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

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