The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

  • David Cruz-UribeEmail author
  • Estefanía Dalmasso
  • Francisco J. Martín-Reyes
  • Pedro Ortega Salvador


We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces \(L_w^{p(\cdot )}(0,\infty )\), assuming that the exponent function \({p(\cdot )}\) is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals \(\{ (0,b) : b>0\}\) on \((0,\infty )\). Our results extend those in Duoandikoetxea et al. (Indiana Univ Math J 62(3):891–910, 2013) for the constant exponent \(L^p\) spaces with weights. We also give two applications: the first is a weighted version of Hilbert’s inequality on variable Lebesgue spaces, and the second generalizes the results in Soria and Weiss (Indiana Univ Math J 43(1):187–204, 1994) for integral operators to the variable exponent setting.


Calderón operator Hardy operator Stieltjes transform Maximal operator Weighted inequalities Muckenhoupt weights Variable Lebesgue spaces 

Mathematics Subject Classification

Primary 42B25 Secondary 26D15 42B35 



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Copyright information

© Universitat de Barcelona 2019

Authors and Affiliations

  • David Cruz-Uribe
    • 1
    Email author
  • Estefanía Dalmasso
    • 2
  • Francisco J. Martín-Reyes
    • 3
  • Pedro Ortega Salvador
    • 3
  1. 1.Department of MathematicsUniversity of AlabamaTuscaloosaUSA
  2. 2.Instituto de Matemática Aplicada del LitoralUNL, CONICET, FCE/FIQSanta FeArgentina
  3. 3.Facultad de CienciasUniversidad de MálagaMálagaSpain

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