Weighted inequalities for integral operators on Lebesgue and \(BMO^{\gamma }(\omega )\) spaces



We characterize the power weights \(\omega \) for which the fractional type operator \(T_{\alpha ,\beta }\) is bounded from \(L^p (\omega ^p)\) into \(L^q (\omega ^q)\) for \(1< p < n/(n- (\alpha + \beta ))\) and \(1/q = 1/p - (n- (\alpha + \beta ))/n\). If \(n/(n-(\alpha + \beta )) \le p < n/(n -(\alpha +\beta ) -1)^{+}\) we prove that \(T_{\alpha ,\beta }\) is bounded from a weighted weak \(L^p\) space into a suitable weighted \(BMO^\delta \) space for weights satisfying a doubling condition and a reverse Hölder condition. Also, we prove the boundedness of \(T_{\alpha ,\beta }\) from a weighted local space \(BMO_{0}^{\gamma }\) into a weighted \(BMO^\delta \) space, for weights satisfying a doubling condition.


BMO spaces Lebesgue spaces Weighted inequalities Integral operators 

Mathematics Subject Classification

30H35 42B25 42B35 



The authors wish to thank Eleonor Harboure and the referee for helpful comments and suggestions.


  1. 1.
    Bradley, S.: Hardy inequalities with mixed norms. Canad. Math. l. 21(4), 405–408 (1978)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Drábek, P., Heinig, H.P., Kufner, A.: Higher-dimensional ardy inequality, General inequalities, 7 (Oberwolfach: Internat. er. Numer. Math., vol. 123. Birkhäuser, Basel 1997 3–16 (1995)Google Scholar
  3. 3.
    Fefferman, C., Muckenhoupt, B.: Two nonequivalent conditions or weight functions. Proc. Am. Math. Soc. 45, 99–104 (1974)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ferreyra, E., Flores, G.: Weighted estimates for integral perators on local BMO type spaces. Math. Nachr. 288(8–9), 905–916 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Godoy, T., Urciuolo, M.: About the \(L^p\)-boundedness of some integral perators. Rev. Un. Mat. Argent. 38(3–4), 192–195 (1993)MATHGoogle Scholar
  6. 6.
    Harboure, E., Salinas, O., Viviani, B.: Boundedness of the ractional integral on weighted Lebesgue and Lipschitz spaces. Trans. Am. Math. Soc. 349(1), 235–255 (1997)CrossRefMATHGoogle Scholar
  7. 7.
    Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for ractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)CrossRefMATHGoogle Scholar
  8. 8.
    Muckenhoupt, B., Wheeden, R.: Weighted bounded mean oscillation and the Hilbert transform. Stud. Math. 54(3), 221–237 (1976)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ricci, F., Sjögren, P.: Two-parameter maximal functions in the Heisenberg group. Math. Z. 199(4), 565–575 (1988)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Riveros, M.S., Urciuolo, M.: Weighted inequalities or fractional type operators with some homogeneous kernels. Acta Math. Sin. Engl. Ser. 29(3), 449–460 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2018

Authors and Affiliations

  1. 1.CIEM (CONICET), FaMAFUniversidad Nacional de CórdobaCórdobaArgentina

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