Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2012 Dirac Operators, Multipliers and H[superscript P] Spaces of Monogenic Functions Guanghou Wang Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES DIRAC OPERATORS, MULTIPLIERS AND HP SPACES OF MONOGENIC FUNCTIONS By GUANGHONG WANG A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2012 Guanghong Wang defended this dissertation on Apr 25, 2012. The members of the supervisory committee were: Craig Nolder Professor Directing Dissertation Lois Hawkes University Representative Bettye Case Committee Member Eriko Hironaka Committee Member Jack Quine Committee Member Mika Sepp¨al¨a Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with the university requirements. ii ACKNOWLEDGMENTS First of all, I want to express my deep gratitude to Dr. Nolder. I want to thank him for offering me the possibility to do a Ph.D at FSU, also for introducing me in the fasci- nating world of mathematics research. It has being a pleasant experience to study under his guidance during the last four years. All the wisdom he shared with me and the nice conversations we had will always be cherished by me. Without his help, this dissertation would not be possible, my Ph.D would not be possible. It is not easy to get a Ph.D degree in US. Besides the research work, I have to deal with a lot of other issues, all of which are solved with the help of a lot of people from our department. I want to thank Dr. Kirby for helping with my teaching since the very beginning. I started my TA training classes with her back in Aug 2006, and took her solo teaching workshops about four years ago, and her help is always available when I need it. Thanks to her, I have become more and more confident as a teacher. I also want to express my gratitude to Dr. Case, not only because of her suggestions at the beginning of each semester, but also for her faith in me, which I will never forget. Also I want to thank my parents for their constant support and love. The last six years has not been easy for me, and their encouragement and faith in me always make me overcome the difficulties. Finally, my special thanks goes to my wife, Qiong, who stayed with me during the last two years. Her love and support has been my source of strength and inspiration to get my Ph.D. iii TABLE OF CONTENTS ListofSymbols....................................... v Abstract........................................... vii 1 INTRODUCTION 1 2 BASIC FACTS RELATED TO CLIFFORD ANALYSIS 4 2.1 Classes of functions and function spaces . 4 2.2 Clifford algebras . 6 2.3 Grassmann algebras . 7 2.4 Quaternions.................................... 8 2.5 Some important results . 9 2.6 Some operator theory . 13 2.7 Spherical Monogenic functions . 15 3 A-DIRAC EQUATIONS 17 3.1 Introduction . 17 3.2 Preliminaries . 17 3.3 One Caccioppoli estimate . 19 4 MULTIPLIER THEORY 22 4.1 Introduction . 22 4.2 Some properties of operator Π . 23 4.3 Some properties of generalized operator φ,ψΠ . 26 4.4 Iteration of Dirac and ∆ operators . 27 5 MONOGENIC HARDY SPACES 29 5.1 Monogenic Hp-spaces . 29 5.2 Carleson Measures . 31 5.3 A Clifford Riesz Representation Theorem . 33 Bibliography . 37 Biographical Sketch . 43 iv LIST OF SYMBOLS The following list of symbols are used in this dissertation, and they are ordered by their sequence of appearance. Rn+1 Euclidean space with dimension n+1 · Dot product (α0,α1, ···,αn) An n+1-tuple multi-index m Cα (Ω) The functional space with functions’ derivative up to order m are in H(f,α) C Some constant number p L (Ω) The functional space with ||f||p ≤ C W m, p(Ω) Sobolev spaces m, p W0 (Ω) Sobolev spaces with compact supports (V,Q) A quadratic space p, q (R , Qp, q) A real Minkowski space Cℓn A universal Clifford algebra (k) Cℓn The subspace of Cℓn with elements of length k u′ Principal automorphism of u u∗ Principal anti-automorphism of u u Conjugation of u Λ(V ) A Grassmann algebra ∧ Wedge product sgn(σ) The signature of the permutation σ Λk(V ) The vector subspace of Λ(V ) with elements of length k H Quaternions D Dirac operator D Generalized Cauchy Riemann operator M(Ω,Cℓn) The functional space of monogenic functions over domain Ω ∆ Laplacian operator e(x) Cauchy Kernel (FΓu)(x) One Cauchy type operator (SΓu)(x) One singular Cauchy type operator Π Π operator defined by DT v + M (k,Cℓn) The homogeneous monogenic polynomials of degree k − M (k,Cℓn) The homogeneous monogenic polynomials of degree -(k + n) I The inversion operator divA(x, ∇u) = 0 A-harmonic type elliptic equations DA(x, Du) = 0 A-Dirac equations m(ξ) The multiplier of operator T if Tf(ξ)= m(ξ)f(ξ) Tm The m−fold convolution of operator T c b n R Clifford Riesz transform defined by R = R0 + Σj=1Rjej φD Generalized operator of D φT Generalized operator of T φ,ψΠ Generalized operator of Π defined by φDψT φ φ Tk The iteration of operator T by k times ML Left Clifford valued monogenic space MR Right Clifford valued monogenic space p p p H Monogenic Hardy spaces, the union of HL and HR f ⋆ The nontangential maximal function of f vi ABSTRACT I have done a few things under Clifford algebra settings. Firstly, one Caccioppoli type estimate is derived for solutions of A-Dirac equations in the form DA(x, Du) = 0, where D is the Dirac operator. This kind of A-Dirac equations are generalizations of elliptic equations of A-harmonic type, i.e. divA(x, ∇u) = 0. Secondly, the multiplier theory from Fourier analysis is generalized to Clifford analysis. After the multipliers of the operators D, T and Π are identified, some related properties will be very easy to achieve, including two integral representation theorems. The iterations of the operators D and ∆ are also discussed. Thirdly, We achieve one Carleson measure theorem for monogenic Hardy spaces on the unit ball in Rn+1, as well as one Clifford Riesz representation theorem. vii CHAPTER 1 INTRODUCTION The whole story started around 1880’s, when William Clifford first introduced the algebras that are named after him. Basically speaking, these algebras are generalizations of com- plex numbers and quaternions. The idea is embedding the Euclidean space Rn into some algebras, where all the vectors in the Euclidean space satisfies x2 = −||x||2, and it turns out that the minimal algebras satisfying the condition are universal Clifford algebras. For example, if n = 0, real numbers will be given; n = 1, complex numbers; n = 2, quaternions and so on. Because Clifford was a student of Maxwell’s, he tried to put Maxwell’s work on elec- tromagnetism in a more mathematical way. At the same time, some other physicists tried to understand geometric calculus better, they all came to Clifford algebras. We can say Clifford algebras are well connected with geometry and physics from the very beginning [2], [14], [18], [51], [62]. There has been lots of work called Clifford analysis done over Clifford algebras during the last century. One pioneer is Alfred Cardew Dixon, who worked at Queen university, Belfast, from 1901 to 1930. Another one is C. Lanczos, who did some quaternionic analysis in his Ph.D dissertation. Later, Rudolph Fueter wrote over 10 papers on quaternionic analysis around 1940s, for example, [15], [16], [21], [22]. A good summary of all the related materials can be found in [89] by A. Sudbery. About the same time as Fueter worked on this subject, Moisil and Theodorescu were also working on the same subject independently [51]. Around 1970s, Richard Delanghe and some other mathematicians tried to generalize the one variable complex analysis theories to higher Euclidean space under the Clifford algebra settings, and Kelvin transformation is one of them, which can be used to derive Cauchy’s integral formula, Laurent and Taylor series and so on [7], [8],[11], [16], [24],[40], [41], [42], [60], [61], [62]. For another example, Iftimie gave the Cauchy transform and proved Plemelj formulas for H¨older continuous functions defined on compact Liapunov surfaces in Rn. Furthermore, he also showed that the square of this Cauchy transform is the identity map under certain conditions, which is just like the case of the complex analysis in the plane. Not to mention lots of boundary value problems are generated by Cauchy transform and Plemelj formulas under Clifford algebra settings, [9], [10], [11], [25], [41], [50], [52]. The first book about Clifford analysis appeared in 1982 and is written by Brackx, De- langhe, and Sommen, [11]. It talks about many aspects of Clifford algebras, for example, 1 some general theories, monogenic functions and boundary value problems. More specifi- cally, F. Sommen proved that Cauchy-Kowalweski holds in the Clifford case, i.e. monogenic functions can be extended by real analytic functions from the dimension less by one [82], [83]. Later Fourier analysis and complex analysis are also involved in the development of Clifford analysis. Clifford analysis has been one of the fastest growing areas during the last few decades and is a well established mathematics research area now.