Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The perfect local $ Tb$ Theorem and twisted Martingale transforms

Authors: Michael T. Lacey and Antti V. Vähäkangas
Journal: Proc. Amer. Math. Soc. 142 (2014), 1689-1700
MSC (2010): Primary 42B20
Published electronically: February 17, 2014
MathSciNet review: 3168475
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A local $ Tb$ Theorem provides a flexible framework for proving the boundedness of a Calderón-Zygmund operator $ T$. One needs only boundedness of the operator $ T$ on systems of locally pseudo-accretive functions $ \{b_Q\}$, indexed by cubes. We give a new proof of this theorem in the setting of perfect (dyadic) models of Calderón-Zygmund operators, imposing integrability conditions on the $ b_Q$ functions that are the weakest possible. The proof is a simple direct argument, based upon an inequality for transforms of so-called twisted martingale differences, which has been noted by Auscher-Routin.

References [Enhancements On Off] (What's this?)

  • [1] P. Auscher, S. Hofmann, C. Muscalu, T. Tao, and C. Thiele, Carleson measures, trees, extrapolation, and $ T(b)$ theorems, Publ. Mat. 46 (2002), no. 2, 257-325. MR 1934198 (2003f:42019),
  • [2] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $ {\mathbb{R}}^n$, Ann. of Math. (2) 156 (2002), no. 2, 633-654. MR 1933726 (2004c:47096c),
  • [3] P. Auscher and E. Routin, Local Tb theorems and Hardy inequalities, J. Geometric Anal. 23 (2013), no. 1, 303-374. MR 3010282
  • [4] Pascal Auscher and Qi Xiang Yang, BCR algorithm and the $ T(b)$ theorem, Publ. Mat. 53 (2009), no. 1, 179-196. MR 2474120 (2010i:42021),
  • [5] G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms. I, Comm. Pure Appl. Math. 44 (1991), no. 2, 141-183. MR 1085827 (92c:65061),
  • [6] D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647-702. MR 744226 (86b:60080)
  • [7] Donald L. Burkholder, Explorations in martingale theory and its applications, École d'Été de Probabilités de Saint-Flour XIX--1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 1-66. MR 1108183 (92m:60037),
  • [8] Michael Christ, A $ T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601-628. MR 1096400 (92k:42020)
  • [9] R. R. Coifman, Peter W. Jones, and Stephen Semmes, Two elementary proofs of the $ L^2$ boundedness of Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc. 2 (1989), no. 3, 553-564. MR 986825 (90k:42017),
  • [10] Guy David and Jean-Lin Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371-397. MR 763911 (85k:42041),
  • [11] Tadeusz Figiel, Singular integral operators: a martingale approach, Geometry of Banach spaces (Strobl, 1989) London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 95-110. MR 1110189 (94e:42015)
  • [12] Steve Hofmann, Michael Lacey, and Alan McIntosh, The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds, Ann. of Math. (2) 156 (2002), no. 2, 623-631. MR 1933725 (2004c:47096b),
  • [13] Steve Hofmann, Local $ T(b)$ theorems and applications in PDE, Harmonic analysis and partial differential equations, Contemp. Math., vol. 505, Amer. Math. Soc., Providence, RI, 2010, pp. 29-52. MR 2664559 (2011e:42024),
  • [14] Steve Hofmann, Local $ Tb$ theorems and applications in PDE, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1375-1392. MR 2275650 (2007k:42030)
  • [15] Tuomas Hytönen and Henri Martikainen, Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces, J. Geom. Anal. 22 (2012), no. 4, 1071-1107. MR 2965363
  • [16] Tuomas Hytönen and Henri Martikainen, On general local $ Tb$ theorems, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4819-4846. MR 2922611,
  • [17] Tuomas Hytönen and Fedor Nazarov, The local $ Tb$ theorem with rough test functions, available at
  • [18] Tuomas P. Hytönen and Antti V. Vähäkangas, The local non-homogeneous $ Tb$ theorem for vector-valued functions (2012), available at arxiv:1201.0648.
  • [19] Michael T. Lacey, Stefanie Petermichl, and Maria Carmen Reguera, Sharp $ A_2$ inequality for Haar shift operators, Math. Ann. 348 (2010), no. 1, 127-141. MR 2657437 (2011i:42027),
  • [20] Michael T. Lacey, Eric T. Sawyer, and Ignacio Uriarte-Tuero, Astala's conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane, Acta Math. 204 (2010), no. 2, 273-292. MR 2653055 (2011f:30043),
  • [21] Michael T. Lacey, Eric T. Sawyer, Ignacio Uriarte-Tuero, and Chun-Yen Shen, The two weight inequality for Hilbert transform, coronas, and energy conditions, available at
  • [22] Michael T. Lacey, Eric T. Sawyer, Ignacio Uriarte-Tuero, and Chun-Yen Shen, Two weight inequality for the Hilbert transform: A real variable characterization, available at
  • [23] Camil Muscalu, Terence Tao, and Christoph Thiele, Multi-linear operators given by singular multipliers, J. Amer. Math. Soc. 15 (2002), no. 2, 469-496. MR 1887641 (2003b:42017),
  • [24] Stephanie Anne Salomone, $ b$-weighted dyadic BMO from dyadic BMO and associated $ T(b)$ theorems, Collect. Math. 61 (2010), no. 2, 151-171. MR 2666228 (2011e:42031),
  • [25] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42B20

Retrieve articles in all journals with MSC (2010): 42B20

Additional Information

Michael T. Lacey
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Antti V. Vähäkangas
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Received by editor(s): April 29, 2012
Received by editor(s) in revised form: May 14, 2012, and June 19, 2012
Published electronically: February 17, 2014
Additional Notes: This research was supported in part by grant NSF-DMS 0968499 and a grant from the Simons Foundation (#229596) to the first author
The second author was supported by the School of Mathematics, Georgia Institute of Technology, and by the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society