Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 548082
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42B15

Retrieve articles in all journals with MSC: 42B15

Additional Information

Keywords: Disc multiplier on radial functions, failure of weak type estimates
Article copyright: © Copyright 1980 American Mathematical Society