# Covariant differential operators and Green's functions

Annales Polonici Mathematici (1997)

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

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topMiroslav 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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- [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. Zbl0465.33001
- [14] L. Lewin (ed.), Structural Properties of Polylogarithms, Math. Surveys Monographs 37, Amer. Math. Soc., Providence, R.I., 1991. Zbl0745.33009
- [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] 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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