We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in W2,p⁠. The result is improved for p=2 avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

Impera, D., Rimoldi, M., Veronelli, G. (2021). Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2021(14), 10521-10558 [10.1093/imrn/rnz131].

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Impera, D;Rimoldi, M;Veronelli, G
2021

Abstract

We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in W2,p⁠. The result is improved for p=2 avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.
Articolo in rivista - Articolo scientifico
Sobolev spaces on manifolds; distance-like functions; Sobolev inequalities; cut-off functions
English
5-lug-2019
2021
2021
14
10521
10558
none
Impera, D., Rimoldi, M., Veronelli, G. (2021). Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2021(14), 10521-10558 [10.1093/imrn/rnz131].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/256443
Citazioni
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 6
Social impact