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On generalized Fatou theorems for the square root of the Poisson kernel and in rank one symmetric space. (English) Zbl 0868.31001
The author treats the convergence problem for the \(\lambda\)-Poisson integral \[ P_{\lambda}f = P_t^{\lambda + \frac{1}{2}} \ast f , \;\;P_t^{\lambda + \frac{1}{2}} = \left [ P_t(x)\right ]^{\lambda + \frac{1}{2}} \;, \] \(P_t(x)\) being the Poisson kernel. This means the characterization of the regions \(\Omega \subset \mathbb{R}^2\) of points \((x,t), x \in \mathbb{R}^1 , t>0, \) where \(P_{\lambda}f \rightarrow f(x_0)\) for almost all \(x_0 \in \mathbb{R}^1\) whenever \((x,t) \rightarrow (x_0,0)\) in \(\Omega^{x_0}+ (x_0,0)\) . The main attention is paid to the case \(\lambda = 0\).
Another generalization is connected with the consideration of Riemannian symmetric spaces.
Reviewer: S.Samko (Faro)
MSC:
31A10 Integral representations, integral operators, integral equations methods in two dimensions
30A10 Inequalities in the complex plane
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