Density problems on vector bundles and manifolds

Bandara, Lashi

doi:10.1090/S0002-9939-2014-12284-2

Density problems on vector bundles and manifolds
HTML articles powered by AMS MathViewer

by Lashi Bandara PDF

Proc. Amer. Math. Soc. 142 (2014), 2683-2695 Request permission

Abstract:

We study some canonical differential operators on vector bundles over smooth, complete Riemannian manifolds. Under very general assumptions, we show that smooth, compactly supported sections are dense in the domains of these operators. Furthermore, we show that smooth, compactly supported functions are dense in second order Sobolev spaces on such manifolds under the sole additional assumption that the Ricci curvature is uniformly bounded from below.

References

Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on ${\Bbb R}^n$, Ann. of Math. (2) 156 (2002), no. 2, 633–654. MR 1933726, DOI 10.2307/3597201
Andreas Axelsson, Stephen Keith, and Alan McIntosh, The Kato square root problem for mixed boundary value problems, J. London Math. Soc. (2) 74 (2006), no. 1, 113–130. MR 2254555, DOI 10.1112/S0024610706022873
Andreas Axelsson, Stephen Keith, and Alan McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), no. 3, 455–497. MR 2207232, DOI 10.1007/s00222-005-0464-x
Lashi Bandara and Alan McIntosh, The Kato square root problem on vector bundles with generalised bounded geometry, arXiv:1203.0373v2 (2012).
Paul R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973), 401–414. MR 0369890, DOI 10.1016/0022-1236(73)90003-7
Matthew P. Gaffney, The harmonic operator for exterior differential forms, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 48–50. MR 48138, DOI 10.1073/pnas.37.1.48
Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1688256
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Andrew J. Morris, The Kato square root problem on submanifolds, J. Lond. Math. Soc. (2) 86 (2012), no. 3, 879–910. MR 3000834, DOI 10.1112/jlms/jds039
Walter Roelcke, Über den Laplace-Operator auf Riemannschen Mannigfaltigkeiten mit diskontinuierlichen Gruppen, Math. Nachr. 21 (1960), 131–149 (German). MR 151927, DOI 10.1002/mana.19600210303
K\B{o}saku Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition. MR 1336382, DOI 10.1007/978-3-642-61859-8