Advertisement

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
Article
  • 22 Downloads

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.

Keywords

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

Mathematics Subject Classification

Primary 42B25 Secondary 26D15 42B35 

Notes

References

  1. 1.
    Andersen, K.F.: Weighted inequalities for the Stieltjes transformation and Hilbert’s double series. Proc. R. Soc. Edinb. Sect. A 86(1–2), 75–84 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ariño, M.A., Muckenhoupt, B.: Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Trans. Am. Math. Soc. 320(2), 727–735 (1990)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bandaliev, R.A.: The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces. Czechoslov. Math. J. 60(2), 327–337 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bandaliev, R.A.: Corrections to the paper “The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces”. Czechoslov. Math. J. 63(4), 1149–1152 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bastero, J., Milman, M., Ruiz, F.J.: On the connection between weighted norm inequalities, commutators and real interpolation. Mem. Am. Math. Soc. 154, 731 (2001)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Berezhnoi, E.I.: Two-weighted estimations for the Hardy–Littlewood maximal function in ideal Banach spaces. Proc. Am. Math. Soc. 127(1), 79–87 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bernardis, A., Dalmasso, E., Pradolini, G.: Generalized maximal functions and related operators on weighted Musielak–Orlicz spaces. Ann. Acad. Sci. Fenn. Math. 39(1), 23–50 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cruz-Uribe, D.: Two weight inequalities for fractional integral operators and commutators. In: Martin-Reyes, F.J. (ed.) VI International Course of Mathematical Analysis in Andalusia, pp. 25–85. World Scientific, Singapore (2016)Google Scholar
  9. 9.
    Cruz-Uribe, D., Diening, L., Hästö, P.: The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal. 14(3), 361–374 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Heidelberg (2013)zbMATHCrossRefGoogle Scholar
  11. 11.
    Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl. 394(2), 744–760 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Cruz-Uribe, D., Mamedov, F.I.: On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces. Rev. Mat. Complut. 25(2), 335–367 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Cruz-Uribe, D., Wang, L.-A.: Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Am. Math. Soc. 369(2), 1205–1235 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Diening, L.: Maximal function on generalized Lebesgue spaces \(L^{p(\cdot )}\). Math. Inequal. Appl. 7(2), 245–253 (2004)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Diening, L., Harjulehto, P., Hästö, P., R\(\mathring{{\rm u}}\)žička, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, vol. 2011. Springer, Heidelberg (2017)zbMATHCrossRefGoogle Scholar
  16. 16.
    Diening, L., Samko, S.: Hardy inequality in variable exponent Lebesgue spaces. Fract. Calc. Appl. Anal. 10(1), 1–18 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Duoandikoetxea, J.: Fractional integrals on radial functions with applications to weighted inequalities. Ann. Mat. Pura Appl. (4) 192(4), 553–568 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Duoandikoetxea, J., Martín-Reyes, F.J., Ombrosi, S.: Calderón weights as Muckenhoupt weights. Indiana Univ. Math. J. 62(3), 891–910 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Duoandikoetxea, J., Martín-Reyes, F.J., Ombrosi, S.: On the \(A_\infty \) conditions for general bases. Math. Z. 282(3–4), 955–972 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gogatishvili, A., Kufner, A., Persson, L.-E.: The weighted Stieltjes inequality and applications. Math. Nachr. 286(7), 659–668 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gogatishvili, A., Kufner, A., Persson, L.-E., Wedestig, A.: An equivalence theorem for integral conditions related to Hardy’s inequality. Real Anal. Exch. 29(2), 867–880 (2003/2004)Google Scholar
  22. 22.
    Gogatishvili, A., Persson, L.-E., Stepanov, V.D., Wall, P.: Some scales of equivalent conditions to characterize the Stieltjes inequality: the case \(q < p\). Math. Nachr. 287(2–3), 242–253 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Gorosito, O., Pradolini, G., Salinas, O.: Boundedness of fractional operators in weighted variable exponent spaces with non doubling measures. Czechoslov. Math. J. 60(4), 1007–1023 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Gorosito, O., Pradolini, G., Salinas, O.: Boundedness of the fractional maximal operator on variable exponent Lebesgue spaces: a short proof. Rev. Un. Mat. Argent. 53(1), 25–27 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hardy, G.H.: Note on a theorem of Hilbert concerning series of positive terms. Proc. Lond. Math. Soc. 23(2), 1 (1925). (Records of Proc. XLV-XLVI) Google Scholar
  26. 26.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). (Reprint of the 1952 edition) Google Scholar
  27. 27.
    Harman, A., Mamedov, F.I.: On boundedness of weighted Hardy operator in \(L^{p(\cdot )}\) and regularity condition. J. Inequal. Appl. 14, Art. ID 837951 (2010)Google Scholar
  28. 28.
    Hedberg, L.: On certain convolution inequalities. Proc. Am. Math. Soc. 36, 505–510 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Hytönen, T.: The \(A_2\) theorem: remarks and complements. In: Cifuentes, P., García-Cuerva, J., Garrigós, G., Hernández, E., Martell, J.M., Parcet, J., Rogers, K.M., Ruiz, A., Soria, F., Vargas, A. (eds.) Harmonic Analysis and Partial Differential Equations. Contemporary Mathematics, vol. 612, pp. 91–106. American Mathematical Society, Providence (2014)zbMATHCrossRefGoogle Scholar
  30. 30.
    Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslov. Math. J. 41(4), 592–618 (1991)zbMATHGoogle Scholar
  31. 31.
    Lerner, A.K.: On a dual property of the maximal operator on weighted variable \(L^p\) spaces. In: Cwikel, M., Milman, M. (eds.) Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth. Contemporary Mathematics, vol. 693, pp. 283–300. American Mathematical Society, Providence (2017)Google Scholar
  32. 32.
    Mamedov, F.I., Harman, A.: On a weighted inequality of Hardy type in spaces \(L^{p(\cdot )}\). J. Math. Anal. Appl. 353(2), 521–530 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Mamedov, F.I., Harman, A.: On a Hardy type general weighted inequality in spaces \(L^{p(\cdot )}\). Integral Eq. Oper. Theory 66(4), 565–592 (2010)zbMATHCrossRefGoogle Scholar
  34. 34.
    Mamedov, F.I., Mammadova, F.M., Aliyev, M.: Boundedness criterions for the Hardy operator in weighted \(L^{p(.)}(0,l)\) space. J. Convex Anal. 22(2), 553–568 (2015)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Mamedov, F.I., Zeren, Y.: On equivalent conditions for the general weighted Hardy type inequality in space \(L^{p(\cdot )}\). Z. Anal. Anwend. 31(1), 55–74 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Mashiyev, R.A., Çekiç, B., Mamedov, F.I., Ogras, S.: Hardy’s inequality in power-type weighted \(L^{p(\cdot )}(0,\infty )\) spaces. J. Math. Anal. Appl. 334(1), 289–298 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972). (Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Orlicz, W.: Über konjugierte exponentenfolgen. Studia Math. 3, 200–211 (1931)zbMATHCrossRefGoogle Scholar
  41. 41.
    Schur, J.: Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. J. Reine Angew. Math. 140, 1–28 (1911)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Sinnamon, G.: A note on the Stieltjes transformation. Proc. R. Soc. Edinb. Sect. A 110(1–2), 73–78 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Soria, F., Weiss, G.: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43(1), 187–204 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Widder, D.V.: The Laplace Transform. Princeton Mathematical Series, vol. 6, 2nd edn. Princeton University Press, Princeton (1946)Google Scholar

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

Personalised recommendations