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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Inequalities for some maximal functions. II
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by M. Cowling and G. Mauceri PDF
Trans. Amer. Math. Soc. 296 (1986), 341-365 Request permission

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 $\mathcal {S}({{\mathbf {R}}^n}),{\mu ^\# }f$ denotes the maximal function ${\sup _{t > 0}}|{\mu _t}\ast f|$. We seek conditions on $\mu$ which guarantee that the a priori estimate \[ \left \| \mu ^\# f\right \|_p \leq C\left \| f \right \|_p, \quad f \in S(\mathbf {R}^n),\] holds; this estimate entails that the sublinear operator ${\mu ^\# }$ 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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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 296 (1986), 341-365
  • MSC: Primary 42B25; Secondary 42B10
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0837816-0
  • MathSciNet review: 837816