A linear dimensionless bound for the weighted Riesz vector

Overview & Synopsis

We show that the norm of the vector of Riesz transforms as operator in the weighted Lebesgue space $L^2_ω$ is bounded by a constant multiple of the first power of the Poisson-$A_2$ characteristic of $ω$. The bound is free of dimension. We also show that for $n > 1$, the Poisson-$A_2$ class is properly included in the classical $A_2$ class.