Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weighted Hardy spaces associated with elliptic operators. Part I: Weighted norm inequalities for conical square functions


Authors: José María Martell and Cruz Prisuelos-Arribas
Journal: Trans. Amer. Math. Soc. 369 (2017), 4193-4233
MSC (2010): Primary 42B30, 42B25, 35J15, 47A60
DOI: https://doi.org/10.1090/tran/6768
Published electronically: February 13, 2017
MathSciNet review: 3624406
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This is the first part of a series of three articles. In this paper, we obtain weighted norm inequalities for different conical square functions associated with the Heat and the Poisson semigroups generated by a second order divergence form elliptic operator with bounded complex coefficients. We find classes of Muckenhoupt weights where the square functions are comparable and/or bounded. These classes are natural from the point of view of the ranges where the unweighted estimates hold. In doing that, we obtain sharp weighted change of angle formulas which allow us to compare conical square functions with different cone apertures in weighted Lebesgue spaces. A key ingredient in our proofs is a generalization of the Carleson measure condition which is more natural when estimating the square functions below $ p=2$.


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

  • [1] Pascal Auscher, On necessary and sufficient conditions for $ L^p$-estimates of Riesz transforms associated to elliptic operators on $ \mathbb{R}^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75. MR 2292385 (2007k:42025), https://doi.org/10.1090/memo/0871
  • [2] Pascal Auscher, Change of angle in tent spaces, C. R. Math. Acad. Sci. Paris 349 (2011), no. 5-6, 297-301 (English, with English and French summaries). MR 2783323 (2012e:42037), https://doi.org/10.1016/j.crma.2011.01.023
  • [3] 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), https://doi.org/10.2307/3597201
  • [4] Pascal Auscher, Steve Hofmann, and José-María Martell, Vertical versus conical square functions, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5469-5489. MR 2931335, https://doi.org/10.1090/S0002-9947-2012-05668-6
  • [5] Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights, Adv. Math. 212 (2007), no. 1, 225-276. MR 2319768 (2008m:42015), https://doi.org/10.1016/j.aim.2006.10.002
  • [6] Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ. 7 (2007), no. 2, 265-316. MR 2316480 (2008m:47059), https://doi.org/10.1007/s00028-007-0288-9
  • [7] Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. III. Harmonic analysis of elliptic operators, J. Funct. Anal. 241 (2006), no. 2, 703-746. MR 2271934 (2007g:42022), https://doi.org/10.1016/j.jfa.2006.07.008
  • [8] Pascal Auscher, Alan McIntosh, and Emmanuel Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), no. 1, 192-248. MR 2365673 (2009d:42053), https://doi.org/10.1007/s12220-007-9003-x
  • [9] Pascal Auscher, Alan McIntosh, and Andrew Morris, Calderón reproducing formulas and applications to Hardy spaces, Rev. Mat. Iberoam. 31 (2015), no. 3, 865-900. MR 3420479
  • [10] Pascal Auscher and Emmanuel Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $ \mathbb{R}^n$, J. Funct. Anal. 201 (2003), no. 1, 148-184. MR 1986158 (2004c:42049), https://doi.org/10.1016/S0022-1236(03)00059-4
  • [11] Sönke Blunck and Peer Christian Kunstmann, Calderón-Zygmund theory for non-integral operators and the $ H^\infty $ functional calculus, Rev. Mat. Iberoamericana 19 (2003), no. 3, 919-942. MR 2053568 (2005f:42033), https://doi.org/10.4171/RMI/374
  • [12] The Anh Bui, Jun Cao, Luong Dang Ky, Dachun Yang, and Sibei Yang, Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Taiwanese J. Math. 17 (2013), no. 4, 1127-1166. MR 3085503
  • [13] The Anh Bui, Jun Cao, Luong Dang Ky, Dachun Yang, and Sibei Yang, Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces 1 (2013), 69-129. MR 3108869
  • [14] T. A. Bui and X. T. Duong, Weighted Hardy spaces associated to operators and boundedness of singular integrals, Preprint (2012), arXiv:1202.2063.
  • [15] R. R. Coifman, Y. Meyer, and 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), https://doi.org/10.1016/0022-1236(85)90007-2
  • [16] David V. Cruz-Uribe, José Maria Martell, and Carlos Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2797562 (2012f:42001)
  • [17] Javier Duoandikoetxea, Fourier analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. MR 1800316 (2001k:42001)
  • [18] Xuan Thinh Duong and Lixin Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), no. 10, 1375-1420. MR 2162784 (2006i:26012), https://doi.org/10.1002/cpa.20080
  • [19] Xuan Thinh Duong and Lixin Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943-973. MR 2163867 (2006d:42037), https://doi.org/10.1090/S0894-0347-05-00496-0
  • [20] 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)
  • [21] José García-Cuerva, Weighted $ H^{p}$ spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 63. MR 549091 (82a:42018)
  • [22] José García-Cuerva, An extrapolation theorem in the theory of $ A_{p}$ weights, Proc. Amer. Math. Soc. 87 (1983), no. 3, 422-426. MR 684631 (84c:42028), https://doi.org/10.2307/2043624
  • [23] José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
  • [24] Loukas Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR 2463316 (2011d:42001)
  • [25] Steve Hofmann, Guozhen Lu, Dorina Mitrea, Marius Mitrea, and Lixin Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78. MR 2868142, https://doi.org/10.1090/S0065-9266-2011-00624-6
  • [26] Steve Hofmann and José María Martell, $ L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), no. 2, 497-515. MR 2006497 (2004i:35067), https://doi.org/10.5565/PUBLMAT_47203_12
  • [27] Steve Hofmann and Svitlana Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), no. 1, 37-116. MR 2481054 (2009m:42038), https://doi.org/10.1007/s00208-008-0295-3
  • [28] Steve Hofmann, Svitlana Mayboroda, and Alan McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $ L^p$, Sobolev and Hardy spaces, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 5, 723-800 (English, with English and French summaries). MR 2931518
  • [29] Andrei K. Lerner, On sharp aperture-weighted estimates for square functions, J. Fourier Anal. Appl. 20 (2014), no. 4, 784-800. MR 3232586, https://doi.org/10.1007/s00041-014-9333-6
  • [30] Suying Liu and Liang Song, An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators, J. Funct. Anal. 265 (2013), no. 11, 2709-2723. MR 3096987, https://doi.org/10.1016/j.jfa.2013.08.003
  • [31] J. M. Martell and C. Prisuelos-Arribas, Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $ H^1_L(w)$, preprint 2017, arXiv:1701.00920.
  • [32] C. Prisuelos-Arribas, Weighted Hardy spaces associated with elliptic operators. Part III: Characterizations of $ H^p_L(w)$ and the weighted Hardy space associated with the Riesz transform, in preparation.
  • [33] José L. Rubio de Francia, Factorization theory and $ A_{p}$ weights, Amer. J. Math. 106 (1984), no. 3, 533-547. MR 745140 (86a:47028a), https://doi.org/10.2307/2374284
  • [34] 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)
  • [35] Elias M. Stein and Guido Weiss, On the theory of harmonic functions of several variables. I. The theory of $ H^{p}$-spaces, Acta Math. 103 (1960), 25-62. MR 0121579 (22 #12315)
  • [36] Jan-Olov Strömberg and Alberto Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989. MR 1011673 (90j:42053)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42B30, 42B25, 35J15, 47A60

Retrieve articles in all journals with MSC (2010): 42B30, 42B25, 35J15, 47A60


Additional Information

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
Email: chema.martell@icmat.es

Cruz Prisuelos-Arribas
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain
Email: cruz.prisuelos@icmat.es

DOI: https://doi.org/10.1090/tran/6768
Keywords: Hardy spaces, conical square functions, tent spaces, Muckenhoupt weights, extrapolation, elliptic operators, Heat and Poisson semigroup, off-diagonal estimates
Received by editor(s): June 24, 2014
Received by editor(s) in revised form: June 15, 2015
Published electronically: February 13, 2017
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. The first author was supported in part by MINECO Grant MTM2010-16518, ICMAT Severo Ochoa project SEV-2011-0087. The second author was supported in part by ICMAT Severo Ochoa project SEV-2011-0087.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society