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 | References | Similar Articles | Additional Information

Abstract: Let be a collection of measurable sets in . The associated geometric maximal operator is defined on by . If , is said to satisfy a *Tauberian condition with respect to * if there exists a finite constant such that for all measurable sets the inequality holds. It is shown that if is a homothecy invariant collection of convex sets in and the associated maximal operator satisfies a Tauberian condition with respect to some , then must satisfy a Tauberian condition with respect to for all and moreover is bounded on for sufficiently large . As a corollary of these results it is shown that any density basis that is a homothecy invariant collection of convex sets in must differentiate for sufficiently large .

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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

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.