Abstract.
For a certain class of domains Ω⊂ℂ with smooth boundary and Δtilde;Ω=w 2Δ the Laplace–Beltrami operator with respect to the Poincaré metric ds 2=w(z)-2 dz dz on Ω, we (1) show that the Green function for the biharmonic operator Δtilde;Ω 2, with Dirichlet boundary data, is positive on Ω×Ω; and (2) obtain an eigenfunction expansion for the operator Δtilde;Ω, which reduces to the ordinary non-Euclidean Fourier transform of Helgason for Ω=𝔻 (the unit disc). In both cases the proofs go via uniformization, and in (1) we obtain a Myrberg-like formula for the corresponding Green function. Finally, the latter formula as well as the eigenfunction expansion are worked out more explicitly in the simplest case of Ω an annulus, and a result is established concerning the convergence of the series ∑ ω∈G (1-|ω0|2)s for G the covering group of the uniformization map of Ω and 0<s<1.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: August 21, 2000¶Published online: October 30, 2002
RID="*"
ID="*"The first author was supported by GA AV CR grants no. A1019701 and A1019005.
Rights and permissions
About this article
Cite this article
Engliš, M., Peetre, J. Green functions and eigenfunction expansions for the square of the Laplace–Beltrami operator on plane domains. Ann. Mat. Pura Appl. IV. Ser. 181, 463–500 (2002). https://doi.org/10.1007/s10231-002-0051-3
Issue Date:
DOI: https://doi.org/10.1007/s10231-002-0051-3