L1-elliptic regularity and H = W on the whole Lp-scale on arbitrary manifolds

We define abstract Sobolev type spaces on Lp-scales, p ∈ [1,∞), on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families B of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sections in these spaces under a generalised ellipticity condition on the underlying family. In particular, this implies a covariant version of Meyers-Serrin's theorem on the whole Lp-scale, for arbitrary Riemannian manifolds. Furthermore, we prove a new local elliptic regularity result in L1 on the Besov scale, which shows that the above generalised ellipticity condition is satisfied on the whole Lp-scale, if some differential operator from B that has a sufficiently high (but not necessarily the highest) order is elliptic.