Clifford Algebras in Analysis and Related Topics (Studies in Advanced Mathematics)

This new book contains the most up-to-date and focused description of the applications of Clifford algebras in analysis, particularly classical harmonic analysis. It is the first single volume devoted to applications of Clifford analysis to other aspects of analysis. All chapters are written by world authorities in the area. Of particular interest is the contribution of Professor Alan McIntosh. He gives a detailed account of the links between Clifford algebras, monogenic and harmonic functions and the correspondence between monogenic functions and holomorphic functions of several complex variables under Fourier transforms. He describes the correspondence between algebras of singular integrals on Lipschitz surfaces and functional calculi of Dirac operators on these surfaces. He also discusses links with boundary value problems over Lipschitz domains. Other specific topics include Hardy spaces and compensated compactness in Euclidean space; applications to acoustic scattering and Galerkin estimates; scattering theory for orthogonal wavelets; applications of the conformal group and Vahalen matrices; Newmann type problems for the Dirac operator; plus much, much more! Clifford Algebras in Analysis and Related Topics also contains the most comprehensive section on open problems available. The book presents the most detailed link between Clifford analysis and classical harmonic analysis. It is a refreshing break from the many expensive and lengthy volumes currently found on the subject. Cover Half Title Title Page Copyright Page Table of Contents 1: Acknowledgment 2: Conference Participants 3: Introduction 4: Problem Book 5: Clifford Algebras, Fourier Theory, Singular Integrals, and Harmonic Functions on Lipschitz Domains 5.1 Introduction 5.2 Lecture 1 5.2.1 (A) Spectral theory of . . .m 5.2.2 (B) Spectral theory of . . .m 5.3 Lecture 2 5.3.1 (C) Spectral theory of commuting matrice 5.3.2 (D) Spectral theory of the Dirac operator D 5.4 Lecture 3 5.4.1 (E) Monogenic functions 5.4.2 (F) Singular convolution integrals on Lipschitz surfaces 5.5 Lecture 4 5.5.1 (G) Fourier transforms 5.5.2 (H) Correspondence between monogenic functions and holomorphic functions of several complex variables under Fourier transforms 5.6 Lecture 5 5.6.1 (J) Algebras of singular integrals on Lipschitz surfaces, and functional calculi of Dirac operators on these surfaces 5.6.2 (K) Boundary value problems for harmonic functions 5.7 Additional Material 5.7.1 (L) Singular integrals on the boundary of a strongly Lipschitz domain 5.7.2 (M) More about singular convolution integrals on Lipschitz surfaces 6: Clifford Algebras and H. Functional Calculi of Commuting Operators 6.1 Introduction 6.2 The H. functional calculus of a single operator 6.3 Fourier transform between holomorphic functions and monogenic functions 6.4 H. functional calculi of .-tuples of commuting operators 7: Hypercomplex Variable Techniques in Harmonic Analysis 7.1 Introduction 7.2 Clifford Algebra Rudiments 7.3 Elements of Clifford Analysis 7.4 Non-Homogeneous Dirac Operators 7.5 Clifford Algebra-Valued Singular Integral Operators 7.6 Hardy Spaces on Lipschitz Domains 7.7 Rellich Type Formulas for Monogenic Functions 7.8 A Burkholder-Gundy-Silverstein Theorem for Monogenic Functions in Lipschitz Domains 7.9 Some Applications to the Theory of Harmonic Functions in Lipschitz Domains 8: Some Applications of Conformal Covariance in Clifford Analysis 8.1 Introduction 8.2 Preliminaries 8.3 Conformal Invariance of Cells of Harmonicity 8.4 V(n) and Real Clifford Analysis 8.5 V (Cn) and Complex Clifford Analysis 8.6 The Bergman Kernel and Harmonic Measure 8.7 More on Conformal Covariance in Complex Clifford Analysis 9: Singular Integrals with Monogenic Kernels on the m-Torus and their Lipschitz Perturbations 9.1 Introduction 9.2 Preliminary 9.3 Fourier Transforms 9.4 Singular Integral Operators and Fourier Multiplier Operators on Periodic Lipschitz Surfaces 10: Scattering Theory for Orthogonal Wavelets 10.1 Introduction 10.2 Preliminaries 10.3 Scattering Theory 10.4 Wavelets 10.5 Multiresolutions 10.6 The Integral Translation Group 10.7 The Besicovich Spaces 11: Acoustic Scattering, Galerkin Estimates and Clifford Algebras 11.1 Introduction 11.2 Definitions and Preliminary Results 11.3 Inverting Quaternionic-Valued Boundary Operators 11.4 Finite Element Spaces 11.5 Local Galerkin Estimates 12: Clifford Algebras, Hardy Spaces, and Compensated Compactness 12.1 Introducstion 12.2 Preliminaries 12.3 Components of the product of two monogenic functions 12.4 Proof of the main theorem 12.5 Applications in compensated compactness 13: Frame Decompositions of Form-Valued Hardy Spaces 13.1 In troduction 13.2 Dilatation Frames 13.3 The deRham complex and Hardy spaces of forms 13.4 Projections on Hp.r 13.5 Lusin’s representation for Hp.r 13.6 Discrete frames: the case of one-forms 13.7 Concluding remarks and problems 14: Applications o f Clifford Analysis to Inverse Scattering for the Linear Hierarchy in Several Space Dimensions 14.1 Introduction 14.2 Review of Some Clifford Analysis 14.3 Relation between Complex and Clifford Analysis in Dimension One 14.4 Review of the Linear Hierarchy in One Space Dimension 14.5 The Linear Hierarchy in Several Space Dimensions 15: On Riemann-Hilbert Problems for Nonhomogeneous Dirac Equations in a Half Space of Rm(m . 2) 15.1 Introduction 15.2 Preliminaries 15.3 Riemann-Hilbert Problems for Nonhomogeneous Dirac Equations 16: Regularity and Approximation Results for the Maxwell Problem on C1 and Lipschitz Domains 16.1 Introduction 16.2 The vector Helmholtz equation 16.3 The singular integral operator M 16.4 Sobolev-Besov space regularity results 16.5 Some approximation results 17: Continuity o f Calderón-Zygmund Type Operators on the Predual of a Morrey Space 17.1 Introduction 17.2 Definition of the Space Lp. 17.3 The Space Lp,. as the Dual of the Atomic Space Hp,. 17.4 A Notion of a Molecule in the Space Hp,. 17.5 Continuity on the Spaces Hp,. and Lp,. of Calderon-Zygmund Type Operators 18: Neumann Type Problems for the Dirac Operator 18.1 Introduction 18.2 Neumann Problems for Monogenic Functions 18.3 Neumann Problems for Nonhomogeneous Dirac Equations 19: The Hyperholomorphic Bergman Projector and its Properties 19.1 Introduction 19.2 Preliminaries 19.3 The T-operator and the singular integral operator over a domain 19.4 Bergman hyperholomorphic spaces and Bergman hyperholo-morphic projector 19.5 Some algebras generated by the Bergman projector 20: Multivector Solutions to the Hyperholomorphic Massive Dirac Equation 20.1 Introduction 20.2 Algebraic Notatio 20.2.1 Classical Galilean Space and The Pauli Algebra 20.2.2 Minkowski Spacetime and Majorana Algebra 20.2.3 Automorphisms and Conservation Laws 20.3 Functional Solutions of the Massive Dirac Equation 20.3.1 Relativistic Quantum Wave Equations 20.3.2 Meta-Monogenic Functions 20.3.3 Multivectorial Hilbert Space 20.4 Integral Meta-Monogenic Solutions 20.4.1 The Propagating Kernel 20.4.2 Green Function 20.4.3 Path Integral Formulation ( PIF) 20.5 Summary 21 Möbius Transformations, Vahlen Matrices, and their Factorization 21.1 Similarities and transversions 21.2 The counter-example of J. Maks 21.3 Recent mistakes in factorization 21.4 How to factor Vahlen matrices?

This new book contains the most up-to-date and focused description of the applications of Clifford algebras in analysis, particularly classical harmonic analysis. It is the first single volume devoted to applications of Clifford analysis to other aspects of analysis.
All chapters are written by world authorities in the area. Of particular interest is the contribution of Professor Alan McIntosh. He gives a detailed account of the links between Clifford algebras, monogenic and harmonic functions and the correspondence between monogenic functions and holomorphic functions of several complex variables under Fourier transforms. He describes the correspondence between algebras of singular integrals on Lipschitz surfaces and functional calculi of Dirac operators on these surfaces. He also discusses links with boundary value problems over Lipschitz domains.
Other specific topics include Hardy spaces and compensated compactness in Euclidean space; applications to acoustic scattering and Galerkin estimates; scattering theory for orthogonal wavelets; applications of the conformal group and Vahalen matrices; Newmann type problems for the Dirac operator; plus much, much more!
Clifford Algebras in Analysis and Related Topics also contains the most comprehensive section on open problems available. The book presents the most detailed link between Clifford analysis and classical harmonic analysis. It is a refreshing break from the many expensive and lengthy volumes currently found on the subject.

Booknews

Based on a conference held in Fayetteville, Arkansas, April 1993, this proceedings volume contain descriptions of the applications of Clifford algebras in analysis, primarily the impact of Clifford analysis on harmonic analysis. Of particular interest is the paper based on the lectures of principal speaker Alan McIntosh, of Macquarie U., Australia, which provides a detailed account of the links between Clifford algebras, monogenic and harmonic functions, and the correspondence between monogenic functions and holomorphic functions of several complex variables under Fourier transforms. No index. Annotation c. Book News, Inc., Portland, OR (booknews.com)