*preprint*

**Inserted:** 1 dec 2020

**Last Updated:** 7 dec 2020

**Year:** 2020

**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 $\mathrm{Riem}$ 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, when $p>2$ we give the first counterexample to the density property on manifolds whose sectional curvature is bounded from below by a negative constant. 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.

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