## 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 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.

Original language | English |
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Article number | 112429 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 211 |

DOIs | |

Publication status | Published - 2021 Oct |

## Keywords

- Alexandrov space
- Curvature
- Density
- RCD space
- Sampson formula
- Singular point
- Sobolev space

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