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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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