Poisson integrals and nontangential limits
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- by Victor L. Shapiro PDF
- Proc. Amer. Math. Soc. 134 (2006), 3181-3189 Request permission
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.References
- Victor L. Shapiro, On Green’s theorem, J. London Math. Soc. 32 (1957), 261–269. MR 89275, DOI 10.1112/jlms/s1-32.3.261
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- Victor L. Shapiro
- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- Email: shapiro@math.ucr.edu
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 3181-3189
- MSC (2000): Primary 31B25, 35K20; Secondary 35J05, 35K05
- DOI: https://doi.org/10.1090/S0002-9939-06-08331-6
- MathSciNet review: 2231901