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Nonadmissible convergence in symmetric spaces
1996 (English)In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, no 472, p. 53-68Article in journal (Refereed) Published
Abstract [en]

In the disc (or half-plane, or half-space) the well-known Fatou theorem says that the Poisson integral of an $L^1$-function has non-tangential boundary limits a.e. J. Marcinkiewicz and A. Zygmund cleared up the corresponding situation in products of discs (or half-spaces), introducing the notions of restricted and unrestricted non-tangential convergence. For general Riemannian symmetric spaces the corresponding notions ("restricted and unrestricted admissible convergence") have also been found, and the classical results generalized, some time ago. In 1984 A. Nagel and E. M. Stein investigated again the half-space case and found that the convergence theorem could be proved for slightly larger than nontangential approach domains (the exact description is too technical to state here). The subject of the present paper is to extend this investigation to general Riemannian symmetric spaces. The notion of restricted admissible convergence (which is the natural one for $L^1$-functions) is appropriately extended and the convergence theorem proved in a way which seems to be the final word on the subject. Unrestricted convergence (which goes with $L^p$ functions, $p>1$) is also discussed and a theorem proved, but only in the case of one particular example of a space of rank two. The proofs display considerable virtuosity in the application of real-variable methods.

Place, publisher, year, edition, pages
1996. no 472, p. 53-68
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-6327DOI: 10.1515/crll.1996.472.53Scopus ID: 2-s2.0-85026009933Local ID: 48cd0270-b5fd-11db-bf94-000ea68e967bOAI: oai:DiVA.org:ltu-6327DiVA: diva2:979204
Note

Godkänd; 1996; 20070206 (kani)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-08-22Bibliographically approved

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