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The Schwarz lemma in Clifford analysis


Author: Zhongxiang Zhang
Journal: Proc. Amer. Math. Soc. 142 (2014), 1237-1248
MSC (2010): Primary 30G35
DOI: https://doi.org/10.1090/S0002-9939-2014-11854-5
Published electronically: January 6, 2014
MathSciNet review: 3162246
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we first give the Cauchy type integral representation for harmonic functions in the Clifford analysis framework, and by using integral representations for harmonic functions in Clifford analysis, the Poisson integral formula for harmonic functions is represented. As its application, the mean value theorems and the maximum modulus theorem for Clifford-valued harmonic functions are presented. Second, some properties of Möbius transformations are given, and a close relation between the monogenic functions and Möbius transformations is shown. Finally, by using the integral representations for harmonic functions and the properties of Möbius transformations, Schwarz type lemmas for harmonic functions and monogenic functions are established.


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  • [1] Heinrich Begehr, Iterations of Pompeiu operators, Mem. Differential Equations Math. Phys. 12 (1997), 13-21 (English, with English and Georgian summaries). International Symposium on Differential Equations and Mathematical Physics (Tbilisi, 1997). MR 1636853 (99g:47115)
  • [2] H. Begehr, Iterated integral operators in Clifford analysis, Z. Anal. Anwendungen 18 (1999), no. 2, 361-377. MR 1701359 (2000i:30075)
  • [3] H. Begehr, Representation formulas in Clifford analysis, Acoustics, mechanics, and the related topics of mathematical analysis, World Sci. Publ., River Edge, NJ, 2002, pp. 8-13. MR 2059106
  • [4] F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564 (85j:30103)
  • [5] Richard Delanghe, On regular-analytic functions with values in a Clifford algebra, Math. Ann. 185 (1970), 91-111. MR 0265618 (42 #527)
  • [6] Richard Delanghe, On the singularities of functions with values in a Clifford algebra, Math. Ann. 196 (1972), 293-319. MR 0301213 (46 #371)
  • [7] Richard Delanghe, Clifford analysis: history and perspective, Comput. Methods Funct. Theory 1 (2001), no. 1, 107-153. MR 1931607 (2003g:30092)
  • [8] R. Delanghe and F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel, Proc. London Math. Soc. (3) 37 (1978), no. 3, 545-576. MR 512025 (81j:46032), https://doi.org/10.1112/plms/s3-37.3.545
  • [9] R. Delanghe, F. Sommen, and V. Souček, Clifford algebra and spinor-valued functions, A function theory for the Dirac operator. Mathematics and its Applications, vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992. Related REDUCE software by F. Brackx and D. Constales, with 1 IBM-PC floppy disk (3.5 inch). MR 1169463 (94d:30084)
  • [10] Sirkka-Liisa Eriksson-Bique and Heinz Leutwiler, Hypermonogenic functions and Möbius transformations, Adv. Appl. Clifford Algebras 11 (2001), no. S2, 67-76. MR 2075343 (2005b:30050), https://doi.org/10.1007/BF03219123
  • [11] K. Gürlebeck and W. Sprössig, Quaternionic analysis and elliptic boundary value problems, Akademie-Verlag, Berlin, 1989. MR 1056478 (91k:35002a)
  • [12] V. Iftimie, Fonctiones hypercomplexes (French), Bull. Math. Soc. Sci. Math. R. S. Romanie 9(57), 1965, 279-332 (1966). MR 0217312 (36:403)
  • [13] Elena Obolashvili, Higher order partial differential equations in Clifford analysis, Effective solutions to problems, Progress in Mathematical Physics, vol. 28, Birkhäuser Boston Inc., Boston, MA, 2003. MR 1939756 (2003m:30097)
  • [14] Emilio Marmolejo Olea, Morera type problems in Clifford analysis, Rev. Mat. Iberoamericana 17 (2001), no. 3, 559-585. MR 1900895 (2003b:30064), https://doi.org/10.4171/RMI/304
  • [15] Jaak Peetre and Tao Qian, Möbius covariance of iterated Dirac operators, J. Austral. Math. Soc. Ser. A 56 (1994), no. 3, 403-414. MR 1271529 (95j:31007)
  • [16] Tao Qian and John Ryan, Conformal transformations and Hardy spaces arising in Clifford analysis, J. Operator Theory 35 (1996), no. 2, 349-372. MR 1401694 (97d:30064)
  • [17] Yan Yang and Tao Qian, Schwarz lemma in Euclidean spaces, Complex Var. Elliptic Equ. 51 (2006), no. 7, 653-659. MR 2242486 (2007d:30041), https://doi.org/10.1080/17476930600688623
  • [18] Zhongxiang Zhang, On $ k$-regular functions with values in a universal Clifford algebra, J. Math. Anal. Appl. 315 (2006), no. 2, 491-505. MR 2202595 (2006i:30071), https://doi.org/10.1016/j.jmaa.2005.06.053
  • [19] Z. Zhongxiang, Some properties of operators in Clifford analysis, Complex Var. Elliptic Equ. 52 (2007), no. 6, 455-473. MR 2326185 (2008d:30079), https://doi.org/10.1080/17476930701200666

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Additional Information

Zhongxiang Zhang
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
Email: zhxzhang.math@whu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2014-11854-5
Keywords: Clifford algebra, Poisson integral, M\"obius transformation, Schwarz lemma
Received by editor(s): December 19, 2011
Received by editor(s) in revised form: April 29, 2012
Published electronically: January 6, 2014
Additional Notes: The author was supported by the DAAD K. C. Wong Education Foundation and the NNSF for Young Scholars of China (No. 11001206).
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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