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## Annales Polonici Mathematici

1997 | 66 | 1 | 77-103
Tytuł artykułu

### Covariant differential operators and Green's functions

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $Δ^m$ of the ordinary Laplace's operator Δ in the unit disk 𝔻 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 𝜟. 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 $Δ^m$ and 𝜟 : the former is up to conjugation a polynomial of the latter.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
77-103
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-06-10
Twórcy
autor
• Mathematical Institute, Academy of Sciences, Žitná 25, Czech Republic
autor
• Department of Mathematics, Lund University, Box 118 11567 Praha 1, S-22100 Lund, Sweden
Bibliografia
• [1] B. Berndtsson and M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble) 32 (3) (1982), 91-110.
• [2] T. Boggio, Sull'equilibrio delle piastre elastiche incastrate, Atti Reale Accad. Lincei 10 (1901), 197-205.
• [3] T. Boggio, Sulle funzioni di Green d'ordine m, Rend. Circ. Mat. Palermo 20 (1905), 97-135.
• [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).
• [5] M. Engliš and J. Peetre, A Green's function for the annulus, Ann. Mat. Pura Appl., to appear.
• [6] B. Gustafsson and J. Peetre, Notes on projective structures on complex manifolds, Nagoya Math. J. 116 (1989), 63-88.
• [7] W. K. Hayman and B. Korenblum, Representation and uniqueness of polyharmonic functions, J. Anal. Math. 60 (1993), 113-133.
• [8] P. J. H. Hedenmalm, A computation of Green functions for the weighted biharmonic operators $Δ|z|^{-2α}Δ$, with α > -1, Duke Math. J. 75 (1994), 51-78.
• [9] S. Helgason, Groups and Geometrical Analysis, Academic Press, Orlando, 1984.
• [10] C. J. Hill, Über die Integration logarithmisch-rationaler Differentiale, J. Reine Angew. Math. 3 (1828), 101-159.
• [11] C. J. Hill, Specimen exercitii analytici, functionem integralem $∫₀^x dx/x L(1 + 2xCα + x²) = 𝒟x$ tum secundum amplitudinem, tum secundum modulem comparandi modum exhibentis, Typis Berlingianis, Londinii Gothorum (≡ Lund), 1830.
• [12] E. Kamke, Handbook of Ordinary Differential Equations, Nauka, Moscow, 1971 (in Russian).
• [13] L. Lewin, Polylogarithm and Associated Functions, North-Holland, New York, 1981.
• [14] L. Lewin (ed.), Structural Properties of Polylogarithms, Math. Surveys Monographs 37, Amer. Math. Soc., Providence, R.I., 1991.
• [15] J. Peetre and G. Zhang, Harmonic analysis on the quantized Riemann sphere, Internat. J. Math. Math. Sci. 16 (1993), 225-243.
• [16] J. Peetre and G. Zhang, A weighted Plancherel formula III. The case of the hyperbolic matrix ball, Collect. Math. 43 (1992), 273-301.
• [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.
Typ dokumentu
Bibliografia
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