Skip to main content
SpringerLink
Log in

Weights for One–Sided Operators

  • Chapter
  • First Online:
Book cover Recent Developments in Real and Harmonic Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.

    Article  Google Scholar 

  2. K.F. Andersen, Weighted inequalities for maximal functions associated with general measures, Trans. Amer. Math. Soc. 326 (1991), no. 2, 907–920.

    Article  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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. A.L. Bernardis, F.J. Martín-Reyes, Singular integrals in the Cesàro sense, J. Fourier Anal. Appl. 6 (2000), no. 2, 143–152.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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. J. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), no. 4, 405–408.

    Article  Google Scholar 

  13. A.-P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353.

    Article  Google Scholar 

  14. S. Chanillo, Weighted norm inequalities for strongly singular convolution operators, Trans. Amer. Math. Soc. 281 (1984), no. 1, 77–107.

    Article  MathSciNet  Google Scholar 

  15. R. Coifman, C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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. C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36.

    Article  MathSciNet  Google Scholar 

  20. E.V. Ferreyra, Weighted Lorentz norm inequalities for integral operators, Studia Math. 96 (1990), no. 2, 125–134.

    MathSciNet  Google Scholar 

  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. 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.

    Article  Google Scholar 

  23. J. García-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Mathematics Studies. 1985.

    Google Scholar 

  24. G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314–317.

    Article  Google Scholar 

  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. 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.

    Article  MathSciNet  Google Scholar 

  27. R.L. Jones, J. Rosenblatt, Differential and ergodic transforms, Math. Ann. 323 (2002), 525–546.

    Article  MathSciNet  Google Scholar 

  28. W.B. Jurkat, J.L. Troutman, Maximal inequalities related to generalized a.e. continuity, Trans. Amer. Math. Soc. 252 (1979), 49–64.

    Article  MathSciNet  Google Scholar 

  29. C.H. Kan, Ergodic properties of Lamperti operators, Canad. J. Math. 30 (1978), no. 6, 1206–1214.

    Article  MathSciNet  Google Scholar 

  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. A. Kufner, L.E. Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.

    Google Scholar 

  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. 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. 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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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. F.J. Martín-Reyes, L. Pick, A. de la Torre, <$>A_{\infty}^+<$> condition, Canad. J. Math. 45 (1993), no. 6, 1231–1244.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  39. F.J. Martín-Reyes, A. de la Torre, One sided BMO spaces, J. London Math. Soc. (2), 49 (1994), no. 3, 529–542.

    Article  Google Scholar 

  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. W.G. Maz'ya, Sobolev spaces, Springer-Verlag, 1985.

    Google Scholar 

  42. B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31–38.

    MathSciNet  Google Scholar 

  43. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  45. S. Ombrosi, C. Segovia, R. Testoni, An interpolation theorem between one-sided Hardy spaces, Ark. Mat. 44 (2006), no. 2, 335–348.

    Article  MathSciNet  Google Scholar 

  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. J.L. Rubio de Francia, Factorization and extrapolation of weights, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 393–395.

    Article  MathSciNet  Google Scholar 

  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. E. Sawyer, Weighted inequalities for the two-dimensional Hardy operator, Studia Math. 82 (1985), no. 1, 1–16.

    MathSciNet  Google Scholar 

  50. E. Sawyer, A weighted weak type inequality for the maximal function. Proc. Amer. Math. Soc. 93 (1985), no. 4, 610–614.

    Article  MathSciNet  Google Scholar 

  51. E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Trans. Amer. Math. Soc. 297 (1986), no. 1, 53–61.

    Article  MathSciNet  Google Scholar 

  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. C. Segovia, L. de Rosa, Dual spaces for one-sided weighted Hardy spaces, Rev. Un. Mat. Argentina 40 (1997), no. 3–4, 49–71.

    MathSciNet  Google Scholar 

  54. C. Segovia, L. de Rosa,Equivalence of norms in one-sided Hp i>spaces, Collect. Math. 53 (2002), no. 1, 1–20.

    MathSciNet  Google Scholar 

  55. C. Segovia, R. Testoni, One-sided strongly singular operators, Preprint.

    Google Scholar 

  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.

    MathSciNet  Google Scholar 

  57. P. Sjögren,A remark on the maximal function for measures in Rn, Amer. J. Math. 105 (1983), no. 5, 1231–1233.

    Article  Google Scholar 

  58. J. Strömberg, A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, 1381. Springer-Verlag, Berlin, 1989.

    Google Scholar 

  59. G. Talenti,Observazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. e Fis. Milano 39 (1969), 171–185.

    MathSciNet  Google Scholar 

  60. G. Tomaselli, A class of inequalities, Bul. Un. Mat. Ital. 21 (1969), 622–631.

    Google Scholar 

  61. A. de la Torre, J.L. Torrea, One-sided discrete square function, Studia Math. 156 (2003), no. 3, 3243–3260.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071, Málaga, Spain

    Francisco Javier Martín-Reyes, Pedro Ortega & Alberto la de Torre

Authors
  1. Francisco Javier Martín-Reyes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro Ortega
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alberto la de Torre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Javier Martín-Reyes .

Editor information

Editors and Affiliations

  1. Departamento de Matemátics, Universidad de Buenos Aires Fac. de Ciencias Exactas y Naturales, Buenos Aires, Argentina

    Carlos Cabrelli

  2. Depto. Matematicas, Universidad Autonoma de Madrid Facultad de Ciencias, Madrid, Spain

    Jose Luis Torrea

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Birkhäuser Boston

About this chapter

Cite this chapter

Martín-Reyes, F.J., Ortega, P., de Torre, A.l. (2010). Weights for One–Sided Operators. In: Cabrelli, C., Torrea, J. (eds) Recent Developments in Real and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4588-5_6

Download citation

Publish with us

Policies and ethics

Search

Navigation