Advertisement

Weighted norm inequalities for geometric fractional maximal operators

  • David Cruz-Uribe
  • C. J. Neugebauer
  • V. Olesen
Article

Abstract

For 0 ≤α < ∞ let Tαf denote one of the operators
$$M_{\alpha ,0} f(x) = \mathop {\sup }\limits_{I \mathrel\backepsilon x} \left| I \right|^\alpha \exp \left( {\frac{1}{{\left| I \right|}}\int_I {\log \left| f \right|} } \right),M_{\alpha ,0}^* f(x) = \mathop {\lim }\limits_{r \searrow 0} \mathop {\sup }\limits_{I \mathrel\backepsilon x} \left| I \right|^\alpha \left( {\frac{1}{{\left| I \right|}}\int_I {\left| f \right|^r } } \right)^{{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r}} .$$
We characterize the pairs of weights (u, v) for which Tα is a bounded operator from Lp(v) to Lq(u), 0 <p ≤q < ∞. This extends to α > 0 the norm inequalities for α=0 in [4, 16]. As an application we give lower bounds for convolutions ϕ ⋆ f, where ϕ is a radially decreasing function.

Math subject classifications

42B25 

Keywords and phrases

Fractional maximal operator weighted norm inequalities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Andersen, K.F. and Sawyer, E.T. (1988). Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integal operators,Trans. Am. Math. Soc.,308, 547–557.Google Scholar
  2. [2]
    Cruz-Uribe, D. (1997). A new proof of the two weight norm inequality for the one-sided fractional maximal operator,Proc. Am. Math. Soc.,125, 1419–1424.Google Scholar
  3. [3]
    Cruz-Uribe, D. and Neugebauer, C.J. (1995). The structure of the reverse Hölder classes,Trans. Am. Math. Soc.,347, 2941–2960.Google Scholar
  4. [4]
    Cruz-Uribe, D. and Neugebauer, C.J. (1998). Weighted norm inequalities for the geometric maximal operator,Publ. Mat.,42, 239–263.Google Scholar
  5. [5]
    Cruz-Uribe, D., Neugebauer, C. J., and Olesen, V. (1997). Norm Inequalities for the minimal and maximal operator, and differentiation of the integral,Publ. Mat.,41, 577–604.Google Scholar
  6. [6]
    Cruz-Uribe, D., Neugebauer, C.J., and Olesen, V. (1995). The one-sided minimal operator and the one-sided reverse Hölder inequality,Studia Math.,116, 255–270.Google Scholar
  7. [7]
    Cruz-Uribe, D., Neugebauer, C.J., and Olesen, V. (1997). Weighted norm inequalities for a family of one-sided minimal operators,Illinois J. Math.,41, 77–92.Google Scholar
  8. [8]
    García-Cuerva, J. and Rubio de Francia, J.L. (1985).Weighted Norm Inequalities and Related Topics, North-Holland,116.Google Scholar
  9. [9]
    Harboure, E., Macías, R.A., and Segovia, C. (1984). Boundedness of fractional operators on Lp spaces with different weights,TAMS,285, 629–647.Google Scholar
  10. [10]
    Martín-Reyes, F.J. and de la Torre, A. (1993). Two weight norm inequalities for fractional one-sided maximal operators,PAMS,117, 483–489.Google Scholar
  11. [11]
    Muckenhoupt, B. (1972). Weighted norm inequalities for the Hardy maximal function,Trans. Amer. Math. Soc.,165, 207–226.Google Scholar
  12. [12]
    Muckenhoupt, B. and Wheeden, R.L. (1974). Weighted norm inequalities for fractional integrals,Trans. Am. Math. Soc.,192, 261–274.Google Scholar
  13. [13]
    Sawyer, E.T. (1982). A characterization of a two weight norm inequality for maximal operators,Studia Math.,75, 1–11.Google Scholar
  14. [14]
    Stein, E.M. and Weiss, G. (1971).Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ.Google Scholar
  15. [15]
    Strömberg, J.O. and Wheeden, R.L. (1985). Fractional integrals on weighted Hp and Lp spaces,Trans. Am. Math. Soc.,287, 293–321.Google Scholar
  16. [16]
    Yin, X. and Muckenhoupt, B. (1996). Weighted inequalities for the maximal geometric mean operator,Proc. Am. Math. Soc.,124, 75–81.Google Scholar

Copyright information

© Birkhäuser 1999

Authors and Affiliations

  • David Cruz-Uribe
    • 1
  • C. J. Neugebauer
    • 2
  • V. Olesen
    • 2
  1. 1.Department of MathematicsTrinity CollegeHartford
  2. 2.Department of MathematicsPurdue UniversityWest Lafayette

Personalised recommendations