Covariant differential operators and Green's functions

Miroslav Engliš; Jaak Peetre

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 77-103
  • ISSN: 0066-2216

Abstract

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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 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 Δ m and : the former is up to conjugation a polynomial of the latter.

How to cite

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Miroslav Engliš, and Jaak Peetre. "Covariant differential operators and Green's functions." Annales Polonici Mathematici 66.1 (1997): 77-103. <http://eudml.org/doc/269957>.

@article{MiroslavEngliš1997,
abstract = {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 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 $Δ^m$ and : the former is up to conjugation a polynomial of the latter.},
author = {Miroslav Engliš, Jaak Peetre},
journal = {Annales Polonici Mathematici},
keywords = {covariant differential operator; Laplace operator; Green's function; Hayman-Korenblum fomula; Bojarski's theorem; Bol's lemma; covariant Cauchy-Riemann operator; dilogarithm; trilogarithm; general nonsense; Green's functions; Poincaré invariant Laplace operator},
language = {eng},
number = {1},
pages = {77-103},
title = {Covariant differential operators and Green's functions},
url = {http://eudml.org/doc/269957},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Miroslav Engliš
AU - Jaak Peetre
TI - Covariant differential operators and Green's functions
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 77
EP - 103
AB - 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 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 $Δ^m$ and : the former is up to conjugation a polynomial of the latter.
LA - eng
KW - covariant differential operator; Laplace operator; Green's function; Hayman-Korenblum fomula; Bojarski's theorem; Bol's lemma; covariant Cauchy-Riemann operator; dilogarithm; trilogarithm; general nonsense; Green's functions; Poincaré invariant Laplace operator
UR - http://eudml.org/doc/269957
ER -

References

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  10. [10] C. J. Hill, Über die Integration logarithmisch-rationaler Differentiale, J. Reine Angew. Math. 3 (1828), 101-159. 
  11. [11] C. J. Hill, Specimen exercitii analytici, functionem integralem ∫ ₀ x d x / x L ( 1 + 2 x C α + x ² ) = x tum secundum amplitudinem, tum secundum modulem comparandi modum exhibentis, Typis Berlingianis, Londinii Gothorum (≡ Lund), 1830. 
  12. [12] E. Kamke, Handbook of Ordinary Differential Equations, Nauka, Moscow, 1971 (in Russian). 
  13. [13] L. Lewin, Polylogarithm and Associated Functions, North-Holland, New York, 1981. Zbl0465.33001
  14. [14] L. Lewin (ed.), Structural Properties of Polylogarithms, Math. Surveys Monographs 37, Amer. Math. Soc., Providence, R.I., 1991. Zbl0745.33009
  15. [15] J. Peetre and G. Zhang, Harmonic analysis on the quantized Riemann sphere, Internat. J. Math. Math. Sci. 16 (1993), 225-243. Zbl0776.30009
  16. [16] J. Peetre and G. Zhang, A weighted Plancherel formula III. The case of the hyperbolic matrix ball, Collect. Math. 43 (1992), 273-301. Zbl0836.43018
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