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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Poisson integrals and nontangential limits


Author: Victor L. Shapiro
Journal: Proc. Amer. Math. Soc. 134 (2006), 3181-3189
MSC (2000): Primary 31B25, 35K20; Secondary 35J05, 35K05
Published electronically: June 1, 2006
MathSciNet review: 2231901
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Abstract: A new result is established for nontangential limits of the Poisson integral of an $ f\in L^{p}(\mathbf{R}^{N})$ for $ N\geq 2.$ This is accomplished by showing for $ N=2,\exists f$ such that the $ \sigma$-set of $ f$ strictly contains the Lebesgue set of $ f.$ A similar theorem is also proved for Gauss-Weierstrass integrals, giving a new result for solutions of the heat equation.


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Additional Information

Victor L. Shapiro
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: shapiro@math.ucr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08331-6
PII: S 0002-9939(06)08331-6
Keywords: Nontangential limit, Poisson integral, Gauss-Weierstrass integral.
Received by editor(s): September 24, 2004
Received by editor(s) in revised form: April 26, 2005
Published electronically: June 1, 2006
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.