Data di Pubblicazione:
2021
Citazione:
Density and non-density of C∞c↪Wk,p on complete manifolds with curvature bounds / S. Honda, L. Mari, M. Rimoldi, G. Veronelli. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 211:112429(2021 Oct), pp. 1-26. [10.1016/j.na.2021.112429]
Abstract:
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.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Sobolev space; Density; Curvature; Singular point; Sampson formula; Alexandrov space; RCD space;
Elenco autori:
S. Honda, L. Mari, M. Rimoldi, G. Veronelli
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