Skip to main content
SpringerLink
Log in

Weighted Inequalities and Dyadic Harmonic Analysis

  • Chapter
  • First Online:
Excursions in Harmonic Analysis, Volume 2

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

Abstract

We survey the recent solution of the so-called A 2 conjecture, that states: all Calderón–Zygmund singular integral operators are bounded on L 2(w) with a bound that depends linearly on the A 2 characteristic of the weight w. We also survey corresponding results for commutators. We highlight the interplay of dyadic harmonic analysis in the solution of the A 2 conjecture, especially Hytönen’s representation theorem for Calderón–Zygmund singular integral operators in terms of Haar shift operators. We describe Chung’s dyadic proof of the corresponding quadratic bound on L 2(w) for the commutator of the Hilbert transform with a BMO function, and we deduce sharpness of the bounds for the dyadic paraproduct on L p(w) that were obtained extrapolating Beznosova’s linear bound on L 2(w). We show that if an operator T is bounded on the weighted Lebesgue space L r(w) and its operator norm is bounded by a power α of the A r characteristic of the weight, then its commutator [T,b] with a function b in BMO will be bounded on L r(w) with an operator norm bounded by the increased power \(\alpha +\max \{ 1, \frac{1} {r-1}\}\) of the A r characteristic of the weight. The results are sharp in terms of the growth of the operator norm with respect to the A r characteristic of the weight for all 1<r<∞.

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 EPUB and 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

References

  1. Alvarez, J., Bagby, R., Kurtz, D., Perez, C.: Weighted estimates for commutators of linear operators. Studia Mat. 104(2), 195–209 (1994)

    MathSciNet  Google Scholar 

  2. Astala, K., Iwaniec, T., Saksman, E.: Beltrami operators in the plane. Duke Math. J. 107(1), 27–56 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beylkin, G., Coifman, R., Rokhlyn, V., Fast Wavelet Transforms and Numerical Algorithms I. Comm. Pure Appl. Math. 44(2), 141–183 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beznosova, O., Linear bound for dyadic paraproduct on weighted Lebesgue space L 2(w). J. Func. Anal. 255(4), 994–1007 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bony, J.: Calcul symbolique et propagation des singularités pour les equations aux dérivées non-linéaires. Ann. Sci. Ecole Norm. Sup. 14, 209–246 (1981)

    MathSciNet  MATH  Google Scholar 

  6. Buckley, S.: Estimates for operator norms and reverse Jensen’s inequalities. Trans. Amer. Math. Soc. 340(1), 253–272 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burkholder, D.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12(3), 647–702 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chung, D.: Sharp estimates for the commutators of the Hilbert, Riesz and Beurling transforms on weighted Lebesgue spaces. To appear in Indiana U. Math. J. 60(5), (2011)

    Google Scholar 

  9. Chung, D.: Weighted inequalities for multivariable dyadic paraproducts. Pub. Mat. 55(2), 475–499 (2011)

    MATH  Google Scholar 

  10. Chung, D., Pereyra, M.C., Pérez, C.: Sharp bounds for general commutators on weighted Lebesgue spaces. Trans. Amer. Math. Soc. 364(3), 1163–1177 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cotlar, M., Sadosky, C.: On the Helson–Szegö theorem and a related class of modified Toeplitz kernels. Harmonic analysis in Euclidean spaces (Proceedings of Symposium on Pure Mathematics, Williams College, Williamstown, MA, 1978), Part 1, pp. 383–407. Proceedings of Symposium on Pure Mathematics, vol. XXXV, Part, American Mathematical Society, Providence, RI (1979)

    Google Scholar 

  14. Cotlar, M., Sadosky, C.: On some L p versions of the Helson-Szegö theorem. In: Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, IL, 1981), pp. 306–317. Wadsworth Mathematical Series, Wadsworth, Belmont, CA (1983)

    Google Scholar 

  15. Cruz-Uribe, D., Pérez, C.: On the two-weight problem for singular integral operators. Ann. Scuola Norm. Super. Pisa Cl. Sci. (5) I, 821–849 (2002)

    Google Scholar 

  16. Cruz-Uribe, D., SFO, Moen, K., Sharp norm inequalities for commutators of classical operators. Pub. Mat. 56(1), 147–190 (2012)

    Google Scholar 

  17. Cruz-Uribe, D., SFO, Martell, J.M., Pérez, C.: Sharp weighted estimates for approximating dyadic operators. Electron. Res. Announc. Math. Sci. 17, 12–19 (2010)

    Google Scholar 

  18. Cruz-Uribe, D., SFO, Martell, J., Pérez, C.: Sharp weighted estimates for classical operators. Adv. Math. 229(1), 408–441 (2012)

    Google Scholar 

  19. David, G., Journé, J.-L.: A boundedness criterion for generalized Calderón–Zygmund operators. Ann. Math. 120, 371–397 (1984)

    Article  MATH  Google Scholar 

  20. Dragic̆ević, O., Grafakos, L., Pereyra, M.C., Petermichl, S.: Extrapolation and sharp norm estimates for classical operators on weighted Lebegue spaces. Publ. Math 49(1), 73–91 (2005)

    Google Scholar 

  21. Duoandikoetxea, J., Fourier Analysis. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001)

    Google Scholar 

  22. Duoandikoetxea, J.: Extrapolation of weights revisited: new proofs and sharp bounds. J. Funct. Anal. 260(6), 1886–1901 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fefferman, C., Stein, E.: H p spaces of several variables. Acta Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Figiel, T.: Singular intergral operators: a martingale approach. In: Geometry of Banach Spaces (Strbl, 1989), London Mathematical Society, vol. 158. Lecture Notes Series, pp. 95–110. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  25. Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studies, vol. 116. North-Holland, Amsterdam, Holland (1981)

    Google Scholar 

  26. Grafakos, L.: Classical Fourier Analysis. 2nd edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2008)

    Google Scholar 

  27. Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250, 2nd edn. Springer, New York (2009)

    Google Scholar 

  28. Haar, A.: Zur Theorie der orthogonalen Funktionen systeme. Math. Ann. 69, 331–371 (1910)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hernandez, E., Weiss, G.: A First Course on Wavelets. 3rd edn. Academic, New York (2008)

    Google Scholar 

  30. Hukovic, S., Treil, S., Volberg, A.: The Bellman function and sharp weighted inequalities for square functions. In: Complex Analysis, Operators and Related Topics. Operator Theory: Advances and Applications, vol. 113, pp. 97–113. Birkhäuser, Basel (2000)

    Google Scholar 

  31. Hunt, R., Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for the conjugate function and the Hilbert transform. Trans. Amer. Math. Soc. 176, 227–252 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hytönen, T.P.: On Petermichl’s dyadic shift and the Hilbert transform. C. R. Acad. de Sci. Paris, Ser. I 346(21–22), 1133–1136 (2008)

    Google Scholar 

  33. Hytönen, T.P.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. (2) 175(3), 1473–1506 (2012)

    Google Scholar 

  34. Hytönen, T.P.: Representation of singular integrals by dyadic operators, and the A 2 theorem. Preprint (2011) available at arXiv:1108.5119

    Google Scholar 

  35. Hytönen, T.P., Lacey, M.T.: The A p  − A ∞ inequality for general Calderón–Zygmund operators. Preprint (2011) available at arXiv:1106.4797

    Google Scholar 

  36. Hytönen, T.P., Pérez, C.: Sharp weighted bounds involving A ∞ . To appear Journal of Analysis and P.D.E., available at arXiv:1103.5562

    Google Scholar 

  37. Hytönen, T.P., Lacey, M.T., Martikainen, H., Orponen, T., Reguera, M.C., Sawyer, E.T., Uriarte-Tuero, I.: Weak and strong type estimates for maximal truncations of Calderón–Zygmund operators on A p weighted spaces. Preprint (2011) available at arXiv:1103.5229

    Google Scholar 

  38. Hytönen, T.P., Pérez, C., Treil, S., Volberg, A.: Sharp weighted estimates for dyadic shifts and the A 2 conjecture. To appear in Journal für die Reine und Angewandte Mathematik, available at arXiv:1010.0755

    Google Scholar 

  39. John, F., Niremberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415–426 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  40. Katz, N.H., Pereyra, M.C.: On the two weights problem for the Hilbert transform. Rev. Mat. Iberoamericana 13(1), 211–243 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lacey, M.T.: The linear bound in A 2 for Calderón–Zygmund operators: a survey. Marcinkiewicz centenary volume, Banach Center Publ., 95, Polish Acad. Sci. Inst. Math., Warsaw, 97–114 (2011)

    Google Scholar 

  42. Lacey, M.T.: On the A 2 inequality for Calderón–Zygmund operators. Preprint (2011) available at arXiv:1106.4802

    Google Scholar 

  43. Lacey, M.T., Moen, K., Pérez, C., Torres, R.: The sharp bound the fractional operators on weighted L p spaces and related Sobolev inequalities. J. Funct. Anal. 259(5), 1073–1097 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Lacey, M.T., Petermichl, S., Reguera, M.C.: Sharp A 2 inequality for Haar shift operators. Math. Ann. 348(1), 127–141 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lacey, M.T., Sawyer, E., Shen, C.-Y., Uriarte-Tuero, I.: The two weight inequality for the Hilbert transform, coronas and energy conditions. Preprint (2011) available at arXiv:1108.2319

    Google Scholar 

  46. Lerner, A.K.: On some weighted norm inequalities for Littlewood–Paley operators. Illinois J. Math. 52(2), 653–666 (2008)

    MathSciNet  Google Scholar 

  47. Lerner, A.K.: A pointwise estimate for the local sharp maximal function with applications to singular integrals. Bull. London Math. Soc. 42(5), 843–856 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Lerner, A.K.: A “local mean oscillation” and some of its applications. Function spaces, Approximation, Inequalities and Lineability, Lectures of the Spring School in Analysis, pp. 71–106. Matfyzpress, Prague (2011)

    Google Scholar 

  49. Lerner, A.K.: Mixed Ap-Ar inequalities for classical singular integrals and Littlewood–Paley operators. To appear in J. Geom. Anal., available at arXiv:1105.5735

    Google Scholar 

  50. Lerner, A.K.: Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals. Adv. Math. 226, 3912–3926 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Moen, K.: Sharp one-weight and two-weight bounds for maximal operators. Studia Math. 194, 163–180 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Moraes, J.C., Pereyra, M.C.: Weighted estimates for dyadic paraproducts and t-Haar multipliers with complexity (m, n). Submitted to Pub. Mat., available at arXiv:1108.3109

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  54. Nazarov, F., Volberg, A.: A simple sharp weighted estimate of the dyadic shifts on metric spaces with geometric doubling. Preprint (2011) available at arXiv:1104.4893

    Google Scholar 

  55. Nazarov, F., Treil, S., Volberg, A.: The Bellman functions and two-weight inequalities for Haar multipliers. J. Amer. Math. Soc. 12(4), 909–928 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  56. Nazarov, F., Treil, S., Volberg, A.: The Tb- theorem on non-homogeneous spaces. Acta Math. 190(2), 151–239 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Nazarov, F., Treil, S., Volberg, A.: Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures. Preprint (2005) available at arXiv:1003.1596

    Google Scholar 

  58. Nazarov, F., Reznikov, A., Volberg, A.: The proof of A 2 conjecture in a geometrically doubling metric space. Preprint (2011), available at arXiv:1106.1342

    Google Scholar 

  59. Ortiz, C.: Quadratic A 1 bounds for commutators of singular integrals with BMO functions. Indiana U. Math. J., available at arXiv:1104.1069

    Google Scholar 

  60. Pereyra, M.C.: Lecture Notes on Dyadic Harmonic Analysis. Contemporary Mathematics, vol. 289. American Mathematical Society, Providence, RI, Chapter 1, pp. 1–61 (2001)

    Google Scholar 

  61. Pereyra, M.C.: Haar multipliers meet Bellman functions. Rev. Mat. Iberoamericana 25(3), 799–840 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  62. Pérez, C.: Endpoint Estimates for Commutators of Singular Integral Operators. J. Funct. Anal. 128(1), 163–185 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  63. Pérez, C.: Sharp estimates for commutators of singular integrals via iterations of the Hardy–Littlewood maximal function. J. Fourier Anal. Appl. 3, 743–756 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  64. Pérez, C.: A course on Singular Integrals and weights. To appear in Birkhäuser as part of the series “Advanced courses in Mathematics at the C.R.M., Barcelona”.

    Google Scholar 

  65. Pérez, C., Treil, S., Volberg, A.: On A 2 conjecture and corona decomposition of weights. Preprint (2010) available at arXiv:1006.2630

    Google Scholar 

  66. Petermichl, S., Dyadic shift and a logarithmic estimate for Hankel operators with matrix symbol. C. R. Acad. Sci. Paris Sér. I Math. 330(6), 455–460 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  67. Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic. Amer. J. Math. 129, 1355–1375 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  68. Petermichl, S.: The sharp weighted bound for the Riesz transforms. Proc. Amer. Math. Soc. 136, 1237–1249 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  69. Petermichl, S., Pott, S.: An estimate for weighted Hilbert transform via square functions. Trans. Amer. Math. Soc. 354(4), 1699–1703 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  70. Petermichl, S., Volberg, A.: Heating of the Ahlfors-Beurling operator: Weakly quasiregular maps on the plane are quasiregular. Duke Math J. 112, 281–305 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  71. Riesz, M.: Sur le fonctions conjuguées. Math. Zeit. 27, 218–244 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  72. Sawyer, E.T.: A characterization of a two weight norm inequality for maximal operators. Studia Math. 75(1), 1–11 (1982)

    MathSciNet  MATH  Google Scholar 

  73. Stein, E.: Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, vol. 43. Monographs in Harmonic Analysis, vol. III. Princeton University Press, Princeton (1993)

    Google Scholar 

  74. Treil, S.: Sharp A 2 estimates of Haar shifts via Bellman function. Preprint (2011) available at arXiv:1105.2252

    Google Scholar 

  75. Treil, S., Volberg, A.: Wavelets and the angle between past and future. J. Funct. Anal. 143(2), 269–308 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  76. Vagharsyakhan, A.: Recovering singular integrals from the Haar shifts. Proc. Amer. Math. Soc. 138(12), 4303–4309 (2010)

    Article  MathSciNet  Google Scholar 

  77. Vasuynin, V., Volberg, A.: Notes on Bellman functions in harmonic analysis. Preprint (2010), available at http://sashavolberg.wordpress.com/

  78. Volberg, A.: Bellman function technique in harmonic analysis. Lectures of INRIA Summer School in Antibes. France (2011), available at arXiv:1106.3899

    Google Scholar 

  79. Wittwer, J.: A sharp estimate on the norm of the martingale transform. Math. Res. Lett. 7(1), 1–12 (2000)

    MathSciNet  MATH  Google Scholar 

  80. Wittwer, J.: A sharp estimate on the norm of the continuous square function. Proc. Amer. Math. Soc. 130(8), 2335–2342 (2002) (electronic)

    Google Scholar 

  81. Wilson, M.: Weighted Littlewood–Paley theory and exponential-square integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008)

    Google Scholar 

  82. Wojtaszczyk, P.: A mathematical introduction to wavelets. London Mathematical Society Student Texts, vol. 37. Cambridge University Press, Cambridge (2003)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank the organizers of the February Fourier Talks, at The Norbert Wiener Center for Harmonic Analysis and Applications, University of Maryland, for inviting her to deliver a talk in the fifth edition of the FFTs on February 18–19, 2010, that was the seed of this chapter. The author dedicates this chapter to the memory of her friend and mentor Cora Sadosky [1940–2010].

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, MSC03 21501, Albuquerque, NM, 87131-0001, USA

    María Cristina Pereyra

Authors
  1. María Cristina Pereyra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to María Cristina Pereyra .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, 20742-0001, Maryland, USA

    Travis D. Andrews

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Radu Balan

  3. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    Wojciech Czaja

  5. , Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Kasso A. Okoudjou

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Birkhäuser Boston

About this chapter

Cite this chapter

Pereyra, M.C. (2013). Weighted Inequalities and Dyadic Harmonic Analysis. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8379-5_15

Download citation

Publish with us

Policies and ethics

Search

Navigation