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Covariant differential operators and Green's functions

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Engliš M., Peetre J.

Annales Polonici Mathematici

1997

tom 66

nr 1

77-103

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Engliš M.

1 Peetre J.

1 1997

Wyniki wyszukiwania

Covariant differential operators and Green's functions