On Weighted Norms of Riesz Transforms Equal to One

We consider the vector Riesz transform $\nabla ^t\Delta ^{-(t+s)/2}\operatorname {div}^s$ of even order $s+t$ in the weighted space$L_2(\mathbb R^n;|x|^a)$. We establish that for $t\ne s$, $n>3$ its norm is equal to one on some interval of values of $a$, while inside the interval a stronger estimate for a subordinate norm is valid.