Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Limit properties of Poisson kernels of tube domains

Author: Lawrence J. Dickson
Journal: Trans. Amer. Math. Soc. 182 (1973), 383-401
MSC: Primary 43A85; Secondary 31B10, 32A25
MathSciNet review: 0330937
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Abstract: If certain local boundary conditions hold near $ P \in \partial \Gamma $, the Poisson kernel belonging to a proper cone $ \Gamma \subset {{\mathbf{R}}^n}$ converges to a tight $ C_0^\ast$ limit as its parameter converges admissibly to P in $ \Gamma $. This limit can be identified with a lower-dimensional Poisson kernel. The result always works for polytopic and ``rounded'' cones; for these, a result on the decrease at infinity is obtained which in fact implies convergence almost everywhere in the appropriate sense of the Poisson integral to certain of its boundary values.

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Keywords: Szegö and Poisson kernels, proper cones, tight $ C_0^\ast$ limit, limit cones, maximal operator, decrease at infinity
Article copyright: © Copyright 1973 American Mathematical Society