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Revista Matemática Complutense

, Volume 25, Issue 1, pp 43–59 | Cite as

The Riesz-Herz equivalence for capacitary maximal functions

  • Irina Asekritova
  • Joan Cerdà
  • Natan Kruglyak
Article

Abstract

We prove a Riesz-Herz estimate for the maximal function associated to a capacity C on ℝ n , M C f(x)=sup  Qx C(Q)−1 Q |f|, which extends the equivalence (Mf)(t)≃f ∗∗(t) for the usual Hardy-Littlewood maximal function Mf. The proof is based on an extension of the Wiener-Stein estimates for the distribution function of the maximal function, obtained using a convenient family of dyadic cubes. As a byproduct we obtain a description of the norm of the interpolation space \((L^{1},{\mathcal{L}}^{1,C})_{1/p',p}\), where \({\mathcal{L}}^{1,C}\) denotes the Morrey space based on a capacity.

Keywords

Maximal function Capacity Morrey space Dyadic cubes Interpolation spaces 

Mathematics Subject Classification (2000)

42B25 46B70 28A12 42B25 

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Copyright information

© Revista Matemática Complutense 2010

Authors and Affiliations

  1. 1.School of Mathematics and System EngineeringLinnaeus UniversityVäxjöSweden
  2. 2.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  3. 3.Department of MathematicsLinkoping UniversityLinkopingSweden

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