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Sobolev functions without compactly supported approximations

### Giona Veronelli

Vol. 15 (2022), No. 8, 1991–2002
##### Abstract

A basic property and useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space ${W}^{k,p}\left({ℝ}^{n}\right)$ (i.e., the functions with weak derivatives of orders $0$ to $k$ in ${L}^{p}$). On Riemannian manifolds, it is well known that the same property remains valid under suitable geometric assumptions. However, on a complete noncompact manifold it can fail to be true in general, as we prove here. This settles an open problem raised for instance by E. Hebey (Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lect. Notes Math. 5 (1999), 48–49).

##### Keywords
Sobolev spaces on manifolds, manifolds with unbounded geometry, density problems, Calderón–Zygmund inequalities
##### Mathematical Subject Classification
Primary: 46E35, 53C20