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

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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≤ 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.