Tauberian conditions for geometric maximal operators
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- by Paul Hagelstein and Alexander Stokolos PDF
- Trans. Amer. Math. Soc. 361 (2009), 3031-3040 Request permission
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 $M_{\mathcal {B}}f(x) = \sup _{x \in R \in \mathcal {B}}\frac {1}{|R|}\int _{R}|f|$. 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 $|\{x : M_{\mathcal {B}} \chi _{E}(x) > \alpha \}| \leq C|E|$ 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.References
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Additional Information
- Paul Hagelstein
- Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798
- MR Author ID: 683523
- ORCID: 0000-0001-5612-5214
- Email: paul_hagelstein@baylor.edu
- Alexander Stokolos
- Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614
- Email: astokolo@math.depaul.edu
- 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. - © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 3031-3040
- MSC (2000): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9947-08-04563-7
- MathSciNet review: 2485416