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The boundedness of Riesz

Authors:	Merja Vihtilä

Institution:	Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland

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 . If the Riesz -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 is an integer. We also give a related result on the $L^{2}$ -boundedness.

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