论文标题
$ c^\ infty_c \ hookrightarrow w^{k，p} $的密度和非密度，具有曲率界限
Density and non-density of $C^\infty_c \hookrightarrow W^{k,p}$ on complete manifolds with curvature bounds
论文作者
论文摘要
我们研究了完整的Riemannian歧管上的Sobolev空间中紧凑型光滑功能的密度。在本文的第一部分中，我们将整个范围$ p \在[1,2] $中的全范围$ p \ the Hilbertian Case中已知的最一般结果。特别是，我们在二次RICCI下限（$ k = 2 $）或仅订单$ k-3 $（$ k> 2 $时）获得了密度。为此，我们证明了可能具有独立利益的梯度规律性引理。在本文的第二部分中，对于每$ n \ ge 2 $和$ p> 2 $，我们构建了一个完整的$ n $二维歧管，其截面曲率从下面界定为负常数，为此，其密度属性在$ w^{k，p} $中均不适合任何$ k \ ge 2 $。我们还推断出对Calderón-Zygmund不平等的有效性的存在，当$ p> 2 $时，当$ \ m \ mathrm {sec} \ ge 0 $时，在紧凑型设置中，我们表明，我们不可能在calderón-zygmund理论上构建$ p> 2 $仅根据diameter and the the diamoter和a diymeter and A diymeter和a poldy of diamorter和acalderón-zygmund理论。
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.
