Density and non-density of Cc∞↪Wk,p on complete manifolds with curvature bounds

Abstract We investigate the density of compactly supported smooth functions in the Sobolev space W k , p on complete Riemannian manifolds. In the first part of the paper, we extend to the full range p ∈ [ 1 , 2 ] the most general results known in the Hilbertian case. In particular, we obtain the density under a quadratic Ricci lower bound (when k = 2 ) or a suitably controlled growth of the derivatives of the Riemann curvature tensor only up to order k − 3 (when k > 2 ). To this end, we prove a gradient regularity lemma that might be of independent interest. In the second part of the paper, for every n ≥ 2 and p > 2 we construct a complete n -dimensional manifold with sectional curvature bounded from below by a negative constant, for which the density property in W k , p does not hold for any k ≥ 2 . We also deduce the existence of a counterexample to the validity of the Calderon–Zygmund inequality for p > 2 when Sec ≥ 0 , and in the compact setting we show the impossibility to build a Calderon–Zygmund theory for p > 2 with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.