The weak behavior of spherical means
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- by Carlos E. Kenig and Peter A. Tomas
- Proc. Amer. Math. Soc. 78 (1980), 48-50
- DOI: https://doi.org/10.1090/S0002-9939-1980-0548082-8
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Abstract:
C. Fefferman has shown that the disc multiplier is not bounded on ${L^p}({{\mathbf {R}}^n}),n > 1,p \ne 2$. In contrast, C. Herz showed that, when restricted to ${L^p}$ radial functions, it is bounded on ${L^p}({{\mathbf {R}}^n})$ if and only if $2n/(n + 1) < p < 2n/(n - 1)$. We show that it is not weakly bounded for $p = 2n/(n + 1)$ or $p = 2n/(n - 1)$.References
- Charles Fefferman, The multiplier problem for the ball, Ann. of Math. (2) 94 (1971), 330–336. MR 296602, DOI 10.2307/1970864
- Carl S. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996–999. MR 63477, DOI 10.1073/pnas.40.10.996 G. N. Watson, Theory of Bessel functions, Cambridge Univ. Press, London, 1966.
Bibliographic Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 78 (1980), 48-50
- MSC: Primary 42B15
- DOI: https://doi.org/10.1090/S0002-9939-1980-0548082-8
- MathSciNet review: 548082