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Transactions of the American Mathematical Society

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Vertical versus conical square functions

Authors: Pascal Auscher, Steve Hofmann and José-María Martell
Journal: Trans. Amer. Math. Soc. 364 (2012), 5469-5489
MSC (2010): Primary 42B25
Published electronically: May 29, 2012
MathSciNet review: 2931335
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Abstract: We study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator.

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  • [Aus] P. Auscher, On necessary and sufficient conditions for $ L^p$ estimates of Riesz transform associated elliptic operators on $ \mathbb{R}^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871. MR 2292385 (2007k:42025)
  • [AHLMcT] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh & Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $ \mathbb{R}\sp n$, Ann. of Math. (2) 156 (2002), no. 2, 633-654. MR 1933726 (2004c:47096c)
  • [AM] P. Auscher & J.M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights, Adv. Math. 212 (2007), no. 1, 225-276. MR 2319768 (2008m:42015)
  • [Be1] F. Bernicot, Use of abstract Hardy spaces, real interpolation and applications to bilinear operators, Math. Z. 265 (2010), no. 2, 365-400. MR 2609316 (2011c:42058)
  • [Be2] F. Bernicot, Use of Hardy spaces and interpolation, C. R. Math. Acad. Sci. Paris 346 (2008), no. 13-14, 745-748. MR 2427074 (2009h:46041)
  • [BZ] F. Bernicot & J. Zhao, New abstract Hardy spaces, J. Funct. Anal. 255 (2008), no. 7, 1761-1796. MR 2442082 (2009k:46043)
  • [CMS] R. Coifman, Y. Meyer & E.M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304-335. MR 791851 (86i:46029)
  • [CMP1] D. Cruz-Uribe, J. M. Martell & C. Pérez, Extensions of Rubio de Francia's extrapolation theorem, Collect. Math. Vol. Extra (2006), 195-231. MR 2264210 (2008a:42014)
  • [CMP2] D. Cruz-Uribe, J. M. Martell & C. Pérez, Weights, extrapolation and the theory of Rubio de Francia. Operator theory: Advances and Applications, 215 Birkhäuser/Springer Basel AG, Basel, 2011. MR 2797562
  • [DJK] B. Dahlberg, D. Jerison, & C. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat. 22 (1984), no. 1, 97-108. MR 735881 (85h:35021)
  • [DV] O. Dragičević & A. Volberg, Bilinear embedding theorem for real elliptic differential operators in divergence form with potential, to appear in J. Funct. Anal.
  • [FS] C. Fefferman & E.M. Stein, $ H^p$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [Fre] J. Freshe, An irregular complex valued solution to a scalar uniformly elliptic equation, Calc. Var. Partial Differential Equations 33 (2008), no. 3, 263-266. MR 2429531 (2009h:35084)
  • [Gar] J. García-Cuerva, An extrapolation theorem in the theory of $ A_p$-weights, Proc. Amer. Math. Soc. 87 (1983), 422-426. MR 684631 (84c:42028)
  • [Gra] L. Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250
  • [HTV] E. Harboure, J.-L. Torrea & B. Viviani, A vector-valued approach to tent spaces, J. Analyse Math. 56 (1991), 125-140. MR 1243101 (94i:42019)
  • [HM] S. Hofmann & S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), no. 1, 37-116. MR 2481054 (2009m:42038)
  • [HMMc] S. Hofmann, S. Mayboroda & A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $ L^p$, Sobolev and Hardy spaces, Ann. Sci. Ecole Norm. Sup. série 4 44, fascicule 5 (2011).
  • [JK] D. Jerison & C. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. (2) 113 (1981), no. 2, 367-382. MR 607897 (84j:35076)
  • [LeM] C. Le Merdy, On square functions associated to sectorial operators, Bull. Soc. Math. France 132 (2004), no. 1, 137-156. MR 2075919 (2005i:47026)
  • [Rub] J.L. Rubio de Francia, Factorization theory and $ A_p$ weights, Amer. J. Math. 106 (1984), 533-547. MR 745140 (86a:47028a)
  • [St] E.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)

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Additional Information

Pascal Auscher
Affiliation: Laboratoire de Mathématiques, UMR 8628, Université Paris-Sud, Orsay F-91405; CNRS, Orsay, F-91405 France

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

José-María Martell
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain

Keywords: Vertical square functions, conical square functions, extrapolation, elliptic operators
Received by editor(s): December 18, 2010
Published electronically: May 29, 2012
Additional Notes: Part of this work was carried out while the first author was visiting the Centre for Mathematics and its Applications, Australian National University, Canberra ACT 0200, Australia
The second author was partially supported by NSF grant number DMS 0801079.
The third author was supported by MEC Grant MTM2010-16518.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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