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On the A 2 Inequality for Calderón–Zygmund Operators

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Recent Advances in Harmonic Analysis and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 25))

Abstract

We prove that for an \({L}^{2}({\mathbb{R}}^{d})\)-bounded Calderón–Zygmund operator and weight w ∈ A 2, we have the inequality below due to Hytönen:

$$\Vert {T\Vert }_{{L}^{2}(w)\rightarrow {L}^{2}(w)} \leq {C}_{T}{[w]}_{{A}_{2}}\,.$$

Our proof will appeal to a distributional inequality used by several authors, adapted Haar functions, and standard stopping times.

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References

  1. Beznosova, O.V.: Linear bound for the dyadic paraproduct on weighted Lebesgue space L 2(w). J. Funct. Anal. 255(4), 994–1007 (2008). MR2433959

    Google Scholar 

  2. Hytönen, T.: The sharp weighted bound for general Calderon-Zygmund operators. Ann. Math. (2010), to appear. Available at http://arxiv.org/abs/1007.4330

  3. Hytönen, T., 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 (2011). Available at http://www.arxiv.org/abs/1103.5229.

  4. Hytönen, T., Pérez, C.: Sharp weighted bounds involving A ∞ (2011). Available at http://www.arxiv.org/abs/1103.5562

  5. Hytönen, T., Pérez, C., Treil, S., Volberg, A.: Sharp weighted estimates of the dyadic shifts and A 2 conjecture. ArXiv e-prints (2010). Available at http://arxiv.org/abs/1010.0755

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

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  7. 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(5), 1355–1375 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Reznikov, A., Treil, S., Volberg, A.: A sharp weighted estimate of dyadic shifts of complexity 0 and 1. ArXiv e-prints (2011). Available at http://arxiv.org/abs/1103.5347

  9. Treil, S.: Sharp A 2 estimates of Haar shifts via Bellman function (2011). Available at http://arxiv.org/abs/1105.2252

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Acknowledgements

Research supported in part by grant NSF-DMS 0968499.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Michael T. Lacey

Authors
  1. Michael T. Lacey
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Correspondence to Michael T. Lacey .

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Editors and Affiliations

  1. , Department of Mathematics, University of South Carolina, Greene Street 1523, Columbia, 29208, South Carolina, USA

    Dmitriy Bilyk

  2. University Park, Department of Mathematics, Florida International University, SW 8th St., Rm 416 11200, Miami, 33199, Florida, USA

    Laura De Carli

  3. , Department of Mathematics, Universtity of Georgia, University of Georgia, Athens, 30602, Georgia, USA

    Alexander Petukhov

  4. Georgia Southern University, PO Box 8093, Statesboro, 30460, Georgia, USA

    Alexander M. Stokolos

  5. , Department of Mathematics, Georgia Institute of Technology, Cherry Street 686, Atlanta, 30332, Georgia, USA

    Brett D. Wick

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Lacey, M.T. (2012). On the A 2 Inequality for Calderón–Zygmund Operators. In: Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A., Wick, B. (eds) Recent Advances in Harmonic Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4565-4_20

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