Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators

Paul Hagelstein*, Ioannis Parissis

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)

Abstract

Let B be a homothecy invariant basis consisting of convex sets in Rn, and define the associated geometric maximal operator MB by MBf(x):= supx∈R∈B 1|R|∫R|f| and the halo function ϕB(α) on (1, ∞) byϕB(α):=supE∫Rn:0<|E|<∞ 1|E||{x∈Rn:MBχE(x)>1/α}|. It is shown that if ϕB(α) satisfies the Solyanik estimate ϕB(α)-1≤C(1-1α)p for α∈(1, ∞) sufficiently close to 1 then ϕB lies in the Hölder class Cp(1, ∞). As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on Rn lie in the Hölder class C1/n(1, ∞).

Original languageEnglish
Pages (from-to)434-453
Number of pages20
JournalADVANCES IN MATHEMATICS
Volume285
DOIs
Publication statusPublished - 5 Nov 2015
MoE publication typeA1 Journal article-refereed

Keywords

  • Differentiation basis
  • Halo function
  • Maximal function
  • Tauberian conditions

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