A linear dimensionless bound for the weighted Riesz vector

Abstract

We show that the norm of the vector of Riesz transforms as operator in the weighted Lebesgue space L^2(w) is bounded by a constant multiple of the first power of the Poisson-A_2 characteristic of w. The bound is free of dimension. 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 L^p(w) for 1<p<\infty are discussed. Our proof for sharpness requires a martingale extrapolation lemma for which we sketch a proof. We also show that for n>1, the Poisson-A_2 class is properly included in the classical A_2 class.