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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama, Tuscaloosa, AL, 35487, USA

    David Cruz-Uribe

  2. Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FCE/FIQ, Colectora Ruta Nac. No 168, Paraje El Pozo, S3007ABA, Santa Fe, Argentina

    Estefanía Dalmasso

  3. Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071, Málaga, Spain

    Francisco J. Martín-Reyes & Pedro Ortega Salvador

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  1. David Cruz-Uribe
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  2. Estefanía Dalmasso
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  3. Francisco J. Martín-Reyes
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  4. Pedro Ortega Salvador
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Correspondence to David Cruz-Uribe.

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David Cruz-Uribe was supported by research funds provided by the Dean of the College of Arts and Sciences, the University of Alabama. Estefanía Dalmasso was supported by CONICET, ANPCyT and CAI+D (UNL). Francisco J. Martín-Reyes and Pedro Ortega Salvador were supported by Grants MTM2011-28149-C02-02 and MTM2015-66157-C2-2-P (MINECO/FEDER) of the Ministerio de Economía y Competitividad (MINECO, Spain) and Grant FQM-354 of the Junta de Andalucía, Spain. The initial stages of this project were begun while the David Cruz-Uribe and Estefanía Dalmasso were visiting Málaga, and they want to thank the Francisco J. Martín-Reyes and Pedro Ortega Salvador for their hospitality.

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Cruz-Uribe, D., Dalmasso, E., Martín-Reyes, F.J. et al. The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights. Collect. Math. 71, 443–469 (2020). https://doi.org/10.1007/s13348-019-00272-3

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