Tauberian Conditions for Geometric Maximal Operators

Document Type

Article

Publication Date

2009

Publication Title

Transactions of the American Mathematical Society

DOI

10.1090/S0002-9947-08-04563-7

ISSN

1088-6850

Abstract

Let B be a collection of measurable sets in Rn. The associated geometric maximal operator MB is defined on L1(Rn) by MBf(x) = supx∈R∈B 1/|R| ƒR|ƒ|. If α > 0, MB is said to satisfy a Tauberian condition with respect to α if there exists a finite constant C such that for all measurable sets E ⊂ Rn the inequality |{x : MBχE(x) > α}| ≤ C|E| holds. It is shown that if B is a homothecy invariant collection of convex sets in Rn and the associated maximal operator MB satisfies a Tauberian condition with respect to some 0 < α < 1, then MB must satisfy a Tauberian condition with respect to γ for all γ > 0 and moreover MB is bounded on Lp(Rn) 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 Rn must differentiate Lp(Rn) for sufficiently large p.

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