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A remark on the convergence of the eigenfunctions of the Laplacian to a critical eigenvalue. (Une remarque sur la convergence des fonctions propres du Laplacien à valeur propre critique.) (French) Zbl 0563.31003
Théorie du potentiel, Proc. Colloq. J. Deny, Orsay/France 1983, Lect. Notes Math. 1096, 544-548 (1984).

Let P(z,θ), zU, θ𝕋=/2π, be the Poisson kernel on the unit disk U={|z|<1}. Then P(z,θ) 1/2 is an eigenfunction for the critical eigenvalue -1/4 of the hyperbolic Laplacian (1/4)(1-|z| 2 )Δ. In this paper, it is shown that P 0 f(z)= T P(z,θ) 1/2 f(θ)dθ for fL 1 (𝕋) has a strong convergence property at the boundary; namely, for almost all α𝕋, P 0 f(z)/P 0 1(z) tends to f(α) as ze iα along a “weakly tangential” domain at α.

[For the entire collection see Zbl 0543.00004.]

Reviewer: Fumi-Yuki Maeda

MSC:
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
58J50Spectral problems; spectral geometry; scattering theory