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.
L1-elliptic regularity and H = W on the whole Lp-scale on arbitrary manifolds / Guidetti D.; Guneysu B.; Pallara D.. - In: ANNALES ACADEMIAE SCIENTIARUM FENNICAE. MATHEMATICA. - ISSN 1239-629X. - STAMPA. - 42:1(2017), pp. 497-521. [10.5186/aasfm.2017.4234]
L1-elliptic regularity and H = W on the whole Lp-scale on arbitrary manifolds
Guidetti D.;
2017
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.