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

Abstract:

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 \begin{equation*}{\mathcal {C}}_{s,\mu }(x)=\int \frac {y-x}{|y-x|^{s+ 1}} d\mu y \end{equation*} is essentially bounded, then $s$ is an integer. We also give a related result on the $L^{2}$-boundedness.