TY - JOUR

T1 - Density and non-density of Cc∞↪Wk,p on complete manifolds with curvature bounds

AU - Honda, Shouhei

AU - Mari, Luciano

AU - Rimoldi, Michele

AU - Veronelli, Giona

N1 - Funding Information:
M.R. and G.V. are members of INdAM-GNAMPA. We acknowledge that the present research has been partially supported by PRIN project 2017, Italy “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” and by MIUR grant Dipartimenti di Eccellenza 2018–2022 ( E11G18000350001 ), DISMA, Politecnico di Torino. S.H. acknowledges the support of the Grant-in-Aid for Scientific Research (B), Japan of 20H01799 and the Grant-in-Aid for Scientific Research (B), Japan of 18H01118 . We also would like to thank Li Chen and Willie WY Wong for pointing out some literature and the anonymous referee for the careful reading of the paper and for useful suggestions.
Funding Information:
M.R. and G.V. are members of INdAM-GNAMPA. We acknowledge that the present research has been partially supported by PRIN project 2017, Italy “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” and by MIUR grant Dipartimenti di Eccellenza 2018–2022 (E11G18000350001), DISMA, Politecnico di Torino. S.H. acknowledges the support of the Grant-in-Aid for Scientific Research (B), Japan of 20H01799 and the Grant-in-Aid for Scientific Research (B), Japan of 18H01118. We also would like to thank Li Chen and Willie WY Wong for pointing out some literature and the anonymous referee for the careful reading of the paper and for useful suggestions.
Publisher Copyright:
© 2021 Elsevier Ltd

PY - 2021/10

Y1 - 2021/10

N2 - 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.

AB - 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.

KW - Alexandrov space

KW - Curvature

KW - Density

KW - RCD space

KW - Sampson formula

KW - Singular point

KW - Sobolev space

UR - http://www.scopus.com/inward/record.url?scp=85107679855&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85107679855&partnerID=8YFLogxK

U2 - 10.1016/j.na.2021.112429

DO - 10.1016/j.na.2021.112429

M3 - Article

AN - SCOPUS:85107679855

VL - 211

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

M1 - 112429

ER -