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Proceedings of the American Mathematical Society

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

 

 

$ L\sp 2$-boundedness of spherical maximal operators with multidimensional parameter sets


Author: Young-Hwa Ha
Journal: Proc. Amer. Math. Soc. 105 (1989), 401-411
MSC: Primary 42B25; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9939-1989-0955460-X
MathSciNet review: 955460
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Abstract: For $ s > 0$, let $ {M_s}f(x) = \int_{\vert y\vert = 1} {f(x - sy)d\sigma (y)} $ be the spherical mean operator on $ {R^n}$. For a certain class of surfaces $ S$ in $ R_ + ^{n + 1}$ with $ \dim S = n - 2$ or $ \dim S = n - 1$ with an additional condition, the maximal operator

$\displaystyle \mathcal{M}f(x) = \mathop {\sup }\limits_{(u,s) \in S} \vert{M_s}f(x - u)\vert$

is shown to be bounded on $ {L^2}({R^n})$. This extends (on $ {L^2}({R^n})$) the theorem of Stein [7], where $ S = \{ (0,s):s > 0\} $, and its generalizations to $ \dim S = 1$ in Greenleaf [2] and Sogge and Stein [6].

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DOI: https://doi.org/10.1090/S0002-9939-1989-0955460-X
Article copyright: © Copyright 1989 American Mathematical Society