## Abstract

Let B be a collection of measurable sets in ℝ^{n}. The associated geometric maximal operator MB is defined on L^{1} (ℝ^{n}) by M_{B}f(x) = sup _{x∈R∈B} 1/|R| ∫ _{R} |f|. If α > 0, M_{B} 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 ⊂ ℝ^{n} the inequality |{x : M_{BχE}(x) > α}| ≤ C|E| holds. It is shown that if B is a homothecy invariant collection of convex sets in ℝ^{n} and the associated maximal operator M_{B} satisfies a Tauberian condition with respect to some 0 < α < 1, then M_{B} must satisfy a Tauberian condition with respect to γ for all γ > 0 and moreover M_{B} is bounded on L^{p}(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 ℝ^{n} must differentiate L^{p}(ℝ^{n}) for sufficiently large p.

Original language | English |
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Pages (from-to) | 3031-3040 |

Number of pages | 10 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2009 |