas. Dispersion relations for elastic, viscoelastic, electro-elastic and prestressed elastic waveguides are analyzed. The formulas allowing to estimate the influence of residual stresses on the radial resonances points are presented. The effect of various types of homogeneous boundary conditions on the structure of dispersion set is analysed.
Alexander Vatulyan, Victor Yurov

Inverse Spectral Problems for Differential Systems

Inverse problems of spectral analysis for non-selfadjoint systems of ordinary differential equations are studied. We establish properties of the spectral characteristics, give statements of the inverse problems, prove uniqueness theorems, obtain algorithms for the solutions of the inverse problems and provide necessary and sufficient conditions for their solvability.
Vjacheslav Anatoljevich Yurko
Weitere Informationen

Premium Partner