Weighted norm inequalities for geometric fractional maximal operators

  • David Cruz-Uribe
  • C. J. Neugebauer
  • V. Olesen


For 0 ≤α < ∞ let Tαf denote one of the operators
$$M_{\alpha ,0} f(x) = \mathop {\sup }\limits_{I \mathrel\backepsilon x} \left| I \right|^\alpha \exp \left( {\frac{1}{{\left| I \right|}}\int_I {\log \left| f \right|} } \right),M_{\alpha ,0}^* f(x) = \mathop {\lim }\limits_{r \searrow 0} \mathop {\sup }\limits_{I \mathrel\backepsilon x} \left| I \right|^\alpha \left( {\frac{1}{{\left| I \right|}}\int_I {\left| f \right|^r } } \right)^{{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r}} .$$
We characterize the pairs of weights (u, v) for which Tα is a bounded operator from Lp(v) to Lq(u), 0 <p ≤q < ∞. This extends to α > 0 the norm inequalities for α=0 in [4, 16]. As an application we give lower bounds for convolutions ϕ ⋆ f, where ϕ is a radially decreasing function.

Math subject classifications


Keywords and phrases

Fractional maximal operator weighted norm inequalities 


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Copyright information

© Birkhäuser 1999

Authors and Affiliations

  • David Cruz-Uribe
    • 1
  • C. J. Neugebauer
    • 2
  • V. Olesen
    • 2
  1. 1.Department of MathematicsTrinity CollegeHartford
  2. 2.Department of MathematicsPurdue UniversityWest Lafayette

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