Density and non-density of $C^\infty_c \hookrightarrow W^{k,p}$ 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 $p\in [1,2]$ 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 $n \ge 2$ 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 $k \ge 2$. We also deduce the existence of a counterexample to the validity of the Calder\'on-Zygmund inequality for $p>2$ when $\mathrm{Sec} \ge 0$, and in the compact setting we show the impossibility to build a Calder\'on-Zygmund theory for $p>2$ with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.