Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

Paul Hagelstein, Teresa Luque, Ioannis Parissis

Research output: Contribution to journalArticleScientificpeer-review

15 Citations (Scopus)

Abstract

Let B be a homothecy invariant collection of convex sets in ℝn. Given a measure μ, the associated weighted geometric maximal operatorMB,μ is defined by (Formula presented) It is shown that, provided μ satisfies an appropriate doubling condition with respect to B and ν is an arbitrary locally finite measure, the maximal operator MB,μ is bounded on Lp(ν) for sufficiently large p if and only if it satisfies a Tauberian condition of the form (Formula presented) As a consequence of this result we provide an alternative characterization of the class of Muckenhoupt weights A∞,B for homothecy invariant Muckenhoupt bases B consisting of convex sets. Moreover, it is immediately seen that the strong maximal function MR,μ, defined with respect to a product-doubling measure μ, is bounded on Lp(ν) for some p > 1 if and only if (Formula presented) holds for all ν-measurable sets E in ℝn. In addition, we discuss applications in differentiation theory, in particular proving that a μ-weighted homothecy invariant basis of convex sets satisfying appropriate doubling and Tauberian conditions must differentiate L∞(ν).

Original languageEnglish
Pages (from-to)7999-8032
Number of pages34
JournalTransactions of the American Mathematical Society
Volume367
Issue number11
DOIs
Publication statusPublished - 1 Nov 2015
MoE publication typeA1 Journal article-refereed

Keywords

  • Automorphism
  • Muckenhoupt weight
  • Strong maximal function
  • Tauberian condition

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