Hörmander spaces on manifolds, and their application to elliptic boundary-value problems

Abstract

We introduce an extended Sobolev scale on a smooth compact manifold with boundary. The scale is formed by inner-product Hörmander spaces for which an RO-varying radial function serves as a regularity index. These spaces do not depend on a choice of local charts on the manifold. The scale consists of all Hilbert spaces that are interpolation ones for pairs of inner-product Sobolev spaces, is obtained by the interpolation with a function parameter of these pairs, and is closed with respect to this interpolation. As an application of the scale introduced, we give a theorem on the Fredholm property of a general elliptic boundary-value problem on appropriate Hörmander spaces and find sufficient conditions under which its generalized solutions belong to the space of $p\ge 0$ times continuously differential functions.

7 pages, Ukrainian