The Kato square root problem on vector bundles with generalised bounded geometry

TL;DR

This paper proves a Kato square root estimate for operators on vector bundles over manifolds with generalized bounded geometry, extending previous results to a broader geometric setting.

Contribution

It introduces the concept of generalized bounded geometry for vector bundles and establishes quadratic estimates for Dirac-type operators under this framework.

Findings

Established Kato square root estimates on vector bundles with generalized bounded geometry.

Proved quadratic estimates for perturbations of Dirac operators.

Extended classical results to manifolds with exponential local doubling and bounded geometry conditions.

Abstract

We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call generalised bounded geometry. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions.