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.
Table of Contents
Introduction, J. Ryan Section on Open Problems Clifford Algebras, Fourier Theory, Singular Integrals, and Harmonic Analysis on Lipschitz Domains, A. McIntosh Clifford Algebras and Ho Functional Calculus of n-Tuple Commuting Operators, C. Li and A. McIntosh Hypercomplex Variables Techniques in Harmonic Analysis, M. Mitrea Applications of Conformal Covariance in Clifford Analysis, J. Ryan Singular Integrals with Monogenic Kernels on the n-Torus and its Lipschitz Perturbations, T. Qian Scattering Theory for Orthogonal Wavelets, P.E.T. Jorgensen Acoustic Scattering, Galerkin Estimates, and Clifford Algebras, B. Jawerth and M. Mitrea Clifford Algebra, Hardy Space, and Compensated Compactness, Z. Wu Frame Decomposition of Form-valued Hardy Spaces, J. Gilbert , J. Lakey , and J. Hogan Applications of Clifford Analysis to Inverse Scattering for Linear Hierarchy in Several Space Dimensions, D.B. Dix On Riemann-Hilbert Problems for the Dirac Operator in a Half Space of Rn, Z. Xu and C. Zhou Regularity and Approximation Results for the Maxwell Problem on C1 and Lipschitz Domains, M. Mitrea, R.H. Torres, and G.V. Welland Continuity of Calderon-Zygmund Type Operators on the Predual of a Morrey Space, J. Alvarez Neumann Type Problems for the Dirac Operator, C. Zhou The Hyperholomorphic Bergman Projector and its Properties, E. Ramirez de Arellano, M. Shapiro, and N. Vasilevski Multivector Solutions to the Meta-Holomorphic Massive Dirac Equation, W. Pizzaglia, Jr. Mobius Transformations, Vahlen Matrices, and their Factorizations, P. Lounesto