Tauberian conditions for geometric maximal operators

Paul Hagelstein, Alexander Stokolos

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

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 languageEnglish
Pages (from-to)3031-3040
Number of pages10
JournalTransactions of the American Mathematical Society
Volume361
Issue number6
DOIs
StatePublished - Jun 2009

Fingerprint

Dive into the research topics of 'Tauberian conditions for geometric maximal operators'. Together they form a unique fingerprint.

Cite this