The boundedness of Riesz $s$-transforms of measures in $\mathbb {R}^n$
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- by Merja Vihtilä PDF
- Proc. Amer. Math. Soc. 124 (1996), 3797-3804 Request permission
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.References
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Additional Information
- Merja Vihtilä
- Affiliation: Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
- Email: vihtila@math.jyu.fi
- Received by editor(s): June 22, 1994
- Received by editor(s) in revised form: June 19, 1995
- Communicated by: Christopher D. Sogge
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 3797-3804
- MSC (1991): Primary {28A75, 42B20}
- DOI: https://doi.org/10.1090/S0002-9939-96-03522-8
- MathSciNet review: 1343727