Journal d'Analyse Mathématique

, Volume 133, Issue 1, pp 71–89 | Cite as

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

  • Michael T. Lacey
  • Henri Martikainen


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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Copyright information

© Hebrew University Magnes Press 2017

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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