| Autor: |
Honda S., Mari L., Rimoldi M., Veronelli G. |
| Přispěvatelé: |
Honda, S, Mari, L, Rimoldi, M, Veronelli, G |
| Jazyk: |
angličtina |
| Rok vydání: |
2021 |
| Předmět: |
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| Popis: |
We investigate the density of compactly supported smooth functions in the Sobolev space Wk,p on complete Riemannian manifolds. In the first part of the paper, we extend to the full range p∈[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≥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 Wk,p does not hold for any k≥2. We also deduce the existence of a counterexample to the validity of the Calderón–Zygmund inequality for p>2 when Sec≥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. |
| Databáze: |
OpenAIRE |
| Externí odkaz: |
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