  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, 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 of functions with
values in a Banach space with the unconditionality property of martingale
differences (UMD). The new proof deals directly with all Lebesgue exponents p
in the range 1<p<infinity, and relies on delicate estimates for the
non-homogenous "Haar" functions, as well as McConnell's (1989) decoupling
inequality for tangent martingale differences.
