arXiv subjects

Aleksandr Murach

Publications and source records attributed to Aleksandr Murach.

At least 19 recordsLinked to original sources

arXiv2026

Matrix parabolic problems in Sobolev spaces of generalized smoothness

We study a general linear parabolic problem for Petrovskii parabolic differential system in Sobolev anisotropic distribution spaces of generalized smoothness. Slowly varying functions are used to characterize supplementary generalized smoothness that cannot be determined by number indexes. We prove that this problem induces topological isomorphisms on appropriate pairs of such spaces. As an application, we give sufficient and necessary conditions for the problem solutions to have prescribed generalized regularity expressed in terms of these spaces. Their use allows obtaining exact conditions for indicated generalized partial derivatives of the solutions to be continuous.

math.AP↗

arXiv2025

Elliptic boundary-value problems in some distribution spaces of generalized smoothness

We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $p>1$. The smoothness is given by a number parameter and a supplementary function parameter that varies slowly at infinity. These spaces are obtained by a combination of the methods of the complex interpolation with number parameter between Banach spaces and the quadratic interpolation with function parameter between Hilbert spaces applied to classical Sobolev spaces. We show that the spaces under study admit localization near a smooth boundary and describe their trace spaces in terms of Besov spaces with the same supplementary function parameter. We prove that a general differential elliptic problem induces Fredholm bounded operators on appropriate pairs of the spaces under study. We also find exact sufficient conditions for solutions of the problem to have a prescribed generalized or classical smoothness on a given set and establish corresponding a priori estimates of the solution. These results are specified for parameter-elliptic problems.

math.AP↗

arXiv2024

Distribution spaces associated with elliptic operators

We study complex distribution spaces given over a bounded Lipschitz domain $\Omega$ and associated with an elliptic differential operator $A$ with $C^{\infty}$-coefficients on $\overline{\Omega}$. If $X$ and $Y$ are quasi-Banach distribution spaces over $\Omega$, then the space $X(A,Y)$ under study consists of all distributions $u\in X$ such that $Au\in Y$ and is endowed with the graph quasi-norm. Assuming $X$ to be an arbitrary Besov space or Triebel--Lizorkin space over $\Omega$, we find sufficient conditions for $Y$ under which the interpolation between the spaces $X(A,Y)$ preserves their structure, these spaces are separable, and the set $C^{\infty}(\overline{\Omega})$ is dense in them. We then explicitly describe the spaces obtained by the real, complex, and $\pm$ interpolation between the spaces under study. We apply these spaces to general elliptic problems with rough boundary data by proving the Fredholm property for bounded operators induced by these problems and defined on certain spaces $X(A,Y)$. Specifically, we establish the maximal regularity of solutions to some elliptic problems with Gaussian white noise in boundary conditions. Quasi-Banach distribution spaces are involved in the concept of $X(A,Y)$ for the first time. Our results are new even for inner product Sobolev spaces of integer-valued order.

math.FA↗

arXiv2024

Extended Sobolev scale for vector bundles, and its applications

We study an extended Sobolev scale for smooth vector bundles over a smooth closed manifold. This scale is built on the base of inner product distribution spaces of generalized smoothness given by an arbitrary positive function OR-varying at infinity. We show that this scale is obtained by the quadratic interpolation (with a function parameter) between inner product Sobolev spaces, is closed with respect to the quadratic interpolation, and consists of all Hilbert spaces that are interpolation spaces between inner product Sobolev spaces. Embedding theorems and a duality theorem are proved for this scale. We give applications of the extended Sobolev scale to mixed-order (Douglis--Nirenberg) elliptic pseudodifferential operators acting between vector bundles of the same rank. We prove their Fredholm property on appropriate pairs of spaces on the scale, give a sufficient and necessary condition for the local generalized smoothness of solutions to a mixed-order elliptic system and provide a corresponding a priori estimate of the solutions. We also give a sufficient condition for a chosen component of the solution to be $q$ times continuously differentiable on a subset of the manifold.

math.AP↗

arXiv2023

Unconditional convergence of eigenfunction expansions for abstract and elliptic operators

We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with two norms and also estimate the degree of this convergence. Our result essentially generalizes and complements the known theorems of M. Krein and of Krasnosel'ski\u{\i} and Pustyl'nik. We apply it to normal elliptic pseudodifferential operators on compact boundaryless $C^{\infty}$-manifolds. We find generic conditions for eigenfunction expansions induced by such operators to converge unconditionally in the Sobolev spaces $W^{\ell}_{p}$ with $p>2$ or in the spaces $C^{\ell}$ (specifically, for the $p$-th mean or uniform convergence on the manifold). These conditions are sufficient and necessary for the indicated convergence on Sobolev or H\"ormander function classes and are given in terms of parameters characterizing these classes. We also find estimates for the degree of the convergence on such function classes. These results are new even for differential operators on the circle and for multiple Fourier series.

math.FA↗

arXiv2023

A new look at old theorems of Fejér and Hardy

The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Cesàro summable series in abstract normed spaces. This theorem generalizes some known results of Hardy and Littlewood for number series. We find sufficient conditions for the convergence of trigonometric Fourier series in homogeneous Banach spaces over the circle. These conditions are expressed in terms of the Fourier coefficients and are weaker than Hardy's condition. We give a description of all Banach function spaces given over the circle and endowed with a norm been equivalent to a norm in a homogeneous Banach space. We study interpolation properties of such spaces and give new examples of them. We extend the classical Fejér theorem on the uniform Cesàro summability of the Fourier series on sets by means of a refined version of Cantor's theorem on the uniform continuity of a mapping between metric spaces. We also generalize the classical Hardy theorem on the uniform convergence of the Fourier series on sets.

math.CA↗

arXiv2023

Interpolation spaces of generalized smoothness and their applications to elliptic equations

We introduce and investigate classes of normed or quasinormed distribution spaces of generalized smoothness that can be obtained by various interpolation methods applied to classical Sobolev, Nikolskii-Besov, and Triebel-Lizorkin spaces. An arbitrary positive function O-regularly varying at infinity serves as the order of regularity for the spaces introduced. They are broad generalizations of the above classical spaces and allow being well defined on smooth manifolds. We give applications of the spaces under investigation to elliptic equations and elliptic problems on smooth manifolds.

math.AP↗

arXiv2021

An extended Hilbert scale and its applications

We propose a new viewpoint on Hilbert scales extending them by means of all Hilbert spaces that are interpolation ones between spaces on the scale. We prove that this extension admits an explicit description with the help of $\mathrm{OR}$-varying functions of the operator generating the scale. We also show that this extended Hilbert scale is obtained by the quadratic interpolation (with function parameter) between the above spaces and is closed with respect to the quadratic interpolation between Hilbert spaces. We give applications of the extended Hilbert scale to interpolational inequalities, generalized Sobolev spaces, and spectral expansions induced by abstract and elliptic operators.

math.FA↗

arXiv2020

Elliptic problems with unknowns on the boundary and irregular boundary data

We consider an elliptic problem with unknowns on the boundary of the domain of the elliptic equation and suppose that the right-hand side of this equation is square integrable and that the boundary data are arbitrary (specifically, irregular) distributions. We investigate local (up to the boundary) properties of generalized solutions to the problem in Hilbert distribution spaces that belong to the refined Sobolev scale. These spaces are parametrized with a real number and a function that varies slowly at infinity. The function parameter refines the number order of the space. We prove theorems on local regularity and a local a priori estimate of generalized solutions to the problem under investigation. These theorems are new for Sobolev spaces as well.

math.AP↗

arXiv2020

Elliptic problems with rough boundary data in generalized Sobolev spaces

We investigate regular elliptic boundary-value problems in bounded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikoskii spaces, which allows us to treat some cases of white noise on the boundary.

math.AP↗

arXiv2019

Parabolic problems in generalized Sobolev spaces

We consider a general inhomogeneous parabolic initial-boundary value problem for a $2b$-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers $s$ and $s/(2b)$ and with a function $φ:[1,\infty)\to(0,\infty)$ that varies slowly at infinity. The function parameter $φ$ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of the solution are continuous on a given set.

math.AP↗

arXiv2018

Sobolev-like Hilbert spaces induced by elliptic operators

We investigate properties of function spaces induced by the inner product Sobolev spaces $H^{s}(Ω)$ over a bounded Euclidean domain $Ω$ and by an elliptic differential operator $A$ on $\overlineΩ$. The domain and the coefficients of $A$ are of the class $C^{\infty}$. These spaces consist of all distributions $u\in H^{s}(Ω)$ such that $Au\in H^λ(Ω)$ and are endowed with the corresponding graph norm, with $s,λ\in\mathbb{R}$. We prove an interpolation formula for these spaces and discuss their application to elliptic boundary-value problems.

math.AP↗

arXiv2018

Nonregular elliptic boundary-value problems and Hörmander spaces

We investigate nonregular elliptic problems with boundary conditions of higher orders. We prove that these problems are Fredholm on appropriate pairs of inner product Hörmander spaces that form a two-sided refined Sobolev scale. We also prove a theorem on the regularity of generalized solutions to the problems in these spaces.

math.AP↗

arXiv2018

Elliptic problems with boundary operators of higher orders in Hörmander-Roitberg spaces

We investigate elliptic boundary-value problems for which the maximum of the orders of the boundary operators is equal to or greater than the order of the elliptic differential equation. We prove that the operator corresponding to an arbitrary problem of this kind is bounded and Fredholm between appropriate Hilbert spaces which form certain two-sided scales and are built on the base of isotropic Hörmander spaces. The differentiation order for these spaces is given by an arbitrary real number and positive function which varies slowly at infinity in the sense of Karamata. We establish a local a priori estimate for the generalized solutions to the problem and investigate their local regularity (up to the boundary) on these scales. As an application, we find sufficient conditions under which the solutions have continuous classical derivatives of a given order.

math.AP↗

arXiv2017

Petrovskii elliptic systems in the extended Sobolev scale

Petrovskii elliptic systems of linear differential equations given on a closed smooth manifold are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. Theorems on the solvability of the elliptic systems on the extended Sobolev scale are proved. An a priori estimate for solutions is obtained, and their regularity is studied.

math.AP↗

arXiv2017

Localization principles for Schrödinger operator with a singular matrix potential

We study the spectrum of the one-dimensional Schrödinger operator $H_0$ with a matrix singular distributional potential $q=Q'$ where $Q\in L^{2}_{\mathrm{loc}}(\mathbb{R},\mathbb{C}^{m})$. We obtain generalizations of Ismagilov's localization principles, which give necessary and sufficient conditions for the spectrum of $H_0$ to be bounded below and discrete.

math.AP↗

arXiv2017

Elliptic problems with boundary conditions of high orders in Hörmander spaces

In a class of inner product Hörmander spaces, we investigate a general elliptic problem for which the maximum of orders of boundary conditions is grater than or equal to the order of elliptic equation. The order of regularity for these spaces is an arbitrary radial positive function RO-varying at infinity in the sense of Avakumović. We prove that the operator of the problem under investigation is bounded and Fredholm on appropriate pairs of Hörmander spaces indicated. A theorem on isomorphism generated by this operator is proved. For generalized solutions to this problem, we establish a local a priory estimate and prove a theorem about their local regularity in Hörmander spaces. As application, we obtain new sufficient conditions under which given derivatives of the solutions are continuous.

math.AP↗

arXiv2017

Parameter-dependent one-dimensional boundary-value problems in Sobolev spaces

We consider the most general class of linear boundary-value problems for higher-order ordinary differential systems whose solutions and right-hand sides belong to the corresponding Sobolev spaces. For parameter-dependent problems from this class, we obtain a constructive criterion under which their solutions are continuous in the Sobolev space with respect to the parameter. We also obtain a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem. These results are applied to a new broad class of parameter-dependent multipoint boundary-value problems.

math.CA↗