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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 ( 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 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
Milestones
Received: 25 June 2020
Revised: 25 February 2021
Accepted: 6 April 2021
Published: 10 February 2023