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Function theory for Laplace and Dirac-Hodge operators in hyperbolic space. (English) Zbl 1135.58013
Summary: We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace’s equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula for \(C^{1}\) functions and a Green’s formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

MSC:
58J05 Elliptic equations on manifolds, general theory
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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[1] L. V. Ahlfors,Möbius Transformations in Several Dimensions, Ordway Lecture Notes, University of Minnesota, 1981.
[2] L. V. Ahlfors,Möbius transformations in R n expressed through 2\(\times\)2 matrices of Clifford numbers, Complex Variables5 (1986), 215–224. · Zbl 0597.30062
[3] Ö. Akin and H. Leutwiler,On the invariance of the solutions of the Weinstein equation under Möbius transformations, inClassical and Modern Potential Theory and Applications (K. Gowrisankran et al. eds.), Kluwer Dodrecht, 1994, pp. 19–29. · Zbl 0869.31005
[4] D. Calderbank,Dirac operators and Clifford analysis on manifolds, Max Plank Institute for Mathematics, Bonn, preprint 96-131, 1996.
[5] C. Cao and P. Waterman,Conjugacy invariants of Möbius groups, inQuasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 109–139. · Zbl 0894.30027
[6] P. Cerejeiras and J. Cnops,Hodge-Dirac operators for hyperbolic space, Complex Variables41 (2000), 267–278. · Zbl 1020.30054
[7] J. Cnops,An Introduction to Dirac Operators on Manifolds, Progress in Mathematical Physics, Birkhäuser, Boston, 2002. · Zbl 1023.58019
[8] S.-L. Eriksson-Bique,Möbius transformations and k-hypermonogenic functions, inClifford algebras and potential theory, Univ. Joensuu Dept. Math. Rep. Ser., 7, Univ. Joensuu, Joensuu, 2004, pp. 213–226.
[9] S.-L. Eriksson,Integral formulas for hypermonogenic functions, Bull. Belg. Math. Soc.11 (2004), 705–707. · Zbl 1071.30047
[10] S.-L. Eriksson-Bique,k-hypermonogenic functions, inProgress in Analysis (H. Begehr et al., eds.) World Scientific, New Jersey, 2003, pp. 337–348.
[11] S.-L. Eriksson-Bique and H. Leutwiler,Hypermonogenic functions, inClifford Algebras and their Applications in Mathematical Physics, Volume 2, (J. Ryan and W. Sprößig, eds.), Birkhäuser, Boston, 2000, pp. 287–302. · Zbl 0965.30020
[12] S.-L. Eriksson and H. Leutwiler,Hypermonogenic functions and their Cauchy-type theorems, inTrends in Mathematics: Advances in Analysis and Geometry, Birkhäuser, Basel, 2003, pp. 1–16.
[13] S.-L. Eriksson and H. Leutwiler,Some integral formulas for hypermonogenic functions, to appear. · Zbl 1121.30024
[14] G. Gaudry, R. Long and T. Qian,A martingale proof of L 2 -boundedness of Clifford valued singular integrals, Ann. Mat. Pura Appl.165 (1993), 369–394. · Zbl 0814.42009 · doi:10.1007/BF01765857
[15] K. Gowrisankran and D. Singman,Minimal fine limits for a class of potentials, Potential Anal.13 (2000), 103–114. · Zbl 0965.31005 · doi:10.1023/A:1008697632471
[16] M. Habib,Invariance des fonctions \(\alpha\)-harmoniques par les transformations de Möbius, Exposition Math.13 (1995), 469–480. · Zbl 0954.31005
[17] L. K. Hua,Starting with the Unit Circle, Springer-Verlag, Heidelberg, 1981. · Zbl 0481.43005
[18] A. Huber,On the uniqueness of generalized axially symmetric potentials, Ann. of Math. (2),60 (1954), 351–358. · Zbl 0057.08802 · doi:10.2307/1969638
[19] V. Iftimie,Fonctions hypercomplexes, Bull. Math. Soc. Sci. Math. R. S. Roumanie9 (1965), 279–332. · Zbl 0177.36903
[20] R. S. Krausshar and J. Ryan,Clifford and harmonic analysis on spheres and hyperbolas Revista Matemática Iberoamericana21 (2005), 87–110.
[21] R. S. Krausshar, J. Ryan and Q. Yuying,Harmonic, monogenic and hypermonogenic functions on some conformally flat manifolds in R n arising from special arithmetic groups of the Vahlen group, Contemporary Mathematics, Contemporary Mathematics370 (2005), 159–173. · Zbl 1084.30056
[22] R. S. Krausshar and J. Ryan,Some conformally flat spin manifolds, Dirac operators and automorphic forms, to appear in Journal of Mathematical Analysis and Applications.
[23] H. Leutwiler,Best constants in the Harnack inequality for the Weinstein equation, Aequationes Math.34 (1987), 304–315. · Zbl 0653.31002 · doi:10.1007/BF01830680
[24] H. Leutwiler,Modified Clifford analysis, Complex Variables17 (1992), 153–171. · Zbl 0758.30037
[25] H. Liu, and J. Ryan,Clifford analysis techniques for spherical pde’s, J. Fourier Analysis Appl.8 (2002), 535–564. · Zbl 1047.53023 · doi:10.1007/s00041-002-0026-1
[26] A. McIntosh,Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains, inClifford Algebras in Analysis and Related Topics, (J. Ryan, ed.), CRC Press, Boca Raton, 1996, pp. 33–87. · Zbl 0886.42011
[27] M. Mitrea,Singular Integrals, Hardy Spaces, and Clifford Wavelets, Lecture Notes in Mathematics, No 1575, Springer-Verlag, Heidelberg, 1994. · Zbl 0822.42018
[28] M. Mitrea,Generalized Dirac operators on non-smooth manifolds and Maxwell’s equations, J. Fourier Analysis Appl.7 (2001), 207–256. · Zbl 0979.31006 · doi:10.1007/BF02511812
[29] I. Porteous,Clifford Algebras and the Classical Groups, Cambridge University Press, cambridge, 1995. · Zbl 0855.15019
[30] J. Ryan,Dirac operators on spheres and hyperbolae, Boletín Sociedad Matemática Mexicana.3 (1996), 255–270.
[31] K. Th. Vahlen,Über Bewegungen und Complexe Zahlen, Math. Ann.55 (1902), 585–593. · JFM 33.0721.01 · doi:10.1007/BF01450354
[32] P. Van Lancker,Clifford analysis on the sphere, inClifford Algebras and their Applications in Mathematical Physics (V. Dietrich et al., eds.), Kluwer, Dordrecht, 1998, pp. 201–215. · Zbl 0896.15015
[33] A. Weinstein,Generalized axially symmetric potential theory, Bull. Amer. Math. Soc.59 (1953), 20–38. · Zbl 0053.25303 · doi:10.1090/S0002-9904-1953-09651-3
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