The Riesz transform on a complete Riemannian manifold with Ricci curvature bounded from below

Master thesis (2016)

Authors

Contributors

Department
Delft Institute of Applied Mathematics (EEMCS) (TU Delft)
Copyright: © 2016 Versendaal, R.
Riemannian manifold Riesz transform Hodge-Dirac operator Ricci curvature
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Published Date

08-09-2016

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Abstract

We study the Riesz transform and Hodge-Dirac operator on a complete Riemannian manifold with Ricci curvature bounded from below. We define the Hodge-Dirac operator ∏ on Lp(ΛTM) as the closure of d + d* on smooth, compactly supported k-forms for 1 < p < ∞. Given the boundedness of the Riesz transform on Lp(ΛTM), we show that ∏ is R-bisectorial on Lp(ΛTM). From this we conclude that ∏ has a bounded H∞-functional calculus on a bisector under mild assumptions which we conjecture to be true when the Ricci curvature is nonnegative. We conclude by showing that from this bounded H∞-functional calculus for the Hodge-Dirac operator we can retrieve the boundedness of the Riesz transform, thus giving us that the mentioned assertions are equivalent when the Ricci curvature is nonnegative.