Abstract
Substituting the complex structure by the paracomplex structure plays an important role in para-geometry and para-analysis. In this article we shall introduce the paracomplex structure into the realm of Clifford analysis and establish paracomplex Hermitean Clifford analysis by constructing a paracomplex Hermitean Dirac operator \({\mathcal {D}}\) and establishing the corresponding Cauchy integral formula. The theory of paracomplex Hermitean Clifford analysis turns out to be similar to that of complex Hermitean Clifford analysis which recently emerged as a refinement of the theory of several complex variables. It deserves to be pointed out that the introduction of a single operator \({\mathcal {D}}\) in the paracomplex setting has an advantage over the complex setting where complex Hermitean monogenic functions are described by a system of equations instead of being given as null-solution of a single Dirac operator as in the case of classic monogenic functions.
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References
Brackx, F., Bureš, J., De Schepper, H., Eelbode, D., Sommen, F., Souček, V.: Fundaments of Hermitean Clifford analysis Part I: complex structure. Complex Anal. Oper. Theory 1(3), 341–365 (2007)
Brackx, F., Bureš, J., De Schepper, H., Eelbode, D., Sommen, F., Souček, V.: Fundaments of Hermitean Clifford analysis Part II: Splitting of h-monogenic equations. Complex Var. Elliptic Equ. 52(10–11), 1063–1079 (2007)
Brackx, F., De Knock, B., De Schepper, H., Sommen, F.: On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis. Bull. Braz. Math. Soc. 40(3), 395–416 (2009)
Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. In: Research Notes in Mathematics, vol. 76. Pitman (Advanced Publishing Program), Boston (1982)
Brackx, F., De Schepper, H., Sommen, F.: The Hermitian Clifford analysis toolbox. Adv. Appl. Cliff. Alg. 18(3–4), 451–487 (2008)
Cortes, V.: Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lectures in Mathematics and Theoretical Physics, vol. 16, European Mathematical Society, Zürich (2010)
Cruceanu, V., Fortuny, P., Gadea, P.: A survey on paracomplex geometry. Rocky Mountain J. Math. 26, 83–115 (1996)
Damiano, A., Eelbode, D., Sabadini, I.: Algebraic analysis of Hermitian monogenic functions. C. R. Acad. Sci. Paris Ser. I(346), 139–142 (2008)
Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer, Dordrecht (1992)
Gilbert, J., Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)
Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic Functions in the Plane and n-Dimensional Space. Birkhäuser Verlag, Basel (2008)
Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997)
Khrennikov, A., Segre, G.: An introduction to hyperbolic analysis. (2005) (arXiv: math-ph/0507053v2)
Kytmanov, A.: The Bochner-Martinelli Integral and its Applications. Birkhäuser, Basel (1995)
Libine, M.: Hyperbolic Cauchy integral formula for the split complex Numbers. (2007) (arXiv: math-ph/0712.0375v1)
Mohaupt, T.: Special Geometry, Black Holes and Euclidean Supersymmetry. Handbook of pseudo-Riemannian geometry and supersymmetry. IRMA Lect. Math. Theor. Phys., vol. 16, pp. 149–181. Eur. Math. Soc., Zürich (2010)
Price, G.B.: An Introduction to Multicomplex Spaces and Functions. Marcel Dekker, Inc., New York (1991)
Rocha-Chávez, R., Shapiro, M., Sommen, F.: Integral theorems for functions and differential forms in \({\mathbb{C}}^m\). In: Research Notes in Mathematics, vol. 428. Chapman &Hall/CRC, Boca Raton (2002)
Rosenfeld, B.: Geometry of Lie groups. Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1997)
Sabadini, I., Sommen, F.: Hermitian Clifford analysis and resolutions. Math. Meth. Appl. Sci. 25(16–18), 1395–1413 (2002)
Sommen, F., Peña-Peña, D.: A Martinelli-Bochner formula for the Hermitian Dirac equation. Math. Methods Appl. Sci. 30(9), 1049–1055 (2007)
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Communicated by Vladimir Soucek.
This work was supported by the NNSF of China (11071230, 11371337), RFDP (20123402110068).
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Ren, G., Wang, H. & Chen, L. Paracomplex Hermitean Clifford Analysis. Complex Anal. Oper. Theory 8, 1367–1382 (2014). https://doi.org/10.1007/s11785-013-0341-3
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DOI: https://doi.org/10.1007/s11785-013-0341-3