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Let      μ  \mu       be a finite nonzero Borel measure in      R n \mathbb {R}^{n}       satisfying      0 &gt; c −  1 r s ≤  μ  B ( x , r ) ≤  c r s &gt; ∞  0 &gt;c^{-1}r^{s}\le \mu B(x,r)\le cr^{s} &gt;\infty       for all      x ∈  spt ⁡  μ  x\in \operatorname {spt}\mu       and      0 &gt; r ≤  1 0 &gt; r\le 1       and some      c &gt; 0 c &gt;0      . If the Riesz      s s      -transform      C s , μ  ( x ) = ∫  y −  x | y −  x | s + 1 d μ  y \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 s       is an integer. We also give a related result on the      L 2 L^{2}      -boundedness.

The boundedness of Riesz 𝑠-transforms of measures in ℝⁿ