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EnglishWe 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 $pin [1,2]$ 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 Calder'on-Zygmund inequality for $p>2$ when $Sec ge 0$, and in the compact setting we show the impossibility to build a Calder'on-Zygmund theory for $p>2$ with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.
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. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - STAMPA. - 211(2021). [10.1016/j.na.2021.112429]
| Titolo: | Density and non-density of $C^infty_c hookrightarrow W^{k,p}$ on complete manifolds with curvature bounds | |
| Autori: | ||
| Data di pubblicazione: | 2021 | |
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| Digital Object Identifier (DOI): | http://dx.doi.org/10.1016/j.na.2021.112429 | |
| Appare nelle tipologie: | 1.1 Articolo in rivista |
File in questo prodotto:
| File | Descrizione | Tipologia | Licenza | |
|---|---|---|---|---|
| HondaMariRimoldiVeronelli_Density_Rev.pdf | Articolo principale | 2. Post-print / Author's Accepted Manuscript |  | Embargo: 03/06/2023 Richiedi una copia |
| 1-s2.0-S0362546X21001176-main.pdf | Articolo principale | 2a Post-print versione editoriale / Version of Record | Non Pubblico - Accesso privato/ristretto | Administrator Richiedi una copia |
http://hdl.handle.net/11583/2900752
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