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Generalized local Fatou theorems and area integrals. (English) Zbl 0707.31010
A theorem of Fatou states that certain classes of harmonic functions on \({\mathbb{R}}^ n\times]0,\infty [\) have limits almost everywhere on \({\mathbb{R}}^ n\times \{0\}\), provided that the approach is restricted to cones. More recently, A. Nagel and E. M. Stein [Adv. Math. 54, 83-106 (1984; Zbl 0546.42017)] showed that the restriction on the approach regions could be considerably relaxed. In the paper under review, the restriction is relaxed still further, and it is shown that the existence of limits through cones implies almost everywhere the existence of the less restricted limits, for completely arbitrary functions. The work is carried out in a general space of the form \(X\times]0,\infty [\), where X is a group which is also a space of homogeneous type.
Reviewer: N.A.Watson

MSC:
31B25 Boundary behavior of harmonic functions in higher dimensions
42B25 Maximal functions, Littlewood-Paley theory
35B99 Qualitative properties of solutions to partial differential equations
Keywords:
Fatou theorems
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