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

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.Comment: Slight technical modification of the notion of "GBG constant section" on page 7, and a few technical modifications to Proposition 8.4, 8.6, 8.

Quadratic estimates for perturbed Dirac type operators on doubling measure metric spaces

We consider perturbations of Dirac type operators on complete, connected metric spaces equipped with a doubling measure. Under a suitable set of assumptions, we prove quadratic estimates for such operators and hence deduce that these operators have a bounded functional calculus. In particular, we deduce a Kato square root type estimate.Comment: To appear in Proceedings of the AMSI International Conference on Harmonic Analysis and Applications, Proc. Centre Math. Appl. Austral. Nat. Uni

Continuity of solutions to space-varying pointwise linear elliptic equations

We consider pointwise linear elliptic equations of the form Lα uα = ŋα on a smooth compact manifold where the operators Lα are in divergence form with real, bounded, measurable coefficients that vary in the space variableα. We establish L2-continuity of the solutions at α whenever the coefficients of Lα are L∞ -continuous at α and the initial datum is L2 -continuous at α. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a time evolving family of metrics gt that is tangential to the Ricci flow almost-everywhere along geodesics when starting with a smooth initial metric. Under the assumption that our initial metric is a rough metric on ʍ with a C1 heat kernel on a "non-singular" nonempty open subset Ɲ, we show that α à gt (α) is continuous whenever α € Ɲ