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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 \(P(z,\theta)\), \(z\in U\), \(\theta\in {\mathbb T}={\mathbb R}/2\pi {\mathbb Z}\), be the Poisson kernel on the unit disk \(U=\{| z| <1\}\). Then \(P(z,\theta)^{1/2}\) is an eigenfunction for the critical eigenvalue \(-1/4\) of the hyperbolic Laplacian \((1/4)(1-| z|^ 2)\Delta\). In this paper, it is shown that \(P_ 0f(z)=\int_{T}P(z,\theta)^{1/2}f(\theta) \,d\theta\) for \(f\in L^ 1({\mathbb T})\) has a strong convergence property at the boundary; namely, for almost all \(\alpha\in {\mathbb T}\), \(P_ 0f(z)/P_ 0 1(z)\) tends to \(f(\alpha)\) as \(z\to e^{i\alpha}\) along a “weakly tangential” domain at \(\alpha\).
[For the entire collection see Zbl 0543.00004.]
Reviewer: Fumi-Yuki Maeda

MSC:
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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