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Transactions of the American Mathematical Society

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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
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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$.


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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

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