Geodesic
Green's Functions for Powers of the Invariant Laplacian
Canadian journal of mathematics, Tome 50 (1998) no. 1, pp. 40-73

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

The aim of the present paper is the computation of Green's functions for the powers ${{\mathbf{\Delta }}^{m}}$ of the invariant Laplace operator on rank-one Hermitian symmetric spaces. Starting with the noncompact case, the unit ball in ${{\mathbb{C}}^{d}}$ , we obtain a complete result for $m\,=\,1,\,2$ in all dimensions. For $m\,\ge \,3$ the formulas grow quite complicated so we restrict ourselves to the case of the unit disc $(d\,=\,1)$ where we develop a method, possibly applicable also in other situations, for reducing the number of integrations by half, and use it to give a description of the boundary behaviour of these Green functions and to obtain their (multi-valued) analytic continuation to the entire complex plane. Next we discuss the type of special functions that turn up (hyperlogarithms of Kummer). Finally we treat also the compact case of the complex projective space ${{\mathbb{P}}^{d}}$ (for $d\,=\,1$ , the Riemann sphere) and, as an application of our results, use eigenfunction expansions to obtain some new identities involving sums of Legendre $(d\,=\,1)$ or Jacobi $(d\,>\,1)$ polynomials and the polylogarithm function. The case of Green's functions of powers of weighted (no longer invariant, but only covariant) Laplacians is also briefly discussed.
DOI : 10.4153/CJM-1998-004-8
Mots-clés : 35C05, 33E30, 33C45, 34B27, 35J40 
Engliš, Miroslav; Peetre, Jaak. Green's Functions for Powers of the Invariant Laplacian. Canadian journal of mathematics, Tome 50 (1998) no. 1, pp. 40-73. doi: 10.4153/CJM-1998-004-8
@article{10_4153_CJM_1998_004_8,
     author = {Engli\v{s}, Miroslav and Peetre, Jaak},
     title = {Green's {Functions} for {Powers} of the {Invariant} {Laplacian}},
     journal = {Canadian journal of mathematics},
     pages = {40--73},
     year = {1998},
     volume = {50},
     number = {1},
     doi = {10.4153/CJM-1998-004-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1998-004-8/}
}
TY  - JOUR
AU  - Engliš, Miroslav
AU  - Peetre, Jaak
TI  - Green's Functions for Powers of the Invariant Laplacian
JO  - Canadian journal of mathematics
PY  - 1998
SP  - 40
EP  - 73
VL  - 50
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1998-004-8/
DO  - 10.4153/CJM-1998-004-8
ID  - 10_4153_CJM_1998_004_8
ER  - 
%0 Journal Article
%A Engliš, Miroslav
%A Peetre, Jaak
%T Green's Functions for Powers of the Invariant Laplacian
%J Canadian journal of mathematics
%D 1998
%P 40-73
%V 50
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1998-004-8/
%R 10.4153/CJM-1998-004-8
%F 10_4153_CJM_1998_004_8

Bateman, H. and Erdélyi, A., Higher transcendental functions II.McGraw-Hill, New York-Toronto- London, 1953. Google Scholar

Engliš, M. and Peetre, J., Covariant differential operators and Green's functions. Ann. Polon. Math. 66 (1997), 77–103. Google Scholar

Hayman, W.K. and Korenblum, B., Representation and uniqueness of polyharmonic functions. J. Anal. Math. 60 (1993), 113–133. Google Scholar

Kummer, E.E., Über die Transzendenten, welche aus wiederholten Integrationen rationaler Funktionen entstehen. J. Reine Angew. Math. 21 (1840), 74–90. Google Scholar

Lewin, L., Polylogarithm and associated functions.North Holland, New York, 1981. Google Scholar

Lewin, L., Structural properties of polylogarithms (Ed.: Lewin, L.).Math. SurveysMonographs ,AmericanMathematical Society, Providence, RI, 1991. Google Scholar

Minakshisundaram, S. and Å. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Can. J. Math. 1 (1949), 242–256. Google Scholar

Nielsen, N., Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Leopoldina 90 (1909), 121–211.(=Abhandlungen der Kaiserlichen Leopoldinisch-Carolinischen Deutschen Akademie der Naturforscher, Halle, 1909). Google Scholar

Peetre, J. and Zhang, G., Harmonic analysis on the quantized Riemann sphere. Internat. J. Math. Math. Sci. 16 (1993), 225–243. Google Scholar

Rudin, W., Function theory in the unit ball of ℂn .Springer Verlag, Berlin-Heidelberg-New York, 1980. Google Scholar

Vilenkin, N.Ya., Special functions and the theory of group representations.Nauka, Moscow, 1965. Google Scholar

Wechsung, G., Functional equations of hyperlogarithms. In: [Le2], 171–184.(Chapter 8). Google Scholar

Cité par Sources :