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We introduce an extended Sobolev scale on a smooth compact manifold withboundary. The scale is formed by inner-product H\"ormander spaces for which anRO-varying radial function serves as a regularity index. These spaces do notdepend on a choice of local charts on the manifold. The scale consists of allHilbert spaces that are interpolation ones for pairs of inner-product Sobolevspaces, is obtained by the interpolation with a function parameter of thesepairs, and is closed with respect to this interpolation. As an application ofthe scale introduced, we give a theorem on the Fredholm property of a generalelliptic boundary-value problem on appropriate H\"ormander spaces and findsufficient conditions under which its generalized solutions belong to the spaceof $p\geq0$ times continuously differential functions.

H\"ormander spaces on manifolds, and their application to elliptic boundary-value problems