# A. A. Murach

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### A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems

Vladimir Mikhailets, Aleksandr Murach, Vitalii Soldatov

Methods Funct. Anal. Topology **22** (2016), no. 4, 375-386

We consider the most general class of linear boundary-value problems for ordinary differential systems, of order $r\geq1$, whose solutions belong to the complex space $C^{(n+r)}$, with $0\leq n\in\mathbb{Z}.$ The boundary conditions can contain derivatives of order $l$, with $r\leq l\leq n+r$, of the solutions. We obtain a constructive criterion under which the solutions to these problems are continuous with respect to the parameter in the normed space $C^{(n+r)}$. We also obtain a two-sided estimate for the degree of convergence of these solutions.

### Elliptic problems in the sense of B. Lawruk on two-sided refined scales of spaces

Iryna S. Chepurukhina, Aleksandr A. Murach

Methods Funct. Anal. Topology **21** (2015), no. 1, 6-21

We investigate elliptic boundary-value problems with additional unknown functions on the boundary of a Euclidean domain. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on two-sided refined scales built on the base of inner product isotropic H\"ormander spaces. The regularity of the distributions forming these spaces are characterized by a real number and an arbitrary function that varies slowly at infinity in the sense of Karamata. For the generalized solutions to the problem, we prove theorems on a priori estimates and local regularity in these scales. As applications, we find new sufficient conditions under which the solutions have continuous classical derivatives of a prescribed order.

### Parameter-elliptic problems and interpolation with a function parameter

Anna V. Anop, Aleksandr A. Murach

Methods Funct. Anal. Topology **20** (2014), no. 2, 103-116

Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to a Hilbert Sobolev scale. The latter are the Hörmander spaces $B_{2,k}$ for which the smoothness index $k$ is an arbitrary radial function RO-varying at $+\infty$. We prove that the operator corresponding to this problem sets isomorphisms between appropriate Hörmander spaces provided the absolute value of the parameter is large enough. For solutions to the problem, we establish two-sided estimates, in which the constants are independent of the parameter.

### Parabolic problems and interpolation with a function parameter

Valerii Los, Aleksandr A. Murach

Methods Funct. Anal. Topology **19** (2013), no. 2, 146-160

We give an application of interpolation with a function parameter to parabolic differential operators. We introduce a refined anisotropic Sobolev scale that consists of some Hilbert function spaces of generalized smoothness. The latter is characterized by a real number and a function varying slowly at infinity in Karamata's sense. This scale is connected with anisotropic Sobolev spaces by means of interpolation with a function parameter. We investigate a general initial--boundary value parabolic problem in the refined Sobolev scale. We prove that the operator corresponding to this problem sets isomorphisms between appropriate spaces pertaining to this scale.

### Parameter-elliptic operators on the extended Sobolev scale

Aleksandr A. Murach, Tetiana Zinchenko

Methods Funct. Anal. Topology **19** (2013), no. 1, 29-39

Parameter--elliptic pseudodifferential operators 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 a Hilbert--Sobolev scale. We prove that these operators set isomorphisms between appropriate spaces of the scale provided the absolute value of the parameter is large enough. For solutions to the corresponding parameter--elliptic equations, we establish two-sided a priori estimates, in which the constants are independent of the parameter.

### General forms of the Menshov-Rademacher, Orlicz, and Tandori theorems on orthogonal series

Vladimir A. Mikhailets, Aleksandr A. Murach

Methods Funct. Anal. Topology **17** (2011), no. 4, 330-340

We prove that the classical Menshov--Rademacher, Orlicz, and Tandori theorems remain true for orthogonal series given in the direct integrals of measurable collections of Hilbert spaces. In particular, these theorems are true for the spaces $L_{2}(X,d\mu;H)$ of vector-valued functions, where $(X,\mu)$ is an arbitrary measure space, and $H$ is a real or complex Hilbert space of an arbitrary dimension.

### Extension of some Lions-Magenes theorems

Methods Funct. Anal. Topology **15** (2009), no. 2, 152-167

A general form of the Lions-Magenes theorems on solvability of an elliptic boundary-value problem in the spaces of nonregular distributions is proved. We find a general condition on the space of right-hand sides of the elliptic equation under which the operator of the problem is bounded and has a finite index on the corresponding couple of Hilbert spaces. Extensive classes of the spaces satisfying this condition are constructed. They contain the spaces used by Lions and Magenes and many others spaces.

### Douglis-Nirenberg elliptic systems in the refined scale of spaces on a closed manifold

Methods Funct. Anal. Topology **14** (2008), no. 2, 142-158

Douglis-Nirenberg elliptic systems of linear pseudodifferential equations are studied on a smooth closed manifold. We prove that the operator generated by the system is a Fredholm one on the refined two-sided scale of the functional Hilbert spaces. Elements of this scale are the special isotropic spaces of H\"{o}rmander--Volevich--Paneah. The refined smoothness of a solution of the system is studied. The elliptic systems with a parameter are investigated as well.

### Interpolation with a function parameter and refined scale of spaces

Vladimir A. Mikhailets, Aleksandr A. Murach

Methods Funct. Anal. Topology **14** (2008), no. 1, 81-100

The interpolation of couples of separable Hilbert spaces with a function parameter is studied. The main properties of the classical interpolation are proved. Some applications to the interpolation of isotropic Hörmander spaces over a closed manifold are given.