Delft University of Technology Faculty of Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics The Riesz transform on a complete Riemannian manifold with Ricci curvature bounded from below A thesis submitted to the Delft Institute of Applied Mathematics in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in APPLIED MATHEMATICS by RIK VERSENDAAL Delft, the Netherlands September 2016 Copyright c 2016 by Rik Versendaal. All rights reserved. MSc THESIS APPLIED MATHEMATICS \The Riesz transform on a complete Riemannian manifold with Ricci curvature bounded from below" RIK VERSENDAAL Delft University of Technology Daily Supervisor and responsible professor Prof. dr. J.M.A.M van Neerven Other thesis committee members Prof. dr. F.H.J. Redig Dr. J.L.A Dubbeldam September, 2016 Delft iii Summary In this thesis we study the Riesz transform and Hodge-Dirac operator on a complete Riemannian manifold with Ricci curvature bounded from below. The motivation for this is the paper `Etude´ des transformation de Riesz dans les vari´et´esriemanniennes `acourbure de Ricci minor´ee' by D. Bakry ([6]). In this paper, Bakry proves the boundedness of the Riesz transform acting on k-forms under the assumption that the Ricci-curvature, as well as related quadratic forms, are bounded from below. The analysis of this paper is one of two main goals in this thesis. In the second part we extend the operators defined by Bakry beyond L2 to Lp-spaces for arbitrary 1 ≤ p < 1 and analyse the Hodge-Dirac operator Π = d + δ on Lp. For this, we will follow the lines of the paper `Quadratic estimates and functional calculi of perturbed Dirac operators' by A. Axelsson, S. Keith and A. McIntosh ([4]) and also of the paper `Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces' by J.Maas and J.M.A.M van Neerven ([25]). Before we turn to the analysis of the paper however, we first need to introduce some basic theory of differential geometry and strongly continuous semigroups. We collect the necessary definitions and results in these areas to create a basic understanding of these subjects. For a more detailed discussion of these subjects one should look in the references made in chapters 2 and 3. After the basic theory is discussed, we thoroughly discuss the paper of Bakry up to and including the section on the Riesz transform on k-forms. We first start out by introducing the Witten-Laplacian on smooth functions and 1-forms. These turn out to be self-adjoint on L2, and via the spectral theory one can define the strongly continuous semigroups they generate. The lower bound for the Ricci-curvature is used to get useful estimates for these semigroups. Next, we define subordinated semigroups, the generators of which turn out to be useful in the proof of the boundedness of the Riesz transform. The final tools needed are two estimates, one of which is proved in a probabilistic manner, while the other is purely analytic. These tools are then combined to prove the boundedness of the Riesz transform on functions. In the final section we show that under minor adjustments, one can follow a similar approach in proving the boundedness of the Riesz transform on k-forms. It is this result that is most important for the remainder of the thesis. We then present a general discussion of the theory of sectorial and bisectorial operators. We give the definitions of such operators, and also introduce the concept of R-(bi)sectoriality. We furthermore construct the H1-functional calculus for sectorial and bisectorial operators, which is based on the Dunford functional calculus. We will introduce the concept of a bounded H1-functional calculus and collect some results that we wish to use. Finally, we extend the operators defined by Bakry only for smooth functions and k-forms to Lp for 1 ≤ p < 1 and introduce the Hodge-Dirac operator Π = d + δ on Lp. We then show that the Riesz transform on k-forms is also bounded on Lp for 1 < p < 1. From this, we deduce gradient bounds, which in turn imply the R-bisectoriality of the Hodge-Dirac operator. From the R-bisectoriality we deduce that Π has a bounded H1-functional calculus. We finish our results by showing that this again implies the boundedness of the Riesz-transform. iv Preface This thesis is the result of my work for my graduation for the master Applied Mathematics at TU Delft. The subject of this thesis is based on the seminar 'Stochastic analysis on manifolds' organized by the Analysis department of DIAM. I attended all meetings so far, and also spoke at two of them. It really helped me to grasp the theory I was studying, and I also gained some experience in presenting mathematical material. Additionally, the work done in this thesis is a preparation for the future PhD research that I intend to carry out at the TU Delft. This opportunity is offered to me by the Peter Paul Peterich fund, which presented me with a scholarship for PhD research, for which I am extremely thankful. Although the final results found in this thesis are not directly related to the projected PhD research, the theory studied to obtain the results form a firm basis for the future research. In this way, I was able to lay a good foundation for the coming years, while also broadening my knowledge even further. I want to thank my supervisor Prof. Dr. J.M.A.M. van Neerven for guiding me through the project and teaching me about the many things one should think about when working in the field of (unbounded) operators. I furthermore want to thank Prof. Dr. F.H.J. Redig for quite some hours helping me out filling in details in proofs that I studied. Last, but not least, I want to thank family and friends for the mental support, allowing me to finish this thesis. Now it only remains to say that I hope it is an enjoyable read! Kind regards, Rik Versendaal Contents 1 Introduction 1 1.1 The Riesz transform . 1 1.2 Hodge-Dirac operator . 2 1.3 Relating our work . 3 2 Differential geometry 5 2.1 Manifolds and tangent vectors . 5 2.1.1 Smooth manifolds . 6 2.1.2 Tangent vectors . 7 2.2 Differential k-forms . 8 2.2.1 Differential of a function . 8 2.2.2 Alternating tensors and the wedge product . 9 2.2.3 Differential k-forms . 10 2.3 Riemannian manifolds . 11 2.4 Covariant derivative and curvature . 12 2.4.1 Covariant derivative . 12 2.4.2 Levi-Civita connection and normal coordinates . 14 2.4.3 Curvature . 15 2.5 Volume measure and the Laplace-Beltrami operator . 17 2.5.1 Volume measure . 18 2.5.2 Divergence and the Laplace-Beltrami operator . 18 3 Semigroups of linear operators 21 3.1 Strongly continuous semigroups . 21 3.1.1 Strong equals weak . 25 3.1.2 Analytic semigroups . 25 3.2 Resolvents and the Hille-Yosida theorem . 25 3.3 Markovian semigroups . 26 4 Study of the paper of Bakry 29 4.1 Generalities . 30 4.1.1 Heat semigroup Pt corresponding to L .................... 34 4.1.2 The case for 1-forms . 35 4.2 Subordinated semigroups and harmonic extensions . 41 4.2.1 Subordinated semigroups . 41 4.2.2 Harmonic extensions . 47 4.3 Inequalities of the type of Littlewood-Paley-Stein . 50 4.3.1 Stopping time and martingales . 51 4.3.2 Main estimates . 57 v vi CONTENTS 4.4 Riesz transform on functions . 61 4.5 Riesz transform for k-forms . 68 4.5.1 Boundedness of the Riesz transform on k-forms . 70 4.6 Concluding remarks . 76 5 R-sectoriality and H1-calculus 77 5.1 R-boundedness . 77 5.2 R-sectorial operators . 79 5.2.1 Bisectorial operators . 80 5.3 H1-functional calculus . 80 5.3.1 Extending the functional calculus to f 2 H1 . 82 5.3.2 Some additional results . 83 6 The Hodge-Dirac operator 85 6.1 Extension of d and the Hodge-Dirac operator . 86 6.1.1 The Hodge-Dirac operator . 87 k p k 6.2 Extending the semigroup Pt to L (Λ TM) ..................... 88 6.2.1 R-sectoriality of −Lk .............................. 90 6.3 Boundedness of the Riesz transform . 91 6.3.1 R-gradient bounds . 93 6.3.2 The Hodge-Dirac operator . 94 6.4 Nonzero lower bounds . 98 6.5 Remarks and conjecture . 101 7 Conclusion 103 7.1 Boundedness of the Riesz transform on a complete Riemannian manifold . 103 7.2 Hodge-Dirac operator . 104 7.3 Future considerations . 104 Appendices A 105 A.1 Identities from differential geometry . 105 A.2 Some analytic and algebraic results . 108 Chapter 1 Introduction In this thesis we study the Riesz transform and Hodge-Dirac operator on a complete Riemannian manifold M with Ricci curvature bounded from below. This gives us a natural way to divide the thesis into two parts. In the first part we discuss the boundedness of the Riesz transform acting on so called differential forms. For this, we analyse the paper `Etude´ des transformation de Riesz dans les vari´et´esriemanniennes `acourbure de Ricci minor´ee' by D. Bakry ([6]) in which this is ultimately proved. In the second part we turn to the analysis of the Hodge-Dirac operator on Lp(ΛTM) for 1 < p < 1. We prove various properties such as the R-sectoriality and the fact that it has a bounded H1-functional calculus.