The Perfect Local $Tb$ Theorem and Twisted Martingale Transforms

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 a new inequality for transforms of so-called twisted martingale differences.

12 pages v2: typos corrected in section 5 v3: The twisted martingale transform inequalities appeared in Auscher Routin (arXiv:1011.1747), which is now reflected in the text

References in corpus (2)

• The local non-homogeneous Tb theorem for vector-valued functions
• Local Tb theorems and Hardy inequalities

Cited by in corpus (1)

• On the Local $Tb$ Theorem: A Direct Proof under Duality Assumption