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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Inequalities for some maximal functions. II

Authors: M. Cowling and G. Mauceri
Journal: Trans. Amer. Math. Soc. 296 (1986), 341-365
MSC: Primary 42B25; Secondary 42B10
MathSciNet review: 837816
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Abstract: Let $ S$ be a smooth compact hypersurface in $ {{\mathbf{R}}^n}$, and let $ \mu $ be a measure on $ S$, absolutely continuous with respect to surface measure. For $ t$ in $ {{\mathbf{R}}^ + },{\mu _t}$ denotes the dilate of $ \mu $ by $ t$, normalised to have the same total variation as $ \mu $: for $ f$ in $ \mathcal{S}({{\mathbf{R}}^n}),{\mu ^\char93 }f$ denotes the maximal function $ {\sup _{t > 0}}\vert{\mu _t}\ast f\vert$. We seek conditions on $ \mu $ which guarantee that the a priori estimate

$\displaystyle \left\Vert \mu^\char93 f\right\Vert _p \leq C\left\Vert f \right\Vert _p, \quad f \in S(\mathbf{R}^n),$

holds; this estimate entails that the sublinear operator $ {\mu ^\char93 }$ extends to a bounded operator on the Lebesgue space $ {L^p}({{\mathbf{R}}^n})$. Our methods generalise E. M. Stein's treatment of the "spherical maximal function" [5]: a study of "Riesz operators", $ g$-functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of $ \mu $. These depend on the geometry of $ S$ and the relation between $ \mu $ and surface measure on $ S$. In particular, we find that there are natural geometric maximal operators limited on $ {L^p}({{\mathbf{R}}^n})$ if and only if $ p \in (q,\infty ];q$ is some number in $ (1,\infty)$, and may be greater than $ 2$. This answers a question of S. Wainger posed by Stein [6]>.

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Keywords: Maximal function, hypersurface, Fourier transform
Article copyright: © Copyright 1986 American Mathematical Society