Covariant differential operators and Green's functions

Miroslav Engliš; Jaak Peetre

ArticleOriginal scientific text

Title

Covariant differential operators and Green's functions

Authors ¹, ²

Affiliations

Mathematical Institute, Academy of Sciences, Žitná 25, Czech Republic
Department of Mathematics, Lund University, Box 118 11567 Praha 1, S-22100 Lund, Sweden

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.

Keywords

ENG

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

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Title

Covariant differential operators and Green's functions

Affiliations

Abstract

Keywords

Bibliography