The Schwarz lemma in Clifford analysis

Zhang, Zhongxiang

doi:10.1090/S0002-9939-2014-11854-5

The Schwarz lemma in Clifford analysis
HTML articles powered by AMS MathViewer

by Zhongxiang Zhang PDF

Proc. Amer. Math. Soc. 142 (2014), 1237-1248 Request permission

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.

References

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
H. Begehr, Iterated integral operators in Clifford analysis, Z. Anal. Anwendungen 18 (1999), no. 2, 361–377. MR 1701359, DOI 10.4171/ZAA/887
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
F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
Richard Delanghe, On regular-analytic functions with values in a Clifford algebra, Math. Ann. 185 (1970), 91–111. MR 265618, DOI 10.1007/BF01359699
Richard Delanghe, On the singularities of functions with values in a Clifford algebra, Math. Ann. 196 (1972), 293–319. MR 301213, DOI 10.1007/BF01428219
Richard Delanghe, Clifford analysis: history and perspective, Comput. Methods Funct. Theory 1 (2001), no. 1, [On table of contents: 2002], 107–153. MR 1931607, DOI 10.1007/BF03320981
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, DOI 10.1112/plms/s3-37.3.545
R. Delanghe, F. Sommen, and V. Souček, Clifford algebra and spinor-valued functions, Mathematics and its Applications, vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992. A function theory for the Dirac operator; Related REDUCE software by F. Brackx and D. Constales; With 1 IBM-PC floppy disk (3.5 inch). MR 1169463, DOI 10.1007/978-94-011-2922-0
Sirkka-Liisa Eriksson-Bique and Heinz Leutwiler, Hypermonogenic functions and Möbius transformations, Adv. Appl. Clifford Algebras 11 (2001), no. S2, 67–76. MR 2075343, DOI 10.1007/BF03219123
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
Elena Obolashvili, Higher order partial differential equations in Clifford analysis, Progress in Mathematical Physics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 2003. Effective solutions to problems. MR 1939756, DOI 10.1007/978-1-4612-0015-4
Emilio Marmolejo Olea, Morera type problems in Clifford analysis, Rev. Mat. Iberoamericana 17 (2001), no. 3, 559–585. MR 1900895, DOI 10.4171/RMI/304
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
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
Yan Yang and Tao Qian, Schwarz lemma in Euclidean spaces, Complex Var. Elliptic Equ. 51 (2006), no. 7, 653–659. MR 2242486, DOI 10.1080/17476930600688623
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, DOI 10.1016/j.jmaa.2005.06.053
Z. Zhongxiang, Some properties of operators in Clifford analysis, Complex Var. Elliptic Equ. 52 (2007), no. 6, 455–473. MR 2326185, DOI 10.1080/17476930701200666