Geodesic
Boundary Value Problems for Harmonic Functions on the Heisenberg Group
Canadian journal of mathematics, Tome 38 (1986) no. 2, pp. 478-512

Voir la notice de l'article provenant de la source Cambridge University Press

Analysis on the Heisenberg group has become an important area with strong connections to Fourier analysis, group representations, and partial differential operators. We propose to show in this work that special functions methods can also play a significant part in this theory. There is a one-parameter family of second-order hypoelliptic operators Lγ , (γ ∊ C), associated to the Laplacian L 0 (also called the subelliptic or Kohn Laplacian). These operators are closely related to the unit ball for reasons of homogeneity and unitary group invariance. The associated Dirichlet problem is to find functions with specified boundary values and annihilated by Lγ inside the ball (that is, Lγ -harmonic). This is the topic of this paper.Gaveau [9] proved the first positive result, showing that continuous functions on the boundary can be extended to L 0-harmonic functions in the ball, by use of diffusion-theoretic methods. Jerison [15] later gave another proof of the L 0-result. Hueber [14] has recently obtained some results dealing with special values of the Poisson kernel for L 0. 
Dunkl, Charles F. Boundary Value Problems for Harmonic Functions on the Heisenberg Group. Canadian journal of mathematics, Tome 38 (1986) no. 2, pp. 478-512. doi: 10.4153/CJM-1986-024-9
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[1] 1. Cartwright, M. L., The zeros of certain integral functions II, Quart. J. Math Series (1), 2 (1931), 113–129. Google Scholar

[2] 2. Dunkl, C. F., An addition theorem for Heisenberg harmonics in Conference on harmonic analysis in honor of Antoni Zygmund, Wadsworth International, (1983), 690–707. Google Scholar

[3] 3. Dunkl, C. F., The Poisson kernel for Heisenberg polynomials on the disk, Math. Z. 187 (1984), 527–547. Google Scholar

[4] 4. Dunkl, C. F., Orthogonal polynomials and a Dirichlet problem related to the Hilbert transform, Indug. Math. 47 (1985), 147–171. Google Scholar

[5] 5. Erdélyi, A., Higher transcendental functions I (Bateman Manuscript Project); (McGraw-Hill, New York, 1953). Google Scholar

[6] 6. Folland, G. B. and Stein, E. M., Estimates for the complex and analysis on the Heisenberg group, Comm. Pure and Appl. Math. 27 (1974), 429–522. Google Scholar | DOI

[7] 7. Freud, G., Orthogonale polynome (Birkhäuser Verlag, Basel, 1969). Google Scholar | DOI

[8] 8. Gasper, G., Orthogonality of certain functions with respect to complex valued weights, Can. J. Math. 33 (1981), 1261–1270. Google Scholar

[9] 9. Gaveau, B., Principe de moindre action, propagation de chaleur et estimées sous elliptiques sur certain groupes nilpotents, Acta Math. 139 (1977), 95–153. Google Scholar

[10] 10. Greiner, P. C., Spherical harmonies on the Heisenberg group, Can. Math. Bull. 23 (1980), 383–396. Google Scholar

[11] 11. Greiner, P. C. and Koornwinder, T. H., Variations on the Heisenberg spherical harmonics, Report ZW 186/83, C.W.I. Amsterdam, (1983). Google Scholar

[12] 12. Helffer, B. and Nourrigat, J., Caractérisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. P.D.E. 4 (1979), 899–958. Google Scholar

[13] 13. Henrici, P., Applied and computational complex analysis, Vol. 2 (Wiley-Interscience, New York, 1977). Google Scholar

[14] 14. Hueber, H., The Poisson space of the Koranyi ball, Math. Annalen 268 (1984), 221–232. Google Scholar

[15] 15. Jerison, D. S., The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, I, II, J. Functional Anal. 43 (1981), 97–142, 224–257. Google Scholar

[16] 16. Koranyi, A., Kelvin transforms and harmonic polynomials on the Heisenberg group, J. Functional Anal. 49 (1982), 177–185. Google Scholar

[17] 17. Pollaczek, F., Sur une famille de polynômes orthogonaux qui contient les polynômes d'Hermite et de Laguerre comme cas limites, C.R. Acad. Sci. Paris 230 (1950), 1563–1565. Google Scholar

[18] 18. Slater, L. J., Generalized hypergeometric functions (Cambridge University Press, Cambridge, 1966). Google Scholar

[19] 19. Titchmarsh, E. C., The zeros of certain integral functions, Proc. London Math. Soc. (2) 25 (1926), 283–302. Google Scholar

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