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Proceedings of the American Mathematical Society

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The weak behavior of spherical means


Authors: Carlos E. Kenig and Peter A. Tomas
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
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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 [Enhancements On Off] (What's this?)

  • [1] C. Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971), 330-336. MR 0296602 (45:5661)
  • [2] C. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996-999. MR 0063477 (16:127b)
  • [3] G. N. Watson, Theory of Bessel functions, Cambridge Univ. Press, London, 1966.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0548082-8
Keywords: Disc multiplier on radial functions, failure of weak type estimates
Article copyright: © Copyright 1980 American Mathematical Society

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