The vector-valued nonhomogeneous Tb Theorem

This paper gives a Banach space-valued extension of the Tb theorem of Nazarov et al. [20] concerning the boundedness of singular integral operators with respect to a measure μ, which only satisfies an upper control on the size of balls. Under the same assumptions, as in their result, such operators are shown to be bounded on the Bochner spaces L^p(μ;X) of functions with values in X—a Banach space with the unconditionality property of martingale differences. The new proof deals directly with all $p\in(1,\infty)$ and relies on delicate estimates for the nonhomogenous “Haar” functions, as well as McConnell's [18] decoupling inequality for tangent martingale differences.