Density and non-density of \(C_c^\infty\hookrightarrow W^{k,p}\) on complete manifolds with curvature bounds. (English) Zbl 1481.46028
Summary: 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 \(n\geq 2\) 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 \(k\geq 2\). We also deduce the existence of a counterexample to the validity of the Calderón-Zygmund inequality for \(p>2\) when \(\mathrm{Sec}\geq 0\), 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.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
Keywords:
Sobolev space; density; curvature; singular point; Sampson formula; Alexandrov space; RCD spaceReferences:
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