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Discrete analogs of pseudo-differential operators and equations in discrete Sobolev–Slobodetsky spaces are considered. Using suitable discrete boundary conditions, we prove the unique solvability of the discrete boundary-value problem. [ABSTRACT FROM AUTHOR]

The paper investigates connections and relations between different types (concepts) of hypoelliptic differential operators (symbols (characteristic polynomials) of these operators). [ABSTRACT FROM AUTHOR]

We consider the discrete Dirichlet boundary value problem for a discrete elliptic pseudodifferential equation in the quadrant and study its solvability in discrete counterparts of the Sobolev–Slobodetskii space. The study is based on a special factorization of the elliptic symbol. We compare the solutions to the discrete Dirichlet problem and its continuous counterpart. [ABSTRACT FROM AUTHOR]

We introduce integral transforms to study problems for the singular Bessel differential operators B−γ with negative parameter −γ < 0. A solution to the singular Bessel equation B−γu+u = 0 is a function u = 핁μ expressed in terms of the Bessel functions of the first kind with positive parameter μ = (γ + 1)/2. Based on the notion of a 핋-pseudoshift, we formulate the Levitan addition theorem. We construct the Bessel 핁-transform and the corresponding class of singular 핁-pseudodifferential operators. We prove a theorem on the order of 핁-pseudodifferential operators in the class of Sobolev–Kipriyanov functions. [ABSTRACT FROM AUTHOR]

For a pair of smooth transversally intersecting submanifolds in some enveloping smooth manifold, we study the algebra generated by pseudodifferential operators and (co)boundary operators corresponding to submanifolds. We establish that such an algebra has 18 types of generating elements. For operators from this algebra, we define the concept of symbol and obtain the composition formula. [ABSTRACT FROM AUTHOR]

We study the main operations (argument shift, differentiation, etc.) in the generalized spaces of type S and some classes of analytic functions and pseudodifferential operators in spaces of this kind, as well as the properties of the Fourier transforms of generalized functions, convolutions, convolvers, and multiplicators. The correct solvability of the nonlocal time problem is proved for one class of pseudodifferential equations in generalized spaces of type S. Its solution is presented in the form of convolution of the fundamental solution with an initial function, which is an element of the space of generalized functions of the ultradistribution type. [ABSTRACT FROM AUTHOR]

In this paper, we propose an abstract scheme for the study of special operators and apply this scheme to examining elliptic pseudo-differential operators and related boundary-value problems on manifolds with nonsmooth boundaries. In particular, we consider cases where boundaries may contain conical points, edges of various dimensions, and even peak points. Using the constructions proposed, we present well-posed formulations of boundary-value problems for elliptic pseudo-differential equations on manifolds discussed in Sobolev–Slobodecky spaces. [ABSTRACT FROM AUTHOR]

We consider a computation of a group of stable homotopy classes for pseudodifferential elliptic boundary problems. We study this problem in terms of topological K-groups of spaces in the following cases: boundary-value problems on manifolds with boundary, transmission problems with conditions on a closed submanifold of codimension one, and nonlocal problems with contractions. [ABSTRACT FROM AUTHOR]

We consider discrete pseudo-differential equations with elliptic symbols and the corresponding discrete boundary-value problems in special canonical domains of multidimensional spaces. The solvability of such equations and boundary-value problems in discrete analogs of Sobolev–Slobodetsky spaces is examined. [ABSTRACT FROM AUTHOR]

We prove the correct solvability of the nonlocal multipoint (in time) problem for the evolutionary equation with pseudodifferential operator in the space of generalized periodic functions and an initial function, which is an element of the space of periodic ultradistributions. [ABSTRACT FROM AUTHOR]