Density and non-density of $C^\infty_c \hookrightarrow W^{k,p}$ on complete manifolds with curvature bounds
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| Format: | Preprint |
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2020
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| author | Honda, Shouhei Mari, Luciano Rimoldi, Michele Veronelli, Giona |
| author_facet | Honda, Shouhei Mari, Luciano Rimoldi, Michele Veronelli, Giona |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_14630 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Density and non-density of $C^\infty_c \hookrightarrow W^{k,p}$ on complete manifolds with curvature bounds Honda, Shouhei Mari, Luciano Rimoldi, Michele Veronelli, Giona Differential Geometry Analysis of PDEs 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. |
| title | Density and non-density of $C^\infty_c \hookrightarrow W^{k,p}$ on complete manifolds with curvature bounds |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2011.14630 |