The perfect local $Tb$ Theorem and twisted Martingale transforms
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- by Michael T. Lacey and Antti V. Vähäkangas PDF
- Proc. Amer. Math. Soc. 142 (2014), 1689-1700 Request permission
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
- 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, DOI 10.5565/PUBLMAT_{4}6202_{0}1
- 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 ${\Bbb R}^n$, Ann. of Math. (2) 156 (2002), no. 2, 633–654. MR 1933726, DOI 10.2307/3597201
- Pascal Auscher and Eddy Routin, Local $Tb$ theorems and Hardy inequalities, J. Geom. Anal. 23 (2013), no. 1, 303–374. MR 3010282, DOI 10.1007/s12220-011-9249-1
- Pascal Auscher and Qi Xiang Yang, BCR algorithm and the $T(b)$ theorem, Publ. Mat. 53 (2009), no. 1, 179–196. MR 2474120, DOI 10.5565/PUBLMAT_{5}3109_{0}8
- 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, DOI 10.1002/cpa.3160440202
- D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226
- 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, DOI 10.1007/BFb0085167
- 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, DOI 10.4064/cm-60-61-2-601-628
- 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, DOI 10.1090/S0894-0347-1989-0986825-6
- 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, DOI 10.2307/2006946
- 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
- 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, DOI 10.2307/3597200
- 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, DOI 10.1090/conm/505/09914
- 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
- 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, DOI 10.1007/s12220-011-9230-z
- Tuomas Hytönen and Henri Martikainen, On general local $Tb$ theorems, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4819–4846. MR 2922611, DOI 10.1090/S0002-9947-2012-05599-1
- Tuomas Hytönen and Fedor Nazarov, The local $Tb$ theorem with rough test functions, available at http://www.arxiv.org/abs/1206.0907.
- Tuomas P. Hytönen and Antti V. Vähäkangas, The local non-homogeneous $Tb$ theorem for vector-valued functions (2012), available at arxiv:1201.0648.
- 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, DOI 10.1007/s00208-009-0473-y
- 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, DOI 10.1007/s11511-010-0048-5
- 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 http://www.arxiv.org/abs/1108.2319.
- 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 http://www.arxiv.org/abs/1201.4319 .
- 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, DOI 10.1090/S0894-0347-01-00379-4
- 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, DOI 10.1007/BF03191239
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Additional Information
- Michael T. Lacey
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- MR Author ID: 109040
- Email: lacey@math.gatech.edu
- Antti V. Vähäkangas
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- Email: antti.vahakangas@helsinki.fi
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1689-1700
- MSC (2010): Primary 42B20
- DOI: https://doi.org/10.1090/S0002-9939-2014-11930-7
- MathSciNet review: 3168475