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

Abstract

We give a direct proof of the local $Tb$ Theorem, in the Euclidean setting, and under the assumption of dual exponents. This Theorem provides a flexible framework for proving the boundedness of a Calderón-Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form $1/p+1/q\le 1$, and provide a direct proof. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also utilize certain twisted martingale transform inequalities.

References in corpus (3)

• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, I
• The Perfect Local $Tb$ Theorem and Twisted Martingale Transforms
• The local non-homogeneous Tb theorem for vector-valued functions