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We show sufficient conditions on matrix weights $U$ and $V$ for themartingale transforms to be uniformly bounded from $L^2(V)$ to $L^2(U)$. Wealso show that these conditions imply the uniform boundedness of the dyadicshifts as well as the boundedness of the Hilbert transform between thesespaces.

Martingale transforms, the dyadic shift and the Hilbert transform: a sufficient condition for boundedness between matrix weighted spaces