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

We investigate the density of compactly supported smooth functions in the Sobolev space Wk,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 Wk,p does not hold for any k≥2. We also deduce the existence of a counterexample to the validity of the Calderón–Zygmund inequality for p>2 when Sec≥0, and in the compact setting we show the impossibility to build a Calderón–Zygmund theory for p>2 with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.