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$\mathsf{L}^1$-elliptic regularity and $H=W$ on the whole $\mathsf{L}^p$-scale on arbitrary manifolds

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