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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Tauberian conditions for geometric maximal operators


Authors: Paul Hagelstein and Alexander Stokolos
Journal: Trans. Amer. Math. Soc. 361 (2009), 3031-3040
MSC (2000): Primary 42B25
Published electronically: December 29, 2008
MathSciNet review: 2485416
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Abstract: Let $ \mathcal{B}$ be a collection of measurable sets in $ \mathbb{R}^{n}$. The associated geometric maximal operator $ M_{\mathcal{B}}$ is defined on $ L^{1}(\mathbb{R}^n)$ by $ M_{\mathcal{B}}f(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{\vert R\vert}\int_{R}\vert f\vert$. If $ \alpha > 0$, $ M_\mathcal{B}$ is said to satisfy a Tauberian condition with respect to $ \alpha$ if there exists a finite constant $ C$ such that for all measurable sets $ E \subset \mathbb{R}^n$ the inequality $ \vert\{x : M_{\mathcal{B}} \chi_{E}(x) > \alpha\}\vert \leq C\vert E\vert$ holds. It is shown that if $ \mathcal{B}$ is a homothecy invariant collection of convex sets in $ \mathbb{R}^{n}$ and the associated maximal operator $ M_{\mathcal{B}}$ satisfies a Tauberian condition with respect to some $ 0 < \alpha < 1$, then $ M_\mathcal{B}$ must satisfy a Tauberian condition with respect to $ \gamma$ for all $ \gamma > 0$ and moreover $ M_{\mathcal{B}}$ is bounded on $ L^{p}(\mathbb{R}^{n})$ for sufficiently large $ p$. As a corollary of these results it is shown that any density basis that is a homothecy invariant collection of convex sets in $ \mathbb{R}^{n}$ must differentiate $ L^{p}(\mathbb{R}^{n})$ for sufficiently large $ p$.


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Additional Information

Paul Hagelstein
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798
Email: paul_hagelstein@baylor.edu

Alexander Stokolos
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
Email: astokolo@math.depaul.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04563-7
PII: S 0002-9947(08)04563-7
Received by editor(s): September 12, 2006
Received by editor(s) in revised form: May 30, 2007
Published electronically: December 29, 2008
Additional Notes: The first author’s research was partially supported by the Baylor University Summer Sabbatical Program.
The second author’s research was partially supported by the DePaul University Research Council Leave Program.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.