ar X iv : 0 80 9 . 30 97 v 1 [ m at h . FA ] 1 8 Se p 20 08 THE VECTOR - VALUED NON - HOMOGENEOUS Tb THEOREM TUOMAS

Abstract. The paper gives a Banach space -valued extension of the “Tb theorem” of Nazarov, Treil and Volberg (2003) 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(μ;X) of functions with values in X — a Banach space with the unconditionality property of martingale differences (UMD) and a certain maximal function property, which holds for all typical examples of UMD spaces. The new proof deals directly with all p ∈ (1,∞) and relies on delicate estimates for the non-homogenous “Haar” functions, as well as McConnell’s (1989) decoupling inequality for tangent martingale differences.