Density and non-density of Cc∞↪Wk, p on complete manifolds with curvature bounds. (October 2021)

• Record Type: Journal Article
• Authors: Honda, Shouhei; Mari, Luciano; Rimoldi, Michele; Veronelli, Giona
• Is Part Of: Nonlinear analysis. Volume 211(2021)
• Publication Date: 2021-10
• Subjects: 46E35 -- 53C21; Sobolev space -- Density -- Curvature -- Singular point -- Sampson formula -- Alexandrov space -- RCD space; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112429
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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 ∈ [ 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 W k, 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.