The Kato Square Root Problem on Submanifolds

Abstract

We solve the Kato square root problem for divergence form operators on complete Riemannian manifolds that are embedded in Euclidean space with a bounded second fundamental form. We do this by proving local quadratic estimates for perturbations of certain first-order differential operators that act on the trivial bundle over a complete Riemannian manifold with at most exponential volume growth and on which a local Poincaré inequality holds. This is based on the framework for Dirac type operators that was introduced by Axelsson, Keith and McIntosh.

34 pages

References in corpus (1)

• Local Hardy Spaces of Differential Forms on Riemannian Manifolds

Cited by in corpus (12)

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• Continuity of solutions to space-varying pointwise linear elliptic equations
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• Rough metrics on manifolds and quadratic estimates
• Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data
• Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary
• Functional calculus and harmonic analysis in geometry
• On square root domains for non-self-adjoint Sturm-Liouville operators
• The Kato square root problem on an arbitrary domain of ${\mathbb{R}}^d$