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Local Tb theorem with L 2 testing conditions and general measures: square functions

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Abstract

Local Tb theorems with L p type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. In the non-homogeneous world local Tb theorems have only been proved assuming scale invariant (L ∞ or BMO) testing conditions. In this paper, for the first time, we overcome these obstacles in the non-homogeneous world, and prove a nonhomogeneous local Tb theorem with L 2 type testing conditions. This paper is in the setting of the vertical and conical square functions defined using general measures and kernels. On the technique side, we demonstrate a trick of inserting Calderón–Zygmund stopping data of a fixed function into the construction of the twisted martingale difference operators. This built-in control of averages is an alternative to Carleson embedding.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Michael T. Lacey

  2. Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68, FI-00014, Helsinki, Finland

    Henri Martikainen

Authors
  1. Michael T. Lacey
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  2. Henri Martikainen
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Corresponding author

Correspondence to Michael T. Lacey.

Additional information

Research of M. T. L. is supported in part by grant NSF-DMS 0968499 and the Australian Research Council through grant ARC-DP120100399.

This paper was completed while H. M. was still at Université Paris-Sud 11, Orsay. During this period the research of H. M. was supported by the Emil Aaltonen Foundation.

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Lacey, M.T., Martikainen, H. Local Tb theorem with L 2 testing conditions and general measures: square functions. JAMA 133, 71–89 (2017). https://doi.org/10.1007/s11854-017-0028-1

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  • DOI: https://doi.org/10.1007/s11854-017-0028-1

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