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The monogenic Fischer decomposition: two vector variables. (English) Zbl 1275.30026

Summary: We present two proofs of the monogenic Fischer decomposition in two vector variables. The first one is based on the so-called “Harmonic Separation of Variables Theorem” while the second one relies on some simple dimension arguments. We also show that these decompositions are still valid under milder assumptions than the usual stable range condition. In the process, we derive an explicit formula for the summands in the monogenic Fischer decomposition of harmonics.

MSC:

30G35 Functions of hypercomplex variables and generalized variables
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[1] Brackx F., De Schepper H., Eelbode D., Souček V.: The Howe dual pair in Hermitean Clifford analysis. Rev. Mat. Iberoamericana 26(2), 449–479 (2010) · Zbl 1201.30061
[2] Brackx F., Delanghe R., Sommen F.: Clifford Analysis. Pitman, London (1982)
[3] Brackx, F., Eelbode, D., Raeymaekers, T., Van de Voorde, L.: Triple monogenic functions and higher spin Dirac operators (submitted) · Zbl 1230.30031
[4] Brackx, F., Eelbode, D., Van de Voorde, L.: Higher spin Dirac operators between spaces of simplicial monogenics in two vector variables (submitted) · Zbl 1318.30078
[5] Bureš J., Sommen F., Souček V., Van Lancker P.: Rarita–Schwinger type operators in Clifford analysis. J. Funct. Anal. 185, 425–456 (2001) · Zbl 1078.30041
[6] Bureš J., Sommen F., Souček V., Van Lancker P.: Symmetric analogues of Rarita–Schwinger equations. Ann. Glob. Anal. Geom. 21(3), 215–240 (2001) · Zbl 1025.58013
[7] Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Separation of variables in Clifford analysis and its application to Rarita–Schwinger field. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) Extended abstracts of the ICNAAM 2006 Conference, Official Conference of the European Society of Computational Methods in Sciences and Engineering, Crete, Greece, pp. 630–633 (2006)
[8] Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac Systems and Computational Algebra. Progress in Mathematical Physics, vol. 39. Birkhäuser, Basel (2004) · Zbl 1064.30049
[9] Constales, D.: The relative position of L 2-domains in complex and Clifford analysis, Ph. D. thesis, State University Ghent (1989–1990)
[10] Debarre O., Ton-That T.: Representations of $${SO(k, {\(\backslash\)mathbb C})}$$ on harmonic polynomials on a null cone. Proc. Am. Math. Soc. 112(1), 31–44 (1991)
[11] Delanghe, R., Lavicka, R., Soucek, V.: The Fischer decomposition for Hodge-de Rham systems in Euclidean spaces. arXiv:1012.4994v1
[12] Delanghe R., Sommen F., Souček V.: Clifford Analysis and Spinor Valued Functions. Kluwer Academic Publishers, Dordrecht (1992) · Zbl 0747.53001
[13] Eelbode, D., Smid, D., Van de Voorde, L.: A note on polynomial solutions for higher spin Dirac operators (in preparation)
[14] Gilbert J., Murray M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, London (1991) · Zbl 0733.43001
[15] Goodman, R.: Multiplicity-Free Spaces and Schur–Weyl–Howe Duality. In: Eng-Chye, T., Chen-Bo, Z. (eds.) Representations of Real and P-Adic Groups. World Scientific Publishing Company, Singapore (2004–06)
[16] Gürlebeck K., Sprössig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997) · Zbl 0897.30023
[17] Helgason S.: Invariants and fundamental functions. Acta Math. 109, 241–258 (1963) · Zbl 0112.26303
[18] Howe, R., Tan, E.C., Willenbring, J.: Reciprocity Algebras and Branching for Classical Symmetric Pairs, Groups and analysis. London Mathematical Society. Lecture Notes Series in Mathematics, vol. 354, pp. 191–231. Cambridge University Press, Cambridge (2008) · Zbl 1176.22012
[19] Kashiwara M., Vergne M.: On the Segal–Shale–Weil representations and harmonic polynomials. Inventiones Math. 44, 1–47 (1978) · Zbl 0375.22009
[20] Kostant B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963) · Zbl 0124.26802
[21] Sabadini I., Sommen F., Struppa D.C., Van Lancker P.: Complexes of dirac operators in clifford algebras. Math. Zeit. 239(2), 293–320 (2002) · Zbl 1078.30045
[22] Sommen F.: Clifford analysis in two and several vector variables. Appl. Anal. 73, 225–253 (1999) · Zbl 1054.30050
[23] Sommen F.: Functions on the spin group. Adv. Appl. Clifford algebras 6(1), 37–48 (1996) · Zbl 0863.15011
[24] Sommen F.: An algebra of abstract vector variables. Portugal. Math. 54(3), 287–310 (1997) · Zbl 0886.15037
[25] Sommen F., Van Acker N.: SO(m)-invariant differential operators on Clifford algebra valued functions. Found. Phys. 23(11), 1491–1519 (1993)
[26] Sommen F., Van Acker N.: Monogenic differential operators. Results Math. 22(3-4), 781–798 (1992) · Zbl 0765.30033
[27] Sommen F., Van Acker N.: Functions of two vector variables. Adv. Appl. Clifford Algebras 4(1), 65–72 (1994) · Zbl 0820.30029
[28] Sommen, F., Van Acker, N.: Invariant differential operators on polynomial-valued functions, In: Clifford Algebras and their Applications in Mathematical Physics, Fund. Theories Phys. vol. 55, pp. 203–212. Kluwer Academic Publishers, Dordrecht (1993) · Zbl 0841.30037
[29] Stein E.W., Weiss G.: Generalization of the Cauchy–Riemann equations and representations of the rotation group. Am. J. Math. 90, 163–196 (1968) · Zbl 0157.18303
[30] Van Lancker P., Sommen F., Constales D.: Models for irreducible representations of Spin(m). Adv. Appl. Clifford Algebras 11(S1), 271–289 (2001) · Zbl 1221.22016
[31] Weyl H.: The Classical Groups, their Invariants and Representations. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997) · Zbl 1024.20501
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