Advertisement

Weights for One–Sided Operators

  • Francisco Javier Martín-Reyes
  • Pedro Ortega
  • Alberto la de Torre
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We present a survey about weights for one-sided operators, one of the areas in which Carlos Segovia made significant contributions. The classical Dunford–Schwartz ergodic theorem can be considered as the first result about weights for the one-sided Hardy–Littlewood maximal operator. From this starting point, we study weighted inequalities for one-sided operators: positive operators like the Hardy averaging operator, the one-sided Hardy–Littlewood maximal operator, singular approximations of the identity, one-sided singular integrals. We end with applications to ergodic theory and with some recent results in dimensions greater than 1.

Keywords

Maximal Operator Maximal Function Singular Integral Weak Type Hardy Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Aimar, L. Forzani, F.J. Martín-Reyes On weighted inequalities for singular integrals, Proc. Amer. Math. Soc. 125 (1997), no. 7, 2057–2064.CrossRefGoogle Scholar
  2. 2.
    K.F. Andersen, Weighted inequalities for maximal functions associated with general measures, Trans. Amer. Math. Soc. 326 (1991), no. 2, 907–920.CrossRefGoogle Scholar
  3. 3.
    K.F. Andersen, B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9–26.MathSciNetGoogle Scholar
  4. 4.
    K.F. Andersen, E. Sawyer, Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), no. 2, 547–558.MathSciNetGoogle Scholar
  5. 5.
    A.L. Bernardis, M. Lorente, F.J. Martín-Reyes, M.T. Martinez, A. de la Torre, J.L. Torrea, Differential transforms in weighted spaces, J. Fourier Anal. Appl. 12 (2006), no. 1, 83–103.CrossRefGoogle Scholar
  6. 6.
    A.L. Bernardis, M. Lorente, F.J. Martín-Reyes, M.T. Martinez, A. de la Torre, Differences of ergodic averages for Cesàro bounded operators, Q. J. Math, 58 (2007), no. 2, 137–150.Google Scholar
  7. 7.
    A.L. Bernardis, F.J. Martín-Reyes, Singular integrals in the Cesàro sense, J. Fourier Anal. Appl. 6 (2000), no. 2, 143–152.CrossRefGoogle Scholar
  8. 8.
    A.L. Bernardis, F.J. Martín-Reyes, Weighted inequalities for a maximal function on the real line, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 2, 267–277.CrossRefGoogle Scholar
  9. 9.
    A.L. Bernardis, F.J. Martín-Reyes, Two weighted inequalities for convolution maximal operators, Publ. Mat. 46 (2002), no. 1, 119–138.Google Scholar
  10. 10.
    A.L. Bernardis, F.J. Martín-Reyes,Restricted weak type inequalities for convolution maximal operators in weighted Lp spaces, Q. J. Math. 54 (2003), no. 2, 139–157.CrossRefGoogle Scholar
  11. 11.
    A.L. Bernardis, F.J. Martín-Reyes, Differential transforms of Cesàro averages in weighted spaces, Publ. Mat. 52 (2008), no. 1, 101–127.Google Scholar
  12. 12.
    J. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), no. 4, 405–408.CrossRefGoogle Scholar
  13. 13.
    A.-P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353.CrossRefGoogle Scholar
  14. 14.
    S. Chanillo, Weighted norm inequalities for strongly singular convolution operators, Trans. Amer. Math. Soc. 281 (1984), no. 1, 77–107.CrossRefMathSciNetGoogle Scholar
  15. 15.
    R. Coifman, C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.MathSciNetGoogle Scholar
  16. 16.
    D. Cruz-Uribe, The class <$>A_{\infty}^+<$> (g) and the one-sided reverse Hölder inequality, Canad. Math. Bull. 40 (1997), no. 2, 169–173.CrossRefMathSciNetGoogle Scholar
  17. 17.
    D. Cruz-Uribe, C. Neugebauer, V. Olesen, The one-sided minimal operator and the one-sided reverse Hölder inequality, Studia Math. 116 (1995), no. 3, 255–270.MathSciNetGoogle Scholar
  18. 18.
    N. Dunford, J.T. Schwartz, Linear Operators I. General Theory, Pure and Applied Mathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London 1958.Google Scholar
  19. 19.
    C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36.CrossRefMathSciNetGoogle Scholar
  20. 20.
    E.V. Ferreyra, Weighted Lorentz norm inequalities for integral operators, Studia Math. 96 (1990), no. 2, 125–134.MathSciNetGoogle Scholar
  21. 21.
    L. Forzani, S. Ombrosi, F.J. Martín-Reyes, Weighted inequalities for the two-dimensional one-sided Hardy-Littlewood maximal function, preprint.Google Scholar
  22. 22.
    J. García-Cuerva, E. Harboure, C. Segovia, J.L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991), no. 4, 1397–1420.CrossRefGoogle Scholar
  23. 23.
    J. García-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Mathematics Studies. 1985.Google Scholar
  24. 24.
    G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314–317.CrossRefGoogle Scholar
  25. 25.
    R.A. Hunt, D.S. Kurtz, C.J. Neugebauer, A note on the equivalence of A p and Sawyer's condition for equal weights, Conf. Harmonic Analysis in honor of A. Zygmund, Wadsworth Inc. 1981, 156–158.Google Scholar
  26. 26.
    R.A. Hunt, B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251.CrossRefMathSciNetGoogle Scholar
  27. 27.
    R.L. Jones, J. Rosenblatt, Differential and ergodic transforms, Math. Ann. 323 (2002), 525–546.CrossRefMathSciNetGoogle Scholar
  28. 28.
    W.B. Jurkat, J.L. Troutman, Maximal inequalities related to generalized a.e. continuity, Trans. Amer. Math. Soc. 252 (1979), 49–64.CrossRefMathSciNetGoogle Scholar
  29. 29.
    C.H. Kan, Ergodic properties of Lamperti operators, Canad. J. Math. 30 (1978), no. 6, 1206–1214.CrossRefMathSciNetGoogle Scholar
  30. 30.
    A. Kufner, L. Maligranda, L.E. Persson, The Hardy inequality—About its history and some related results, Vydavatelský Servis, Plzen, 2007.Google Scholar
  31. 31.
    A. Kufner, L.E. Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.Google Scholar
  32. 32.
    F.J. Martín-Reyes, New proofs of weighted inequalities for one-sided Hardy–Littlewood maximal functions, Proc. Amer. Math. Soc. 117 (1993), no. 3, 691–698.Google Scholar
  33. 33.
    F.J. Martín-Reyes, On the one-sided Hardy-Littlewood maximal function in the real line and in dimensions greater than one, Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992), Stud. Adv. Math., CRC, Boca Raton, FL, 1995, 237–250.Google Scholar
  34. 34.
    F.J. Martín-Reyes, P. Ortega, On weighted weak type inequalities for modified Hardy operators, Proc. Amer. Math. Soc. 126 (1998), no. 6 1739–1746.CrossRefGoogle Scholar
  35. 35.
    F.J. Martín-Reyes, P. Ortega Salvador, M.D. Sarrión Gavilán, Boundedness of operators of Hardy type in Λp,q spaces and weighted mixed inequalities for singular integral operators, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 1, 157–170.CrossRefGoogle Scholar
  36. 36.
    F.J. Martín-Reyes, P. Ortega Salvador, A. de la Torre, Weighted inequalities for one-sided maximal functions, Trans. Amer. Math. Soc. 319, (1990), no. 2, 517–534.Google Scholar
  37. 37.
    F.J. Martín-Reyes, L. Pick, A. de la Torre, <$>A_{\infty}^+<$> condition, Canad. J. Math. 45 (1993), no. 6, 1231–1244.CrossRefGoogle Scholar
  38. 38.
    F.J. Martín-Reyes, A. de la Torre, The dominated ergodic estimate for mean bounded, invertible, positive operators, Proc. Am. Math. Soc. 104 (1988), no. 1, 69–75.CrossRefGoogle Scholar
  39. 39.
    F.J. Martín-Reyes, A. de la Torre, One sided BMO spaces, J. London Math. Soc. (2), 49 (1994), no. 3, 529–542.CrossRefGoogle Scholar
  40. 40.
    F.J. Martín-Reyes, A. de la Torre, Some weighted inequalities for general onesided maximal operators, Studia Math. 122 (1997), no. 1, 1–14.Google Scholar
  41. 41.
    W.G. Maz'ya, Sobolev spaces, Springer-Verlag, 1985.Google Scholar
  42. 42.
    B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31–38.MathSciNetGoogle Scholar
  43. 43.
    B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.CrossRefMathSciNetGoogle Scholar
  44. 44.
    S. Ombrosi, Weak weighted inequalities for a dyadic one-sided maximal function in Rn, Proc. Amer. Math. Soc. 133 (2005), no. 6, 1769–1775.CrossRefMathSciNetGoogle Scholar
  45. 45.
    S. Ombrosi, C. Segovia, R. Testoni, An interpolation theorem between one-sided Hardy spaces, Ark. Mat. 44 (2006), no. 2, 335–348.CrossRefMathSciNetGoogle Scholar
  46. 46.
    B. Opic, A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Harlow, Essex, England, 1990.Google Scholar
  47. 47.
    J.L. Rubio de Francia, Factorization and extrapolation of weights, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 393–395.CrossRefMathSciNetGoogle Scholar
  48. 48.
    E. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), no. 1, 329–337.Google Scholar
  49. 49.
    E. Sawyer, Weighted inequalities for the two-dimensional Hardy operator, Studia Math. 82 (1985), no. 1, 1–16.MathSciNetGoogle Scholar
  50. 50.
    E. Sawyer, A weighted weak type inequality for the maximal function. Proc. Amer. Math. Soc. 93 (1985), no. 4, 610–614.CrossRefMathSciNetGoogle Scholar
  51. 51.
    E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Trans. Amer. Math. Soc. 297 (1986), no. 1, 53–61.CrossRefMathSciNetGoogle Scholar
  52. 52.
    C. Segovia, L. de Rosa,Weighted Hp spaces for one sided maximal functions, Harmonic analysis and operator theory (Caracas, 1994), 161–183, Contemp. Math., 189, Amer. Math. Soc., Providence, RI, 1995.Google Scholar
  53. 53.
    C. Segovia, L. de Rosa, Dual spaces for one-sided weighted Hardy spaces, Rev. Un. Mat. Argentina 40 (1997), no. 3–4, 49–71.MathSciNetGoogle Scholar
  54. 54.
    C. Segovia, L. de Rosa,Equivalence of norms in one-sided Hp i>spaces, Collect. Math. 53 (2002), no. 1, 1–20.MathSciNetGoogle Scholar
  55. 55.
    C. Segovia, R. Testoni, One-sided strongly singular operators, Preprint.Google Scholar
  56. 56.
    C. Segovia, R. Testoni, A multiplier theorem for one-sided Hardy spaces, Proceedings of the Royal Society of Edinburgh: Section A, 139A, (2009), 209–223.MathSciNetGoogle Scholar
  57. 57.
    P. Sjögren,A remark on the maximal function for measures in Rn, Amer. J. Math. 105 (1983), no. 5, 1231–1233.CrossRefGoogle Scholar
  58. 58.
    J. Strömberg, A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, 1381. Springer-Verlag, Berlin, 1989.Google Scholar
  59. 59.
    G. Talenti,Observazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. e Fis. Milano 39 (1969), 171–185.MathSciNetGoogle Scholar
  60. 60.
    G. Tomaselli, A class of inequalities, Bul. Un. Mat. Ital. 21 (1969), 622–631.Google Scholar
  61. 61.
    A. de la Torre, J.L. Torrea, One-sided discrete square function, Studia Math. 156 (2003), no. 3, 3243–3260.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2010

Authors and Affiliations

  • Francisco Javier Martín-Reyes
    • 1
  • Pedro Ortega
    • 1
  • Alberto la de Torre
    • 1
  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de MálagaMálagaSpain

Personalised recommendations