Advertisement

Positivity

, Volume 23, Issue 1, pp 125–138 | Cite as

Fractional maximal operators with weighted Hausdorff content

  • Hiroki SaitoEmail author
  • Hitoshi Tanaka
  • Toshikazu Watanabe
Article
  • 17 Downloads

Abstract

Let \(n\ge 2\) be the spatial dimension. The purpose of this note is to obtain some weighted estimates for the fractional maximal operator \({{\mathfrak {M}}}_{\alpha }\) of order \(\alpha \), \(0\le \alpha <n\), on the weighted Choquet–Lorentz space \(L^{p,q}(H^{d}_{w})\), where the weight w is arbitrary and the underlying measure is the weighted d-dimensional Hausdorff content \(H^{d}_{w}\), \(0<d\le n\). Concerning a dependence of two parameters \(\alpha \) and d, we establish a general form of the Fefferman–Stein type inequalities for \({{\mathfrak {M}}}_{\alpha }\). Our results contain the works of Adams (Publ Mat 42:3–66, 1998) and of Orobitg and Verdera (Bull Lond Math Soc 30(2):145–150, 1998) as the special cases. Our results also imply the Tang result (Georgian Math J 18(3):587–596, 2011), if we assume the weight w is in the Muckenhoupt \(A_{1}\)-class.

Keywords

Choquet integral Choquet–Lorentz space Fefferman–Stein inequality Maximal operator Weighted Hausdorff content 

Mathematics Subject Classification

42B25 

References

  1. 1.
    Adams, D.R.: Choquet integrals in potential theory. Publ. Mat. 42, 3–66 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Heidelberg (1996)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, London (1988)zbMATHGoogle Scholar
  4. 4.
    Cruz-Uribe, D.: SFO, new proofs of two-weight norm inequalities for the maximal operator. Georgian Math. J. 7, 33–42 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, vol. 249, 2nd edn. Springer, New York (2008)zbMATHGoogle Scholar
  6. 6.
    Hunt, R.A.: On \(L(p, q)\) spaces. Enseign. Math. 12, 249–276 (1966)zbMATHGoogle Scholar
  7. 7.
    Orobitg, J., Verdera, J.: Choquet integrals, Hausdorff content and the Hardy–Littlewood maximal operator. Bull. Lond. Math. Soc. 30(2), 145–150 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Saito, H., Tanaka, H., Watanabe, T.: Abstract dyadic cubes and the dyadic maximal operator with the Hausdorff content. Bull. Sci. Math. 140, 757–773 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sawyer, E.: Weighted norm inequalities for fractional maximal operators. Proc. C.M.S 1, 283–309 (1981)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Tang, L.: Choquet integrals, weighted Hausdorff content and maximal operators. Georgian Math. J. 18(3), 587–596 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics, vol. 1736. Springer, Berlin (2000)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.College of Science and TechnologyNihon UniversityFunabashiJapan
  2. 2.Research and Support Center on Higher Education for the Hearing and Visually ImpairedNational University Corporation Tsukuba University of TechnologyTsukubaJapan
  3. 3.College of Science and TechnologyNihon UniversityTokyoJapan

Personalised recommendations