  We introduce an extended Sobolev scale on a smooth compact manifold with
boundary. The scale is formed by inner-product H\"ormander 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 H\"ormander spaces and find
sufficient conditions under which its generalized solutions belong to the space
of $p\geq0$ times continuously differential functions.
