The basic idea of this paper is to use the covariance of a partial differential operator under a suitable group action to determine suitable associated Green’s functions. For instance, we offer a new proof of a formula for Green’s function of the mth power of the ordinary Laplace’s operator Δ in the unit disk found in a recent paper (Hayman-Korenblum, J. Anal. Math. 60 (1993), 113-133). We also study Green’s functions associated with mth powers of the Poincaré invariant Laplace operator . It turns out that they can be expressed in terms of certain special functions of which the dilogarithm (m = 2) and the trilogarithm (m = 3) are the simplest instances. Finally, we establish a relationship between and : the former is up to conjugation a polynomial of the latter.
@article{bwmeta1.element.bwnjournal-article-apmv66z1p77bwm,
author = {Miroslav Engli\v s and Jaak Peetre},
title = {Covariant differential operators and Green's functions},
journal = {Annales Polonici Mathematici},
volume = {66},
year = {1997},
pages = {77-103},
zbl = {0877.35033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p77bwm}
}
Miroslav Engliš; Jaak Peetre. Covariant differential operators and Green's functions. Annales Polonici Mathematici, Tome 66 (1997) pp. 77-103. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p77bwm/
[000] [1] B. Berndtsson and M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble) 32 (3) (1982), 91-110. | Zbl 0466.32001
[001] [2] T. Boggio, Sull'equilibrio delle piastre elastiche incastrate, Atti Reale Accad. Lincei 10 (1901), 197-205.
[002] [3] T. Boggio, Sulle funzioni di Green d'ordine m, Rend. Circ. Mat. Palermo 20 (1905), 97-135. | Zbl 36.0827.01
[003] [4] B. Bojarski [B. Boyarskiĭ], Remarks on polyharmonic operators and conformal mappings in space, in: Trudy Vsesoyuznogo Simpoziuma v Tbilisi, 21-23 Aprelya 1982 g., 49-56 (in Russian).
[004] [5] M. Engliš and J. Peetre, A Green's function for the annulus, Ann. Mat. Pura Appl., to appear. | Zbl 0871.35097
[005] [6] B. Gustafsson and J. Peetre, Notes on projective structures on complex manifolds, Nagoya Math. J. 116 (1989), 63-88. | Zbl 0667.30038
[006] [7] W. K. Hayman and B. Korenblum, Representation and uniqueness of polyharmonic functions, J. Anal. Math. 60 (1993), 113-133. | Zbl 0793.31002
[007] [8] P. J. H. Hedenmalm, A computation of Green functions for the weighted biharmonic operators , with α > -1, Duke Math. J. 75 (1994), 51-78.
[008] [9] S. Helgason, Groups and Geometrical Analysis, Academic Press, Orlando, 1984.
[009] [10] C. J. Hill, Über die Integration logarithmisch-rationaler Differentiale, J. Reine Angew. Math. 3 (1828), 101-159.
[010] [11] C. J. Hill, Specimen exercitii analytici, functionem integralem tum secundum amplitudinem, tum secundum modulem comparandi modum exhibentis, Typis Berlingianis, Londinii Gothorum (≡ Lund), 1830.
[011] [12] E. Kamke, Handbook of Ordinary Differential Equations, Nauka, Moscow, 1971 (in Russian).
[012] [13] L. Lewin, Polylogarithm and Associated Functions, North-Holland, New York, 1981. | Zbl 0465.33001
[013] [14] L. Lewin (ed.), Structural Properties of Polylogarithms, Math. Surveys Monographs 37, Amer. Math. Soc., Providence, R.I., 1991. | Zbl 0745.33009
[014] [15] J. Peetre and G. Zhang, Harmonic analysis on the quantized Riemann sphere, Internat. J. Math. Math. Sci. 16 (1993), 225-243. | Zbl 0776.30009
[015] [16] J. Peetre and G. Zhang, A weighted Plancherel formula III. The case of the hyperbolic matrix ball, Collect. Math. 43 (1992), 273-301. | Zbl 0836.43018
[016] [17] H. Weyl, Ramifications, old and new, of the eigenvalue problem, Bull. Amer. Math. Soc. 56 (1950), 115-139; also in: Gesammelte Abhandlungen IV, 432-456. | Zbl 0041.21003