Linear bounds for Calderon-Zygmund operators with even kernel on UMD spaces

It is well-known that several classical results about Calderon-Zygmund singular integral operators can be extended to X-valued functions if and only if the Banach space X has the UMD property. The dependence of the norm of an X-valued Calderon-Zygmund operator on the UMD constant of the space X is conjectured to be linear. We prove that this is indeed the case for sufficiently smooth Calderon-Zygmund operators with cancellation, associated to an even kernel. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytonen to extend the result to general Calderon-Zygmund operators. (C) 2013 Elsevier Inc. All rights reserved.