Sobolev functions without compactly supported approximations

Abstract

A basilar property and a useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space $W^{k,p}({\mathbb{R}}^n)$ (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 non-compact manifold it can fail to be true in general, as we prove in this paper. This settles an open problem raised for instance by E. Hebey [\textit{Nonlinear analysis on manifolds: Sobolev spaces and inequalities}, Courant Lecture Notes in Mathematics, vol. 5, 1999, pp. 48-49].

9 pages. Minor typos corrected

Cited by in corpus (2)

• Density and non-density of $C_c^{\infty }\hookrightarrow W^{k,p}$ on complete manifolds with curvature bounds
• Generalized Ornstein--Uhlenbeck Semigroups in weighted $L^p$-spaces on Riemannian Manifolds