Clifford Analysis and Its Applications NATO Science Series ASeriespresentingtheresultsofscientificmeetingssupportedundertheNATOScience Programme. TheSeriesispublishedbyIDSPress,Amsterdam,andKluwerAcademicPublishersinconjunction withtheNATOScientificAffairsDivision Sub-Series I. LifeandBehaviouralSciences IDSPress II. Mathematics,PhysicsandChemistry KluwerAcademicPublishers III.ComputerandSystemsScience IDSPress IV.EarthandEnvironmentalSciences KluwerAcademicPublishers TheNATOScienceSeriescontinuestheseriesofbookspublishedformerlyastheNATOASISeries. 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AsaconsequenceoftherestructuringoftheNATOScienceProgrammein1999,theNATOScience Serieswasre-organizedtothefoursub-seriesnotedabove.Pleaseconsultthefollowingwebsitesfor informationonpreviousvolumespublishedintheSeries. http://www.nato.int/science http://www.wkap.nl http://www.iospress.nl http://www.wtv-books.de/nato-pco.htm SeriesII:Mathematics,PhysicsandChemistry- Vol.25 Clifford Analysis and Its Appl ications edited by F. Brackx Department of Mathematical Analysis, Ghent University, Ghent, Belgium J.S.R. Chisholm Institute of Mathematics and Statistics, University of Kent, Canterbury, United Kingdom and v. Soucek Mathematical Institute, Charles University, Prague, Czech Republic Springer-Science+Business Media, B.V. Proceedings of the NATO Advanced Research Workshop on Clifford Analysis and Its Applications Prague, Czech Republic October 30-November 3, 2000 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-0-7923-7045-1 ISBN 978-94-010-0862-4 (eBook) DOI 10.1007/978-94-010-0862-4 Printed an acid-free paper AII Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1s t edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Table of Contents Preface ix 8 Bernstein, Riemann-Hilbert Problems in Clifford Analysis 1 F Brackx & F 8ommen, The Continuous Wavelet Transform in Clifford Analysis 9 T Branson, Automated Symbolic Computation in Spin Geometry 27 J Bures, Monogenic Forms ofthe Polynomial Type 39 P Cerejeiras & U Kahler, On Beltrami Equations in Clifford Analysis and Its Quasi-Confor mal Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 J8R Chisholm, Parallel transport of Algebraic Spinors on Clifford Manifolds . . 59 8-L Eriksson-Bique, A Correspondence of Hyperholomorphic and Monogenic Functions in JR4 ..•..•..•....•.•..••.••.•..•..••.•...•••••••• 71 K Giirlebeck, On Weighted Spaces ofMonogenic Quatemion-valued Functions81 B Jancewicz, PlanewavesinPremetricElectrodynamics 91 L Kadlcakova, Contact Symplectic Geometry in Parabolic Invariant Theory and Symplectic Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 v vi G Kaiser, Communication via Holomorphic Green Functions 113 G Khimshiashvili, Hyper-holomorphic Cells and Riemann-Hilbert Problems . .. , 123 v VKisiI, Nilpotent Lie Groups in Clifford Analysis and Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 V Kravchenko, V Kravchenko &B Williams, A Quatemionic Generalization of the Riccati Differential Equation 143 L Krump, Invariant Operators for Quatemionic Structures. . . . . . . . . . 155 A KwaSniewski, On Generalized Clifford Algebras - a Survey ofApplications . 163 V Labunets, E Labunets-Rundblad &J Astola, Is the Visual Cortex a "Clifford Algebra Quantum Computer"?173 L Lanzani, The Cfn-Valued Robin Boundary Value Problem on Lipschitz Do- mains in JR7l ..•.....•..••.••..•.....•..•.....••... 183 H Leutwiler, Quatemionic Analysis in JR3 versus Its Hyperbolic Modification193 H R Malonek, Contributions to a Geometric Function Theory in Higher Dimen sions by Clifford Analysis Methods: Monogenic Functions and M-con- formal Mappings 213 N Marchuk The Dirac Type Tensor Equation in Riemannian Space. . . . . 223 A Axelsson, R Grognard, J Hogan & A McIntosh, Harmonic Analysis ofDirac Operators on Lipschitz Domains 231 vii V Kokilashvili & A Meskhi, Weight Problemsfor HigherDimensionalSingularIntegrals via Clif- ford Analysis 247 H Liu &J Ryan, The Conformal Laplacian on Spheres and Hyperbolas via Clifford Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 I Sabadini & Fe Sommen, Combinatorics and Clifford Analysis 267 H Schaeben, W Spr6f3ig & G van den Boogaart, The Spherical X-Ray Transform of Texture Goniometry .... 283 P Somberg, Quatemionic Complexes in Clifford Analysis 293 F Sommen, Clifford Analysis on the Level ofAbstract Vector Variables .. 303 V Soucek, Clifford Analysis as a Study ofInvariant Operators 323 W Spr66ig, Teodorescu Type Transforms in Applications 341 I Sabadini& F Sommen, CombinatoricsandCliffordAnalysis 361 A Trautman, Double Covers ofPseudo-orthogonal Groups 377 P Van Lancker, Higher Spin Fields on Smooth Domains 389 N Vasilevski, Bergman Type Spaces on the Unit Disk 409 List ofParticipants 416 IX Preface Clifford Analysis offers a function theory for the solutions of the Dirac equation for spinor fields and is as such a direct generalization to higher dimensions of the classical function theory of holomorphic functions in the complex plane. The functions considered take their values in a Clifford algebra, a graded non-commutative and associative algebrain whicharbitrarydimensionalrealorcomplexEuclideanspace may beembedded ina natural way. Not onlyfrom the theoretical point ofview but also with respect toapplications, in particularto theoretical physics, Clifford Analysis has proven to be a valuable counterpart to the theory ofseveral complex variables. After a few preliminary attempts in the 1930's and the 1940's, Clifford analysis really took off in the 1960's through the effort of three research groups in Bucarest (Romania), Ghent (Belgium) and Arizona (USA). Meanwhile it has developed into an autonomous disci pline in mathematical analysis with research groups all over the world. Moreover Clifford Algebra and Clifford Analysis turned out to play an important role in applications to theoretical physics. A series of conferences to stimulating the interaction between Clifford algebras, Clifford analysis and mathematical physics, was launched at Canter bury in 1985, JSR Chisholm organizing a NATO ASI-ARW, "Clifford Algebras and Their Applications to Mathematical Physics". Fifteenyearsand manyconferences later we have had the pleasureof organizing in Praha (Czech Republic) the NATO Advanced Research. Workshop "Clifford Analysis and Its Applications". The present vol ume is devoted to this conference and by presenting a selection of the talks delivered, it aims at bringing a state-of-the-art view ofthe actual research in Clifford Analysis and its applications. The field of Clifford analysis is indeed going through a period of quick evolution in directions belonging to the main classical stream of research in the field as well as to applications in various other parts of mathematics and physics. All traditional and new fields are very well represented in this volume, and a lot of papers contribute in a substantialway toaclarificationofnewimportantdirectionsfor further research. x Let us now gofor aquickguidedtour throughthe book. Theclassical topics of the main stream and their recent evolution are described in lectures on the use of the Rellich inequalities in a study of boundary operators (A Axelsson, R Grognard, J Hoganand AMcIntosh) singular integraloperators (V Kokilashviliand A Meskhi), Bergman and Hardy spaces (N Vasilevski), the Robin boundary value problem (L Lanzani), various alternatives for monogenicity equations in dimension eight and basic properties oftheir solutions (I Sabadini and D Struppa), double covers ofpseudo-orthogonal groups (A Trautman), Clifford analysis on spheres and hyperbolic spaces (J Ryan), a geometric characterization of monogenic functions (H Malonek), the Beltrami equation and its relation to locally quasiconformal maps (P Cerejeiras and U Kahler), quaternionic generalization of the Riccati equation (V Kravchenko, V Kravchenko, B Williams), a study ofthe Riemann-Hilbert problem in the setting of Clifford analysis (S Bernstein) and wavelet theory (F Brackx and F Sommen). For several years the solutions of the Hodge operator on hyperbolic spaces have been studied by H Leutwiler and his coworkers (papers by H Leutwiler and SL Eriksson-Bique). Duringthe recent decade there has been a growing interest in a deeper study of Clifford analysis on Riemannian manifolds (or on manifolds with a given conformal structure) and of the relations to the broad and interesting field of invariant differential operators on manifolds with a given geometrical structure. These topics are discussed in the contributionsbyT Bransonand VSoucek; thecontributionsofJ Bures, LKrumpand P Sombergtreat closelyrelatedsubjects. Theimportance of these topics lies in the fact that they introduce very powerful geo metrical methods in Clifford analysis. The theory of several Clifford variables has grown a lot already. In Clifford analysis, the analogues ofthe Dolbeault complex are complexes ofdifferential operators; they have been studied for many years by D Struppa and his coworkers. The paper by I Sabadini and F Sommen reports on new results in this directionand on their relation to finite geometries and Platonic bodies. Moreover a geometrical and representational theoretical interpretation of these complexes and their relations to the Baston complexes are given in the papers of L Krump, P Somberg and V Soucek. General izations of holomorphic.cells were studied for some time in symplectic geometry, similar objects have appeared under the name of D-branes in topological field theory and string theory. A clifford version ofsuch objects is studied in the paper by G Khimshiashvili. In recent years Clifford analysis has broken out of its classical bounds and several very interesting generalizations have emerged. The first one is still inside the framework of invariant operators on conformal manifolds and their solutions. Classical Clifford analysis is devoted to xi thestudyofpropertiesofthe (Euclidean) Diracoperator. Thereexistsa familyoffirst orderinvariantdifferentialoperators for fields with values in more complicated spinor representations, the first one in the series being the Rarita-Schwinger operator. The properties oftheir solutions form a new topic in the field havingvery close interactionwith boththe geometrical methods in Clifford analysisand the theory oftwo Clifford variables. In such questions the role of the representation theory of simple Lie groups is becoming more and more important; it forms a new very important tool in the subject. These topics are treated in the contributionsbyP Van Lancker and J Bures. Thesecondgeneralization is based on a different underlying geometry. It is a supersymmetric analogue ofclassical Clifford analysis based on the symplectic Clifford algebra. The corresponding analogue of the Dirac operator acts on fields with values in the infinite-dimensional analogue of the spinor representation introduced by B Kostant. The first properties of its solutions are formulated by F Sommen in his paper on the abstract variable approach, and its relation to the field of parabolic geometries is described in contributions by L KadlCakova and V Soucek. The third direction concerns the role of nilpotent Lie groups in possible generalizations ofClifford analysis (V Kisil). The applications ofCliffordanalysis to other fields ofmathematics and physics form a traditional part of Clifford analysis. Applications to hyperbolic systems of equations are treated by G Kaiser, relations to integral geometry (in the sense ofI M Gelfand) are given in the paper by H Schaeben, W Sprofiig and G van den Boogaart. Applications to boundary value problems in PDE's and their numerical description are further developed by W Sprofiig and K Giirlebeck. Generaliza tions of Clifford algebras and their applications are described by A Kwasniewski. Applicationsto imagerecognitionand thecomputational complexityofthe correspondingalgorithms are treated by VLabunets, E Labunets-Rundblad and J Astola. A large field of applications to problems in theoretical physics is reported on in contributions by JSR Chisholm, B Jancewicz and N Marchuk. The NATO Advanced Research Workshop 976052 Clifford Analysis and Its Applicationswasorganized with thefinancialsupportofNATO, Charles University Prague and Ghent University. The organizers seize theopportunitytothankheartilythesethreeInstitutionsandespecially theNATO Scientificand EnvironmentalAffairsDivisionandthe NATO Science Committee for their generous support. Last but not least let us mention that the whole ARW with its scientificcontent and thepresenceandcontributionsofso manyfriends,