Parameter-elliptic problems and interpolation with a function parameter

Abstract

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 the Hilbert Sobolev scale. The latter are the H\"ormander spaces $B_{2,k}$ for which the smoothness index $k$ is an arbitrary radial function RO-varying at infinity. We prove that the operator corresponding to this problem sets isomorphisms between appropriate H\"ormander spaces provided that 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.