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

Abstract

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.

27 pages