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

It is well-known that several classical results about Calderón-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 Calderón-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 Calderón-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. Hytönen to extend the result to general Calderón-Zygmund operators.

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