Green functions and eigenfunction expansions for the square of the Laplace–Beltrami operator on plane domains

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 Poincare 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.