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

Master thesis (2016)

Authors

R. Versendaal

Contributors

J.M.A.M. van Neerven (mentor)

Department

Delft Institute of Applied Mathematics () (TU Delft)

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.

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