Potential Analysis

, Volume 41, Issue 3, pp 869–885 | Cite as

Weighted Local Estimates for Fractional Type Operators

  • Alberto Torchinsky


In this note we prove the estimate \(M^{\sharp }_{0,s}(Tf)(x) \le c\,M_{\gamma } f(x)\) for general fractional type operators T, where \(M^{\sharp }_{0,s}\) is the local sharp maximal function and M γ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of T f by M γ f. Similar estimates hold for T replaced by fractional type operators with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and M γ replaced by an appropriate maximal function M T . We also prove two-weight, \({L^{p}_{\text {\textit {v}}}}\)-\({L^{q}_{\text {\textit {w}}}}\) estimates for the fractional type operators described above for 1 < p < q < and a range of q. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.


Fractional operators Maximal fractional function Local maximal function Orlicz-Morrey spaces 

Mathematics Subject Classifications (2010)

Primary 42B25 26A33 Secondary 31B10 42B35 


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  1. 1.
    Adams, D.R.: A note on Riesz potentials. Duke Math J. 42, 765–778 (1975)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Adams, D.R.: Weighted nonlinear potential theory. Trans. Amer. Math. Soc. 297(1), 73–94 (1986)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. 50(2), 201–230 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bernardis, A.L., Lorente, M., Riveros, M.S.: Weighted inequalities for fractional integral operators with kernels satisfying Hörmander type conditions. Math. Inequal. Appl. 14(4), 881–895 (2011)MathSciNetMATHGoogle Scholar
  5. 5.
    Chanillo, S., Watson, D., Wheeden, R.L.: Some integral and maximal operators related to starlike sets. Studia Math. 107, 223–255 (1993)MathSciNetMATHGoogle Scholar
  6. 6.
    Cruz-Uribe, D.: A fractional Muckenhoupt-Wheeden theorem and its consequences. Integr. Equ. Oper. Theory 76(3), 421–446 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ding, Y., Lu, S.: Boundedness of homogenous fractional integrals on L p for n/αp. Nagoya Mat. J. 167, 17–33 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Fujii, N.: A proof of the Fefferman-Stein-Strömberg inequality for the sharp maximal function. Proc. Amer. Math. Soc. 106(2), 371–377 (1989)MathSciNetMATHGoogle Scholar
  9. 9.
    Fujii, N.: A condition for the two-weight norm inequality for singular integral operators. Studia Math. 98(3), 175–190 (1991)MathSciNetMATHGoogle Scholar
  10. 10.
    Fujii, N.: Strong type estimation from weak type estimates for some integral operators. Proceedings of the Second ISAAC Congress, Vol. 1 (Fukuoka, 1999), 25-30, Int. Soc. Anal. Appl. Comput., 7, Kluwer Acad. Publ., Dordrecht (2000)Google Scholar
  11. 11.
    García-Cuerva, J., Martell, J.M.: Two-weight norm inequalities for maximal operators and fractional integrals on non-homogenous spaces. Indiana Univ. Math. J. 50(3), 1241–1280 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gogatishvili, A., Mustafayev, R.: Equivalence of norms of Riesz potential and fractional maximal function in Morrey-type spaces. Collect. Math. 63(1), 11–28 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Guliyev, V.S., Aliyev, S.S., Karaman, T., Shukurov, P.S.: Boundedness of sublinear operators and commutators on generalized Morrey spaces. Integr. Equ. Oper. Theory 71, 327–355 (2011). doi: 10.1007/s00020-011-1904-1 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Guliyev, V.S., Shukurov, P.S.: Adams type result for sublinear operators generated by Riesz potentials on generalized Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 32(1), Mathematics, 61–70 (2012)MathSciNetGoogle Scholar
  15. 15.
    Guliyev, V.S., Shukurov, P.S.: On the boundedness of the fractional maximal operator, Riesz potential and their commutators in generalized Morrey spaces. Oper. Theory Adv. Appl. 229, 175–199 (2013)MathSciNetGoogle Scholar
  16. 16.
    Harboure, E., Macías, R.A., Segovia, C.: Boundedness of fractional operators on L p spaces with different weights. Trans. Amer. Math. Soc. 285, 629–647 (1984)MathSciNetMATHGoogle Scholar
  17. 17.
    Jawerth, B., Torchinsky, A.: Local sharp maximal functions. J. Approx. Theory 43(3), 231–270 (1985)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kurtz, D.S., Wheeden, R.L.: Results for weighted norm inequalities for multipliers. Trans. Amer. Math. Soc. 255, 343–362 (1979)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lerner, A.K.: On the John-Strömberg characterization of BMO for nondoubling measures. Real. Anal. Exch. 28(2), 649–660 (2002/2003)MathSciNetGoogle Scholar
  20. 20.
    Lerner, A.K.: A pointwise estimate for the local sharp maximal function with applications to singular integrals. Bull. Lond. Math. Soc. 42(5), 843–856 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Muckenhoup, B., Wheeden, R.: Weighted norm inequaliteies for fractional integrals. Trans. Amer. Math. Soc. 192, 261–274 (1974)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nakai, E.: On generalized fractional integrals. Taiwan. J. Math. 5(3), 587–602 (2001)MathSciNetMATHGoogle Scholar
  23. 23.
    Pérez, C.: On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted L p-spaces with different weights. Proc. London Math. Soc. (3) 71(1), 135–157 (1995)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pérez, C.: Two weighed inequalities for potential and fractional type maximal operators. Indiana U. Math. J. 43, 1–28 (1994)CrossRefGoogle Scholar
  25. 25.
    Pérez, C.: Sharp L p–weighted Sobolev inequalities. Ann. Inst. Fourier (Grenoble) 45(3), 809–824 (1995)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Poelhuis, J., Torchinsky, A.: Medians, continuity, and vanishing oscillation. Studia Math. 213, 227–242 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Poelhuis, J., Torchinsky, A.: Weighted local estimates for singular integral operators. arXiv:1308.1134v2
  28. 28.
    Rakotondratsimba, Y.: Local weighted inequalities for the fractional integral operator. Kobe J. Math. 17, 153–189 (2000)MathSciNetMATHGoogle Scholar
  29. 29.
    Riveros, M.S.: Weighted Inequalities for generalized fractional operators. Rev. Un. Mat. Argent. 49(2), 29–38 (2009)MathSciNetGoogle Scholar
  30. 30.
    Riveros, M.S., Urciuolo, M.: Weighted inequalities for fractional type operators with some homogenous kernels. Acta Math. Sin. (Engl. Ser.) 29(3), 449–460 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sawano, Y., Sugano, S., Tanaka, H.: Orlicz-Morrey spaces and fractional operators. Potential Anal. 36, 517–556 (2012). doi: 10.1007/s11118-011-9239-8 MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Sawyer, E.: A characterization of a two-weight norm inequality for maximal operators. Studia Math. 75(1), 1–11 (1982)MathSciNetMATHGoogle Scholar
  33. 33.
    Sawyer, E.T., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogenous spaces. Amer. J. Math. 144, 813–874 (1992)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Shi, X.L., Torchinsky, A.: Local sharp maximal functions in spaces of homogeneous type. Sci. Sinica Ser. A 30(5), 473–480 (1987)MathSciNetMATHGoogle Scholar
  35. 35.
    Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms duality of Hardy spaces. Indiana Univ. Math. J. 28(3), 511–544 (1979)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Strömberg, J.-O., Torchinsky, A: Weighted hardy spaces. In: Lecture Notes in Mathematics, pp. 1381. Springer-Verlag, Berlin (1989)Google Scholar
  37. 37.
    Torchinsky, A.: Interpolation of operations and Orlicz classes. Studia Math. 59, 177–207 (1976/77)MathSciNetGoogle Scholar
  38. 38.
    Torchinsky, A.: Real-variable methods in harmonic analysis, Pure and Applied Mathematics, 123, Academic Press, Inc., Orlando, FL 1986, (Reprinted by Dover in 2004)Google Scholar
  39. 39.
    Trujillo-González, R.: Two-weight norm inequalities for fractional maximal operators on spaces of generalized homogenous type. Period. Math. Hungar. 44(1), 101–110 (2002)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

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