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

 

Convergence and divergence almost everywhere of spherical means for radial functions


Author: Yūichi Kanjin
Journal: Proc. Amer. Math. Soc. 103 (1988), 1063-1069
MSC: Primary 42B25
MathSciNet review: 954984
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Abstract: Let $ d > 1$. It will be shown that the maximal operator $ {S^*}$ of spherical means $ {S_R},R > 0$, is bounded on $ {L^p}({{\mathbf{R}}^d})$ radial functions when $ 2d/(d + 1) < p < 2d/(d - 1)$, and it implies that, for every $ {L^p}({{\mathbf{R}}^d})$ radial function $ f(t),{S_R}f(t)$ converges to $ f(t)$ for a.e. $ t \in {{\mathbf{R}}^d}$ when $ 2d/(d + 1) < p \leq 2$. Also, it will be proved that there is an $ {L^{2d/(d + 1)}}({R^d})$ radial function $ f(t)$ with compact support such that $ {S_R}f(t)$ diverges for a.e. $ t \in {R^d}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0954984-8
PII: S 0002-9939(1988)0954984-8
Keywords: Maximal operator of spherical means for radial functions, a.e. convergence, a.e. divergence, transplantation theorem, Hankel transform
Article copyright: © Copyright 1988 American Mathematical Society