The boundedness of Riesz $s$-transforms of measures in $\mathbb {R}^{n}$

Let $\mu$ be a finite nonzero Borel measure in $\mathbb {R}^{n}$ satisfying $0 <c^{-1}r^{s}\le \mu B(x,r)\le cr^{s} <\infty$ for all $x\in \operatorname {spt}\mu$ and $0 < r\le 1$ and some $c >0$. If the Riesz $s$-transform

is essentially bounded, then $s$ is an integer. We also give a related result on the $L^{2}$-boundedness.