The perfect local 𝑇𝑏 Theorem and twisted Martingale transforms

Lacey, Michael; Vähäkangas, Antti

doi:10.1090/S0002-9939-2014-11930-7

The perfect local $Tb$ Theorem and twisted Martingale transforms
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by Michael T. Lacey and Antti V. Vähäkangas PDF

Proc. Amer. Math. Soc. 142 (2014), 1689-1700 Request permission

Abstract:

A local $Tb$ Theorem provides a flexible framework for proving the boundedness of a Calderón-Zygmund operator $T$. One needs only boundedness of the operator $T$ on systems of locally pseudo-accretive functions $\{b_Q\}$, indexed by cubes. We give a new proof of this theorem in the setting of perfect (dyadic) models of Calderón-Zygmund operators, imposing integrability conditions on the $b_Q$ functions that are the weakest possible. The proof is a simple direct argument, based upon an inequality for transforms of so-called twisted martingale differences, which has been noted by Auscher-Routin.

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