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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}
DOI: https://doi.org/10.1090/S0002-9939-96-03522-8
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?)

  • [C] M. Christ, Lectures on Singular Integral Operators, Regional Conference Series in Mathematics 77, Amer. Math. Soc., 1990. MR 92f:42021
  • [D] G. David, Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Math. 1465, Springer-Verlag, Berlin-Heidelberg, 1991. MR 92k:42021
  • [DS1] G. David and S. Semmes, Singular Integrals and Rectifiable Sets in $\mathbb R^n $, Astérisque 193, Soc. Math. France, 1991. MR 92j:42016
  • [DS2] G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Surveys and Monographs 38, Amer. Math. Soc., 1993. MR 94i:28003
  • [J] J.-L. Journé, Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón, Lecture Notes in Math. 994, Springer-Verlag, Berlin-Heidelberg, 1983. MR 85i:42021
  • [M] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge, 1995. CMP 95:13
  • [MP] P. Mattila and D. Preiss, Rectifiable Measures in $\mathbb R^n $ and Existence of Principal Values for Singular Integrals, Preprint.
  • [P] D. Preiss, Geometry of Measures in ${\mathbb {R}}^{n}$. Distributions, Rectifiability, and Densities, Ann. of Math. 125 (1987), 537-643. MR 88d:28008

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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: https://doi.org/10.1090/S0002-9939-96-03522-8
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

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