Remote Access Proceedings of the American Mathematical Society Series B
Gold Open Access

Proceedings of the American Mathematical Society Series B

ISSN 2330-1511

   
 
 

 

Linear and bilinear $ T(b)$ theorems à la Stein


Authors: Árpád Bényi and Tadahiro Oh
Journal: Proc. Amer. Math. Soc. Ser. B 2 (2015), 1-16
MSC (2010): Primary 42B20
DOI: https://doi.org/10.1090/bproc/18
Published electronically: October 9, 2015
MathSciNet review: 3406428
Full-text PDF Open Access
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: In this work, we state and prove versions of the linear and bilinear $ T(b)$ theorems involving quantitative estimates, analogous to the quantitative linear $ T(1)$ theorem due to Stein.


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

  • [1] Árpád Bényi and Tadahiro Oh, Smoothing of commutators for a Hörmander class of bilinear pseudodifferential operators, J. Fourier Anal. Appl. 20 (2014), no. 2, 282-300. MR 3200923, https://doi.org/10.1007/s00041-013-9312-3
  • [2] Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1990. MR 1104656 (92f:42021)
  • [3] Michael Christ and Jean-Lin Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159 (1987), no. 1-2, 51-80. MR 906525 (89a:42024), https://doi.org/10.1007/BF02392554
  • [4] 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), https://doi.org/10.2307/2006946
  • [5] G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1-56 (French). MR 850408 (88f:47024), https://doi.org/10.4171/RMI/17
  • [6] C. Fefferman and E. M. Stein, $ H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. MR 0447953 (56 #6263)
  • [7] Loukas Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR 2463316 (2011d:42001)
  • [8] Loukas Grafakos and Rodolfo H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124-164. MR 1880324 (2002j:42029), https://doi.org/10.1006/aima.2001.2028
  • [9] Loukas Grafakos and Rodolfo H. Torres, On multilinear singular integrals of Calderón-Zygmund type, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), 2002, pp. 57-91. MR 1964816 (2004c:42031), https://doi.org/10.5565/PUBLMAT_Esco02_04
  • [10] Yong Sheng Han, Calderón-type reproducing formula and the $ Tb$ theorem, Rev. Mat. Iberoamericana 10 (1994), no. 1, 51-91. MR 1271757 (95h:42020), https://doi.org/10.4171/RMI/145
  • [11] Jarod Hart, A new proof of the bilinear $ \rm T(1)$ Theorem, Proc. Amer. Math. Soc. 142 (2014), no. 9, 3169-3181. MR 3223373, https://doi.org/10.1090/S0002-9939-2014-12054-5
  • [12] Jarod Hart, A bilinear T(b) theorem for singular integral operators, J. Funct. Anal. 268 (2015), no. 12, 3680-3733. MR 3341962, https://doi.org/10.1016/j.jfa.2015.02.008
  • [13] Alan McIntosh and Yves Meyer, Algèbres d'opérateurs définis par des intégrales singulières, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 8, 395-397 (French, with English summary). MR 808636 (87b:47053)
  • [14] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)
  • [15] Elias M. Stein, 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, III, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)

Similar Articles

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

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


Additional Information

Árpád Bényi
Affiliation: Department of Mathematics, Western Washington University, 516 High Street, Bellingham, Washington 98225
Email: arpad.benyi@wwu.edu

Tadahiro Oh
Affiliation: School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
Email: hiro.oh@ed.ac.uk

DOI: https://doi.org/10.1090/bproc/18
Keywords: $T(b)$ theorem, $T(1)$ theorem, Calder\'on-Zygmund operator, bilinear operator
Received by editor(s): February 8, 2015
Published electronically: October 9, 2015
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)

American Mathematical Society