Book Review
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MathSciNet review:
1568181
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Book Information:
Author:
F. Sommen R. Delanghe, and V. Sou\v cek
Title:
Clifford algebra and spinor-valued functions, a function theory for the Dirac operator
Additional book information:
Kluwer, Dordrecht, 1992, xvi + 485 pp., US$176.00. ISBN 0-7923-0229-X.
P. Auscher and Ph. Tchamitchian, Bases d’ondelettes sur les courbes corde-arc, noyau de Cauchy et espaces de Hardy associés, Rev. Mat. Iberoamericana 5 (1989), no. 3-4, 139–170 (French). MR 1077208, DOI 10.4171/RMI/89
[BBW] B. Boos-Bavnbeck and K. Wojciechowski, Elliptic boundary value problems for Dirac operators, Birkhäuser, Basel and Boston, MA, 1993.
F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
Robert J. Baston and Michael G. Eastwood, The Penrose transform, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Its interaction with representation theory; Oxford Science Publications. MR 1038279
Professor Clifford, Applications of Grassmann’s Extensive Algebra, Amer. J. Math. 1 (1878), no. 4, 350–358. MR 1505182, DOI 10.2307/2369379
[D] P.A.M. Dirac, The quantum theory of the electron, I; II, Proc. Roy. Soc. London Ser. A 117 (1928), 610-624; 118 (1928), 351-361.
[Dix] A.C. Dixon, On the Newtonian potential, Quart. J. Math. 35 (1904), 283-296.
G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56 (French). MR 850408, DOI 10.4171/RMI/17
Run Fueter, Die Funktionentheorie der Differentialgleichungen $\Theta u=0$ und $\Theta \Theta u=0$ mit vier reellen Variablen, Comment. Math. Helv. 7 (1934), no. 1, 307–330 (German). MR 1509515, DOI 10.1007/BF01292723
G. I. Gaudry, R. Long, and T. Qian, A martingale proof of $L_2$ boundedness of Clifford-valued singular integrals, Ann. Mat. Pura Appl. (4) 165 (1993), 369–394. MR 1271427, DOI 10.1007/BF01765857
John E. Gilbert and Margaret A. M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, vol. 26, Cambridge University Press, Cambridge, 1991. MR 1130821, DOI 10.1017/CBO9780511611582
Klaus Gürlebeck and Wolfgang Sprössig, Quaternionic analysis and elliptic boundary value problems, Mathematical Research, vol. 56, Akademie-Verlag, Berlin, 1989. MR 1056478
V. Iftimie, Fonctions hypercomplexes, Bull. Math. Soc. Sci. Math. R. S. Roumanie 9(57) (1965), 279–332 (1966) (French). MR 217312
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
Chun Li, Alan McIntosh, and Tao Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana 10 (1994), no. 3, 665–721. MR 1308706, DOI 10.4171/RMI/164
Chun Li, Alan McIntosh, and Stephen Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc. 5 (1992), no. 3, 455–481. MR 1157291, DOI 10.1090/S0894-0347-1992-1157291-5
Marius Mitrea, Clifford wavelets, singular integrals, and Hardy spaces, Lecture Notes in Mathematics, vol. 1575, Springer-Verlag, Berlin, 1994. MR 1295843, DOI 10.1007/BFb0073556
Alan McIntosh, Clifford algebras and the higher-dimensional Cauchy integral, Approximation and function spaces (Warsaw, 1986) Banach Center Publ., vol. 22, PWN, Warsaw, 1989, pp. 253–267. MR 1097197
[McI] -, Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains, Clifford Algebras in Analysis and Related Topics (John Ryan, ed.), CRC Press, Boca Raton, FL, 1995.
[MT] G.C. Moisil and N. Teodorescu, Fonctions holomorphes dans l'espace, Math. Cluj 5 (1931), 142-150.
Margaret A. M. Murray, The Cauchy integral, Calderón commutators, and conjugations of singular integrals in $\textbf {R}^n$, Trans. Amer. Math. Soc. 289 (1985), no. 2, 497–518. MR 784001, DOI 10.1090/S0002-9947-1985-0784001-6
Marcel Riesz, Clifford numbers and spinors, Fundamental Theories of Physics, vol. 54, Kluwer Academic Publishers Group, Dordrecht, 1993. With the author’s private lectures to E. Folke Bolinder; Edited by Bolinder and Pertti Lounesto and with a preface by Bolinder. MR 1247961, DOI 10.1007/978-94-017-1047-3
John Ryan, Runge approximation theorems in complex Clifford analysis together with some of their applications, J. Funct. Anal. 70 (1987), no. 2, 221–253. MR 874056, DOI 10.1016/0022-1236(87)90112-1
John Ryan, Cells of harmonicity and generalized Cauchy integral formulae, Proc. London Math. Soc. (3) 60 (1990), no. 2, 295–318. MR 1031455, DOI 10.1112/plms/s3-60.2.295
A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 199–224. MR 516081, DOI 10.1017/S0305004100055638
- [AT]
- P. Auscher and Ph. Tchamitchian, Bases d'ondelettes sur les courbes corde-arc, noyau de Cauchy et espaces de Hardy associé, Rev. Mat. Iberoamericana 5 (1989), 139-170. MR 1077208 (92a:42039)
- [BBW]
- B. Boos-Bavnbeck and K. Wojciechowski, Elliptic boundary value problems for Dirac operators, Birkhäuser, Basel and Boston, MA, 1993.
- [BDS]
- F. Brackx, R. Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman, New York and London, 1982. MR 697564 (85j:30103)
- [BE]
- R.J. Baston and M.G. Eastwood, The Penrose Transform: Its interaction with representation theory, Oxford Univ. Press, London and New York, 1989. MR 1038279 (92j:32112)
- [C]
- W.K. Clifford, Applications of Grossman's extensive algebra, Amer. J. Math. 1 (1878), 350-358. MR 1505182
- [D]
- P.A.M. Dirac, The quantum theory of the electron, I; II, Proc. Roy. Soc. London Ser. A 117 (1928), 610-624; 118 (1928), 351-361.
- [Dix]
- A.C. Dixon, On the Newtonian potential, Quart. J. Math. 35 (1904), 283-296.
- [DJS]
- G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), 1-57. MR 850408 (88f:47024)
- [F]
- R. Fueter, Die Functionentheorie der Differentialgleichungen und mit Vier Reallen Variablen, Comment. Math. Helv. 7 (1934-35), 307-330. MR 1509515
- [GLQ]
- G.I. Gaudry, R.L. Long, and T. Qian, A martingale proof of -boundedness of Clifford-valued singular integrals, Ann. Mat. Pura Appl. 165 (1993), 369-394. MR 1271427 (95k:30103)
- [GM]
- J. Gilbert and M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Univ. Press, Cambridge and New York, 1991. MR 1130821 (93e:42027)
- [GS]
- K. Gürlebeck and W. Sprössig, Quaternionic analysis and elliptic boundary value problems, Birkhäuser Verlag, Basel and Boston, MA, 1990. MR 1096955 (91k:35002b)
- [I]
- V. Iftimie, Fonctions hypercomplexes, Bull. Math. Soc. Sci. Math. R. S. Roumanie 9 (1965), 279-332. MR 0217312 (36:403)
- [LM]
- H. Blaine Lawson, Jr., and Marie-Louise Michelsohn, Spin geometry, Princeton Univ. Press, Princeton, NJ, 1989. MR 1031992 (91g:53001)
- [LMcQ]
- C. Li, A. McIntosh, and T. Qian, Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana 10 (1994) (to appear). MR 1308706 (95k:42024)
- [LMcS]
- C. Li, A. McIntosh, and S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc. 5 (1992), 455-481. MR 1157291 (93b:42029)
- [M]
- Marius Mitrea, Clifford wavelets, singular integrals, and Hardy spaces, Lecture Notes in Math., vol. 1575, Springer-Verlag, Berlin and New York, 1994. MR 1295843 (96e:31005)
- [Mc]
- Alan McIntosh, Clifford algebras and the higher dimensional Cauchy integral, Approximation Theory and Function Spaces, Banach Center Publ., vol. 22, PWN, Warsaw, 1989, pp. 253-267. MR 1097197 (92d:30032)
- [McI]
- -, Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains, Clifford Algebras in Analysis and Related Topics (John Ryan, ed.), CRC Press, Boca Raton, FL, 1995.
- [MT]
- G.C. Moisil and N. Teodorescu, Fonctions holomorphes dans l'espace, Math. Cluj 5 (1931), 142-150.
- [Mu]
- Margaret Murray, The Cauchy integral, Calderón commutators and conjugations of singular integrals in , Trans. Amer. Math. Soc. 289 (1985), 497-518. MR 784001 (86f:42009)
- [R]
- M. Riesz, Clifford numbers and spinors, Fund. Theories Phys., vol. 54, Kluwer, Dordrecht, 1993. MR 1247961 (94i:15024)
- [R1]
- John Ryan, Runge approximation theorems arising in complex Clifford analysis, together with some of their applications, J. Funct. Anal. 70 (1987), 221-253. MR 874056 (88h:30073)
- [R2]
- -, Cells of harmonicity and generalized Cauchy integral formulae, Proc. London Math. Soc. 60 (1990), 295-318. MR 1031455 (91g:32014)
- [S]
- A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199-225. MR 516081 (80g:30031)
Review Information:
Reviewer:
Alan McIntosh
Journal:
Bull. Amer. Math. Soc.
32 (1995), 344-348
DOI:
https://doi.org/10.1090/S0273-0979-1995-00594-X