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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Merja Vihtilä
Journal: Proc. Amer. Math. Soc. 124 (1996), 3797-3804
MSC (1991): Primary {28A75, 42B20}
MathSciNet review: 1343727
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

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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

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03522-8
PII: S 0002-9939(96)03522-8
Received by editor(s): June 22, 1994
Received by editor(s) in revised form: June 19, 1995
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1996 American Mathematical Society