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New Weighted Estimates for the Disc Multiplier on Radial Functions

  • María J. Carro
  • Carmen Ortiz-Caraballo
Article
  • 44 Downloads

Abstract

We prove a weighted estimate for the disc multiplier, acting on radial functions, at the extreme points \(p_{-}=\frac{2n}{n+1}\), extending the result of Chanillo (J Funct Anal 55:18–24, 1984). To this end, we prove a restricted weak type weighted estimate for \(p=2\) and then develop a new extrapolation result of independent interest.

Keywords

Rubio de Francia Extrapolation \(A_p\) weights Hardy–Littlewood maximal function Radial functions Disc multiplier 

Mathematics Subject Classification

42B35 46E30 

References

  1. 1.
    Andersen, K.F.: Weighted inequalities for the disc multiplier. Proc. Am. Math. Soc. 83(2), 269–275 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andersen, K.F., Muckenhoupt, B.: Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions. Studia Math. 72(1), 9–26 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: harmonic analysis of elliptic operators. J. Funct. Anal. 241, 703–746 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, vol. 129. Academic Press Inc, Boston, MA (1988)zbMATHGoogle Scholar
  5. 5.
    Carro, M.J., Soria, J.: Restricted weak type Rubio de Francia extrapolation for \(p>p_0\) with applications to exponential integrability estimates. Adv. Math. 290, 888–918 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carro, M.J., Duoandikoetxea, J., Lorente, M.: Weighted estimates in a limited range with applications to the Bochner-Riesz operators. Indiana Univ. Math. J. 61, 1485–1511 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carro, M.J., Grafakos, L., Soria, J.: Weighted weak type \((1,1)\) estimates via Rubio de Francia extrapolation. J. Funct. Anal. 269, 1203–1233 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chanillo, S.: The multiplier for the ball and radial functions. J. Func. Anal. 55, 18–24 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chung, H.M., Hunt, R.A., Kurtz, S.D.: The Hardy-Littlewood maximal function on L(p, q) spaces with weights. Indiana Univ. Math. J. 31(1), 109–120 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cruz-Uribe, D., Martell, J.M., and Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia, No. 223. Operator Theory: Advances and Applications, vol. 215. Springer (2011)Google Scholar
  11. 11.
    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Weighted weak type inequalities and a conjecture of Sawyer. Int. Math. Res. Not. 30, 1849–1871 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Duoandikoetxea, J., Moyua, A., Oruetxebarria, O., Seijo, E.: Radial \(A_p\) weights with applications to the Disc multiplier and the Bochner-Riesz operators. Indiana Univ. Math. J. 57(3), 1261–1281 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fefferman, C.: The multiplier problem for the ball. Ann. Math. (2) 94(4), 330–336 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 116, Notas de Matemática [Mathematical Notes], vol. 104. North-Holland Publishing Co., Amsterdam (1985)Google Scholar
  15. 15.
    García-Cuerva, J.: An extrapolation theorem in the theory of \(A_p\) weights. Proc. Am. Math. Soc. 87, 422–426 (1983)zbMATHGoogle Scholar
  16. 16.
    Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, 2nd edn, vol. 250. Springer, New York (2009)Google Scholar
  17. 17.
    Herz, C.: On the mean inversion of Fourier an Hankel transforms. Proc. Natl. Acad. Sci. 40, 996–999 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jones, P.: Factorization of \(A_p\) weights. Ann. Math. (2) 111(3), 511–530 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kenig, C., Tomas, P.: The weak behavior of spherical means. Proc Am. Math. Soc 78(1), 48–50 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kerman, R., Torchinsky, A.: Integral inequalities with weights for the Hardy maximal function. Studia Math. 71(2), 277–284 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, K., Pérez, C., Ombrosi, S.: Proof of an extension of E. Sawyer’s Conjecture about weighted mixed weak-type estimates. https://arxiv.org/abs/1703.01530
  22. 22.
    Mockenhaupt, G.: On radial weights for the spherical summation operator. J. Funct. Anal. 91, 174–181 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Prestini, E.: Almost everywhere convergence of the spherical partial sums for radial functions. Monatsh. Math. 105(3), 207–216 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Romera, E., Soria, F.: Endpoint estimates for the maximal operator associated to spherical partial sums on radial functions. Proc. Am. Math. Soc. 111(4), 1015–1022 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rubio de Francia, J.L.: Factorization theory and \(A_p\) weights. Am. J. Math. 106(3), 533–547 (1984)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and AnalysisUniversity of BarcelonaBarcelonaSpain
  2. 2.School of Technology of IgualadaBarcelonaTechIgualadaSpain

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