Clifford algebras, fourier theory, singular integrals, and harmonic functions on lipschitz domains

Clifford Algebras in Analysis and Related Topics

Volumn , Issue , 2018, Pages 33-87

  McIntosh, Alan ^a  

^a MACQUARIE UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; HARMONIC FUNCTIONS;

ARKANSAS; CLIFFORD ALGEBRA; CONVOLUTION OPERATORS; FOURIER THEORY; LIPSCHITZ; LIPSCHITZ DOMAIN; SINGULAR INTEGRAL;

FOURIER TRANSFORMS;

EID: 85053509365     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.4324/9781315139548     Document Type: Chapter

References (45)

1. P. Auscher and Ph. Tchamitchian, Bases d’ondelettes sur les courbes corde-arc, noyau de Cauchy et espaces de Hardy associés, Revista Mátematica Iberoamericana 5, (1989), 139-170.
2. F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Research Notes in Mathematics 76, Pitman Advanced Publishing Program, 1982.
3. A.P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. U.S.A. 74, (1977), 1324-1327.
4. W.K. Clifford, Applications of Grassman’s extensive algebra, American Journal of Mathematics 1, (1878), 350-358.
5. 2 boundedness of Cauchy integrals on Lipschitz curves, Journal of the American Mathematical Society 2, (1989), 553-564.
6. R.R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque 57, Société Mathématique de France, 1978.
7. R.R. Coifman and Y. Meyer, Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis on Lipschitz curves, Lecture Notes in Mathematics, Springer-Verlag 779, (1980), 104-122.
8. 2 pour les courbes lipschitziennes, Annals of Mathematics 116 (1982), 361-387.
9. B.E.J. Dahlberg, Estimates of harmonic measure, Arch. Rat. Mech. Anal. 65, (1977), 275-288.
10. B.E.J. Dahlberg, Poisson semigroups and singular integrals, Proceedings of the American Mathematical Society 97, (1986), 41-48.
11. A.C. Dixon, On the Newtonian Potential Quarterly Journal of Mathematics 35, (1904), 283-296.
12. G. David, J.-L. Journé and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Revista Matématica Iberoamericana 1, (1985), 1-57.
13. R. Delanghe, F. Sommen and V. Souček, Clifford Algebra and Spinor-Valued Functions, Kluwer, Academic, Norwell, MA, 1992.
14. 2 -boundedness of Clifford-valued singular integrals, Annali di Matematica, Pura Appl. 165, (1993), 369-394.
15. J. Gilbert and M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge, University Press, Cambridge, 1991.
16. K. Gürlebeck and W. Sprössig, Quaternionic Analysis and Elliptic Boundary Value Problems, Birkhäuser Verlag, Basel, 1990.
17. V. Iftimie, Fonctions hypercomplexes, Bull. Math, de la Soc. Sci. Math, de la R. S. de Roumanie 9, (1965), 279-332.
18. D.S. Jerison and C.E. Kenig, The Neumann problem on Lipschitz domains, Bulletin of the American Mathematical Society 4, (1981), 203-207.
19. Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, New York, 1976.
20. Carlos Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, C.B.M.S., American Mathematical Society 83, 1994.
21. H. Blaine Lawson Jr and Marie-Louise Michelsohn Spin Geometry, Princeton University Press, Princeton, NJ, 1989.
22. Chun Li and Alan Mcintosh, Clifford algebras and H∞ functional calculi of commuting operators, in Clifford Analysis, (John Ryan, ed.), CRC Press, Boca Raton, FL, 1995.
23. C. Li, A. Mcintosh and T. Qian, Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces Revista Matemática Iberoamericana, (to appear), 10, (1994), 665-721.
24. C. Li, A. Mcintosh and S. Semmes Convolution singular integrals on Lipschitz surfaces, Journal of the American Mathematical Society 5, (1992), 455-481.
25. Yves Meyer (vol. III with R.R. Coifman), Ondelettes et Opérateurs, I, II, III, Hermann, Paris editeurs des sciences et des arts, 1990.
26. Marius Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics, Springer-Verlag, New York, 1994, 1575.
27. Marius Mitrea, Hypercomplex variable techniques in harmonic analysis, in Clifford Analysis, (John Ryan, ed.), CRC Press, Boca Raton, FL, 1995.
28. Alan Mcintosh, Clifford algebras and the higher dimensional Cauchy integral, Approximation Theory and Function Spaces, Banach Center Publications 22, (1989), 253-267.
29. Alan Mcintosh and Yves Meyer, Algèbres d’opérateurs définis par des intégrates singulières, Comptes Rendus Acad. Sci., Paris, Sér. I, Math 301, (1985), 395-397.
30. A. Mcintosh and A. Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. Journal 36, (1987), 421-439.
31. A. Mcintosh, A. Pryde and W. Ricker, Comparison of joint spectra for certain classes of commuting operators, Studia Math. 88, (1988), 23-36.
32. A. Mcintosh and T. Qian, Convolution singular integral operators on Lipschitz curves, Proceedings of the Special Year on Harmonic Analysis at Nankai Institute of Mathematics, Tianjin, China Lecture Notes in Mathematics, Springer-Verlag, New York, 1494, (1991), 142-162.
33. m , Transactions of the American Mathematical Society 289, (1985), 497-518.
34. J. Nečas, Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris and Academia, Prague, 1967.
35. J. Peetre and P. Sjölin, Three-line theorems and Clifford analysis, Complex Variables 19, (1992), 151-163.
36. Tao Qian, Singular integrals with monogenic kernels on the m-torus and its Lipschitz perturbations, in Clifford Analysis, (John Ryan, ed.), CRC Press, Boca Raton, FL, 1995.
37. John Ryan, Plemelj formulae and transformations associated to plane wave decomposition in complex Clifford analysis, Proc. London Math. Soc. 64, (1992), 70-94.
38. John Ryan, Conformal covariance in Clifford analysis. To appear in Zeutschrift für Analysis und the Anwendugen.
39. Stephen Semmes, Square function estimates and the T(b) theorem, Proceedings of the American Mathematical Society 110, (1990), 721-726.
40. Fraciscus Sommen, Plane waves, biregular functions and hyper-complex Fourier analysis, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, 9, (1985), 205-219.
41. Fraciscus Sommen, Microfunctions with values in a Clifford algebra II, Scientific Papers of the College of Arts and Sciences, The University of Tokyo 36, (1986), 15-37.
42. p spaces, Acta Math. 103, (1960), 25-62.
43. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, P.U.P., Princeton, NJ, 1970.
44. Terry Tao, Convolution operators generated by right-monogenic and harmonic kernels, Masters thesis, Flinders University, Adelaide, South Australia, 1992.
45. Greg Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, Journal of Functional Analysis 59, (1984), 572-611.