DOCSLIB.ORG

The Riesz Transform on a Complete Riemannian Manifold with Ricci Curvature Bounded from Below

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Riesz Transform on a Complete Riemannian Manifold with Ricci Curvature Bounded from Below 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.
Load more

Recommended publications
  • Applications of Riesz Transforms and Monogenic Wavelet Frames in Imaging and Image Processing
    Applications of Riesz Transforms and Monogenic Wavelet Frames in Imaging and Image Processing By the Faculty of Mathematics and Computer Science of the Technische Universität Bergakademie Freiberg approved Thesis to attain the academic degree of doctor rerum naturalium (Dr. rer. nat.) submitted by Dipl. Math. Martin Reinhardt born on the 1st December, 1987 in Suhl Assessors: Prof. Dr. Swanhild Bernstein Prof. Dr. Konrad Froitzheim Date of the award: Freiberg, 6th March, 2019 Versicherung Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Hilfe eines Promotionsberaters habe ich nicht in Anspruch genommen. Weitere Personen haben von mir keine geldwerten Leistungen für Arbeiten erhalten, die nicht als solche kenntlich gemacht worden sind. Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt. 6. Dezember 2018 Dipl. Math. Martin Reinhardt Declaration I hereby declare that I completed this work without any improper help from a third party and without using any aids other than those cited. All ideas derived directly or indirectly from other sources are identified as such. I did not seek the help of a professional doctorate-consultant. Only those persons identified as having done so received any financial payment from me for any work done for me. This thesis has not previously been published in the same or a similar form in Germany or abroad. 6th March, 2019 Dipl. Math. Martin Reinhardt 5 Contents Preface 7 Introduction 9 1.
    [Show full text]
  • Integral Geometry and Radon Transforms
    TO ARTIE Preface This book deals with a special subject in the wide field of Geometric Anal- ysis. The subject has its origins in results by Funk [1913] and Radon [1917] determining, respectively, a symmetric function on the two-sphere S2 from its great circle integrals and an integrable function on R2 from its straight line integrals. (See References.) The first of these is related to a geometric theorem of Minkowski [1911] (see Ch. III, 1). While the above work of Funk and Radon§ lay dormant for a while, Fritz John revived the subject in important papers during the thirties and found significant applications to differential equations. More recent applications to X-ray technology and tomography have widened interest in the subject. This book originated with lectures given at MIT in the Fall of 1966, based mostly on my papers during 1959–1965 on the Radon transform and its generalizations. The viewpoint of these generalizations is the following. The set of points on S2 and the set of great circles on S2 are both acted on transitively by the group O(3). Similarly, the set of points in R2 and the set P2 of lines in R2 are both homogeneous spaces of the group M(2) of rigid motions of R2. This motivates our general Radon transform definition from [1965a] and [1966a], which forms the framework of Chapter II: Given two homogeneous spaces X = G/K and Ξ = G/H of the same group G, two elements x = gK and ξ = γH are said to be incident (denoted x#ξ) if gK γH = (as subsets of G).
    [Show full text]
  • Decay Properties of Riesz Transforms and Steerable Wavelets∗
    SIAM J. IMAGING SCIENCES c 2013 Society for Industrial and Applied Mathematics Vol. 6, No. 2, pp. 984–998 Decay Properties of Riesz Transforms and Steerable Wavelets∗ † ‡ † John Paul Ward , Kunal Narayan Chaudhury , and Michael Unser Abstract. The Riesz transform is a natural multidimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and, as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay. Key words. Riesz transform, steerable wavelets, singular integrals AMS subject classifications. 32A55, 42C40 DOI. 10.1137/120864143 1. Introduction. The Riesz transform of a real-valued function f(x)onRd is the vector- valued function Rf(x)=(R1f(x),...,Rdf(x)) given by − R x y xi yi y (1.1) if( )=Cd lim f( ) d+1 d , →0 x−y> x − y where Cd is an absolute constant depending on the dimension d [5].
    [Show full text]
  • Merchán, Tomás Ph.D., May 2020 PURE
    Merch´an,Tom´asPh.D., May 2020 PURE MATHEMATICS SINGULAR INTEGRALS AND RECTIFIABILITY (79 pages) Dissertation Advisors: Benjamin Jaye and Fedor Nazarov The goal of this thesis is to shed further light on the connection between two basic properties of singular integral operators, the L2-boundedness and the existence in principal value. We do so by providing geometric conditions that allow us to obtain results which are valid for a wide class of kernels, and sharp for operators of great importance: the Riesz and Huovinen transforms. The content of this thesis has turned out to be essential to prove that, in the case of the Huovinen transform associated to a measure with positive and finite upper density, the existence in principal value implies the rectifiability of the underlying measure. SINGULAR INTEGRALS AND RECTIFIABILITY A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Tom´asMerch´an May 2020 c Copyright All rights reserved Except for previously published materials Dissertation written by Tom´asMerch´an B.S., Universidad de Sevilla, 2015 M.S., Kent State University, 2017 Ph.D., Kent State University, 2020 Approved by Benjamin Jaye , Co-Chair, Doctoral Dissertation Committee Fedor Nazarov , Co-Chair, Doctoral Dissertation Committee Artem Zvavitch , Members, Doctoral Dissertation Committee Feodor Dragan Maxim Dzero Accepted by Andrew M. Tonge , Chair, Department of Mathematical Sciences James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS TABLE OF CONTENTS . iv ACKNOWLEDGMENTS . vi 1 Introduction . 1 1.1 Basic Notation and Concepts . .3 2 Classification of Symmetric measures .
    [Show full text]
  • University of Helsinki Singular Integrals and Beltrami Type
    University of Helsinki Master's thesis Singular Integrals and Beltrami Type Operators in the Plane and Beyond Author: Supervisor: Aleksis Koski Prof. Tadeusz Iwaniec October 4, 2011 Contents 1 Introduction 2 1.1 Acknowledgements . 3 2 Multiplier operators 3 2.1 The operators Tk ......................... 5 2.2 Upper bounds . 7 2.3 Lower bounds . 9 2.4 A curious identity . 13 3 Fredholm theory 16 4 The Beltrami equation 17 4.1 Solving the equations . 24 4.2 Ellipticity . 31 4.3 Higher dimensions . 33 5 Quaternions 38 5.1 Lp-theory of the integral operators . 41 5.2 Spectral results . 47 5.3 Quaternionic Beltrami equations . 51 6 Appendix 54 6.1 Notation and denitions . 54 1 1 Introduction This thesis is intended to complement and extend the authors candidate thesis, The Hilbert transform [Kos10]. This candidate thesis is not neces- sarily a prerequisite, but the reader is expected to be familiar with the basic theory of singular integral operators, as well as with complex and functional analysis up to graduate level. The appendix at the end contains notes on the notation and denitions used throughout the thesis. We recall that the Hilbert transform of a suciently well-behaved func- tion f on R was dened by the principal value integral 1 Z f(x − t) H f(x) = p.v. dt: π R t In [Kos10] we extended the domain of the Hilbert transform to the Lp-spaces for 1 ≤ p < 1, and in fact showed that this operator is bounded in the sense that jjH fjjp ≤ jjH jjpjjfjjp for some constants jjH jjp whenever 1 < p < 1.
    [Show full text]
  • Homothetic Variant of Fractional Sobolev Space with Application to Navier-Stokes System
    Dynamics of PDE, Vol.4, No.3, 227-245, 2007 Homothetic Variant of Fractional Sobolev Space with Application to Navier-Stokes System Jie Xiao Communicated by Y. Charles Li, received October 28, 2006. n Abstract. It is proved that for α 2 (0; 1), Qα(R ), not only as an interme- diate space of W 1;n(Rn) and BMO(Rn) but also as a homothetic variant of 2n _ 2 Rn n−2α Rn Sobolev space Lα( ) which is sharply imbedded in L ( ), is isomor- phic to a quadratic Morrey space under fractional differentiation. At the same n n time, the dot product ∇·(Qα(R )) is applied to derive the well-posedness of the scaling invariant mild solutions of the incompressible Navier-Stokes system R1+n 1 × Rn in + = (0; ) . Contents 1. Introduction and Summary 227 2. Proof of Theorem 1.2 233 3. Proof of Theorem 1.4 239 References 244 1. Introduction and Summary We begin by the square form of John-Nirenberg's BMO space (cf. [13]) which plays an important role in harmonic analysis and applications to partial differen- tial equations. For a locally integrable complex-valued function f defined on the Euclidean space Rn, n 2, with respect to the Lebesgue measure dx, we say that f is of BMO class, denoted≥ f BMO = BMO(Rn), provided 2 1 n 2 2 f = sup `(I) − f(x) f dx < : k kBMO − I 1 I ZI 1991 Mathematics Subject Classification. Primary 35Q30, 42B35; Secondary 46E30. 1;n n n n Key words and phrases. Homothety, W (R ), Qα(R ), BMO(R ), quadratic Morrey space, sharp Sobolev imbedding, incompressible Navier-Stokes system.
    [Show full text]
  • Singular Integrals on Homogeneous Spaces and Some Problems Of
    ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze A. KORÁNYI S. VÁGI Singular integrals on homogeneous spaces and some problems of classical analysis Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 25, no 4 (1971), p. 575-648 <http://www.numdam.org/item?id=ASNSP_1971_3_25_4_575_0> © Scuola Normale Superiore, Pisa, 1971, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SINGULAR INTEGRALS ON HOMOGENEOUS SPACES AND SOME PROBLEMS OF CLASSICAL ANALYSIS A. KORÁNYI and S. VÁGI CONTENTS Introduction. Part I. General theory. 1. Definitions and basic facts. 2. The main result on LP -continuity. 3. An L2-theorem. 4. Preservation of Lipschitz classes. 5. Homogeneous gauges and kernel Part 11. Applications. 6. The Canohy-Szego integral for the generalized halfplane D. 7. The Canchy-Szego integral for the complex unit ball. 8. The functions of Littlewood-Paley and Lusin on D. 9. The Riesz transform on Introduction. In this paper we generalize some classical results of Calderon and Zygmund to the context of homogeneous spaces of locally compact groups and use these results to solve certain problems of classical type which can not be dealt with by the presently existing versions of the theory of sin- gular integrals.
    [Show full text]
  • 2018TOU30188.Pdf
    4 Contents 1 Introduction 11 1.1 Motivation (version française)......................... 11 1.1.1 Contexte historique.......................... 11 1.1.2 Cadre du travail............................ 14 n 1.1.3 La transformée de Riesz sur R .................... 17 n 1.1.4 Représentation analytique de la transformée de Riesz sur R ... 18 n 1.1.5 Représentation probabiliste des transformées de Riesz sur R ... 19 1.1.6 Exemple : l’espace Gaussien...................... 20 1.1.7 Méthodes et résultats......................... 22 1.1.8 Applications et perspectives...................... 26 1.2 Motivation................................... 28 1.2.1 Historical context............................ 28 1.2.2 Context of the work in this thesis................... 30 1.3 Notations.................................... 34 2 Preliminaries 37 2.1 The Hilbert and Riesz transforms....................... 37 2.1.1 The Hilbert transform on T ...................... 37 2.1.2 The Hilbert transform on R ...................... 38 2.1.3 Properties................................ 38 n 2.1.4 The Riesz transform on R ...................... 40 2.2 Semigroups................................... 41 2.3 Analytical representation of the Riesz transform.............. 44 2.4 Stochastic calculus............................... 45 2.4.1 Bases of stochastic calculus...................... 45 2.4.2 Feynman-Kac formula......................... 50 2.5 Probabilistic representation of the Riesz transform............. 51 2.6 Example: the Gaussian space......................... 52 2.7 Riemannian
    [Show full text]
  • On the Theory of Harmonic Functions of Several Variables
    ON THE THEORY OF HARMONIC FUNCTIONS OF SEVERAL VARIABLES I. The theory of HP-spaces BY ELIAS M. STEIN and GUIDO WEISS University of Chicago and De Paul University 1. Introduction This paper is the first of series concerned with certain aspects of the theory of harmonic functions of several variables. Our particular interest will be to extend to n variables some of the deeper properties known to hold in the case of two variables. The study of the more fundamental properties of harmonic functions of two real variables is'linked, by the notion of the conjugate harmonic function, to the study of analytic functions of one complex variable. Therefore, the investigation of the deeper properties of harmonic functions of several variables appears, at first sight, to be connected with either the theory of analytic functions of several complex vari- ables or with an appropriate extension of the notion of conjugate harmonic function. The theory of analytic functions of several complex variables, though widely studied, does not seem to have direct applications to the theory of harmonic functions of several real variables. On the other hand, there are known notions of "conjugacy" of harmonic functions which seem to us to be both more natural and more fruitful for the development of the latter theory. It is these notions that form the starting point for our investigation. We begin by sketching their background. Let us first consider a function u=u(re i~ which is harmonic in the interior of the unit circle 0~r<l. Suppose that, for 0~<r<l and ~0>~1, 2rt f lu(r~o) l~ dO<A < ~.
    [Show full text]
  • Harmonic Analysis and Its Applications
    Harmonic Analysis and Its Applications In these lectures, we concentrate on the motivations, development and applications of the Calderon-Zygmund operator theory. Lecture 1. The differential operators with constant coefficients and the first generation of Calderon-Zygmund operators Consider the following differential operator with constant coefficients: X @®u Lu(x) = a : (1:1) ® @x® ® By taking the Fourier transform, X d ® (Lu)(») = a®(¡2¼i») uˆ(»): (1:2) ® This suggests one to consider the following more general Fourier multiplier: Definition 1.3: An operator T is said to be the Fourier multiplier if (dT f)(») = m(»)fˆ(»): (1:4) (1.2) shows that any classical differential operator is a Fourier multiplier. 2 2 Example 1: Suppose f 2 L (R) and F is an analytic extension of f on R+ given by Z Z i 1 i (x ¡ t) ¡ iy F (x + iy) = ¡ f(t)dt = ¡ f(t)dt ¼ x + iy ¡ t ¼ (x ¡ t)2 + y2 Z Z 1 y i (x ¡ t) = f(t)dt ¡ f(t)dt: (1:5) ¼ (x ¡ t)2 + y2 ¼ (x ¡ t)2 + y2 R Letting y ! 0; then 1 y f(t)dt ! f(x) for a. e. x, and, in general, the second ¼ (x¡t)2+y2 R 1 term above has no limit. However, one can show p.v x¡t f(t)dt exists for a. e. x. Thus limy!0F (x + iy) = f(x) + iH(f)(x) (1:6) where H is called the Hilbert transform defined by Z 1 f(t) H(f)(x) = ¡ dt: (1:7) ¼ x ¡ t 1 Example 2: Consider the Laplacian Xn @2u 4u = : (1:8) @x2 j=1 j By taking the Fourier transform, Xn d 2 2 (¡ 4 u)(») = (4¼j»jj) uˆ(») = 4¼j»j uˆ(»): j=1 Define the Riesz transforms Rj; 1 · j · n; by » (Rdf)(») = j fˆ(»): (1:9) j j»j Then d2 @ u d ( )(») = ¡4¼»i»juˆ(») = (RiRj 4 u)(»): (1:10) @xi@xj
    [Show full text]
  • A Renormalized Riesz Potential and Applications
    A renormalized Riesz potential and applications Mihai Putinar Abstract. The convolution in Rn with |x|−n is a very singular operator. Endowed with a proper normalization, and regarded as a limit of Riesz potentials, it is equal to Dirac’s distribution δ. However, a different normalization turns the non-linear operator: −2 −n Ef = exp( |x| ∗ f), |Sn−1| into a remarkable transformation. Its long history (in one dimension) and some of its recent applications in higher dimensions make the subject of this exposition. A classical extremal problem studied by A. A. Markov is related to the operation E in one real variable. Later, the theory of the spectral shift of self-adjoint perturbations was also based on E. In the case of two real variables, the transform E has ap- peared in operator theory, as a determinantal-characteristic function of certain close to normal operators. The interpretation of E in com- plex coordinates reveals a rich structure, specific only to the plane setting. By exploiting an inverse spectral function problem for hy- ponormal operators, applications of this exponential transform to im- age reconstruction, potential theory and fluid mechanics have recently been discovered. In any number of dimensions, the transformation E, applied to characteristic functions of domains Ω, can be regarded as the geometric mean of the distance function to the boundary. This interpretation has a series of unexpected geometric and analytic con- sequences. For instance, for a convex algebraic Ω, it turns out that the operation E is instrumental in converting finite external data (such as field measurements or tomographic pictures) into an equation of the boundary.
    [Show full text]
  • Riesz Transforms on a Sphere
    i ii Abstract Motivated by the Navier-Stokes equations, which are a set of unsolved equations related to fluid motion in R, we explored the incompressibility condition and the Neumann boundary problem. After exploring, we noticed that using iterated Riesz transforms of the boundary data could be used to get information about the velocity field directly from the boundary data. We then used this same logic to look at the same problem in a spherical space, and found no definition for the Riesz transforms. By using the incompressibility condition and solving the Neumann boundary problem on a sphere, using both separation of variables and potential theory, we can define the Riesz transforms of a function on the sphere. This allows us to use boundary data on a sphere to describe the divergence-free field, or in the case of the incompressible Navier-Stokes equation, allows us to solve for the velocity field given initial data. 1/29 1 Introduction 1.1 Motivation and Objective One of the great, unanswered questions in physics and mathematics is related to solving the incompressible Navier-Stokes equations in R3. The Clay Mathematics Institute has put up a one-million dollar prize for a correct, verified solution [5]. As a Millennium Prize Problem, it is an important problem that is currently being used to design a number of practical things by using "approximations" that idealize the systems they're modeling. This includes many sorts of fluid dynamics problems such as designing boat hulls, airplanes, aerodynamic cars, and other systems. The motivation to define the Riesz transforms on the sphere comes from the motion of fluids, which are described by the 3-dimensional incompressible Navier-Stokes equations.
    [Show full text]