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 an inequality for transforms of so-called twisted martingale differences, which has been noted by Auscher-Routin.

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

“…Our argument is direct in the sense that it avoids a reduction to the so-called perfect dyadic case such as that seen in Auscher and Yang [2]. A companion paper [18] addresses a perfect dyadic variant of Theorem 1.2 for the full range 1 < p 1 , p 2 < ∞; it contains many of the features of the argument in the present paper, with significantly fewer technicalities.…”

“…There are also tools in § 3 that are useful, namely martingale transform inequalities for twisted martingale differences, and the associated half-twisted inequalities that are universal in that they hold in all L q -spaces. These inequalities also play a crucial role in [1,Lemma 5.3] and in [18].…”

New Local T1 Theorems on non-homogeneous spaces

“…Since then, research work on this subject has been continuously growing with special focus on more general criteria of boundedness ( [1], [14], [13]), variants that apply to new settings ([12], [11], [15]) and applications to PDEs ( [8], [10]). The articles [7], [9], [2], [17] and the books [4], [5], [16], [23] provide detailed accounts of the evolution of this theory.…”

A local T(b) theorem for perfect multilinear Calderón–Zygmund operators

“…In , Nazarov, Treil and Volberg proved, in a non‐doubling measure setup, that it suffices to assume and uniformly in . Auscher, Hofmann, Muscalu, Tao and Thiele , for dyadic model operators, relaxed these conditions assuming only , , and for any where the different norms are appropriately scaled relative to , see also .…”