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 Calderón–Zygmund inequality for $p>2$ when $\mathrm{Sec}\ge 0$, 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.

我们研究了 Sobolev 空间中紧密支持的平滑函数的密度 ${宽}^{克,磷}$关于完全黎曼流形。在论文的第一部分，我们扩展到全范围$磷\in [1,2]$希尔伯特案例中已知的最一般的结果。特别地，我们获得了二次 Ricci 下界下的密度（当$克=2$) 或 Riemann 曲率张量的导数的适当控制增长仅到阶 $克-3$ （什么时候 $克>2$）。为此，我们证明了一个可能具有独立意义的梯度正则性引理。在论文的第二部分，对于每个$n\ge 2$ 和 $磷>2$ 我们构建了一个完整的 $n$- 截面曲率由负常数限定的维流形，其中的密度属性为 ${宽}^{克,磷}$ 不适合任何 $克\ge 2$. 我们还推导出了 Calderón-Zygmund 不等式有效性的反例：$磷>2$ 什么时候 $\mathrm{秒}\ge 0$，并且在紧凑的设置中，我们展示了建立 Calderón-Zygmund 理论的可能性 $磷>2$ 常数仅取决于直径的界限和截面曲率的下界。