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) = sup x∈R∈B 1/|R| ∫ R |f|. 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.
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 |