  We introduce an extended Sobolev scale for vector bundles over smooth closed
manifolds and investigate its interpolation properties. This scale is built on
the base of the inner product H\"ormander spaces whose order of regularity is
given by an arbitrary positive function OR-varying at infinity in the sense of
Avakumovi\'c. We prove that this scale is obtained by the interpolation with a
function parameter between inner product Sobolev spaces, is closed with respect
to the interpolation with a function parameter, and consists of all Hilbert
spaces that are interpolation spaces between inner product Sobolev spaces. We
also show that the spaces belonging to the scale introduced does not depend (up
to equivalence of norms) on the choice of local charts on manifold and local
trivializations of the bundle. Moreover, we prove embedding theorems and a
duality theorem for this scale. We consider an elliptic pseudodifferential
operators acting between vector bundles of the same rank on the extended
Sobolev scale. We prove that this operator is bounded and Fredholm on pairs of
appropriate H\"ormander spaces. An a priori estimate is established for the
solutions and their local regularity is investigated. We also find new
sufficient conditions for the solutions to have continuous derivatives of a
given order.
