A linear dimensionless bound for the weighted Riesz vector

We show that the norm of the vector of Riesz transforms as operator in the weighted Lebesgue space Lω2 is bounded by a constant multiple of the first power of the Poisson-A2 characteristic of ω. The bound is free of dimension and optimal. Our argument requires an extension of Wittwer's linear estimate for martingale transforms to the vector valued setting with scalar weights, for which we indicate a proof. Extensions to Lωp for 11, the Poisson-A2 class is properly included in the classical A2 class.