Logarithmic Bump with Bilinear T (B) Theorem and Maximal Singular Integral Operators

Abstract

We show that if a pair of weights (u,υ) satisfies a sharp Ap - bump condition in the scale of all log bumps certain loglog bumps , then Haar shifts map L^p (υ) into L^p (u) with a constant quadratic in the complexity of the shift . This in turn implies the two weight boundedness for all Calderón – Zygmund operators. We obtain a generalized version of the former theorem valid for a larger family of Calderón – Zygmund operators in any ambient space . We present a bilinear Tb theorem for singular operators Calderón – Zygmund type. Extending the end point results obtained to maximal singular. Another consequence is a quantitative two weight bump estimate.