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Sobolev functions without compactly supported approximations

Giona Veronelli

Vol. 15 (2022), No. 8, 1991–2002

A basic property and useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space Wk,p(n) (i.e., the functions with weak derivatives of orders 0 to k in Lp). 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).

Sobolev spaces on manifolds, manifolds with unbounded geometry, density problems, Calderón–Zygmund inequalities
Mathematical Subject Classification
Primary: 46E35, 53C20
Received: 25 June 2020
Revised: 25 February 2021
Accepted: 6 April 2021
Published: 10 February 2023
Giona Veronelli
Dipartimento di Matematica e Applicazioni
Università di Milano Bicocca