## Maximal properties of the normalized Cauchy transform

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- by Alexei Poltoratski
- J. Amer. Math. Soc.
**16**(2003), 1-17 - DOI: https://doi.org/10.1090/S0894-0347-02-00403-4
- Published electronically: August 27, 2002
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## Abstract:

We study the normalized Cauchy transform in the unit disk. Our goal is to find an analog of the classical theorem by M. Riesz for the case of arbitrary weights. Let $\mu$ be a positive finite measure on the unit circle of the complex plane and $f\in L^{1}(\mu )$. Denote by $K\mu$ and $Kf\mu$ the Cauchy integrals of the measures $\mu$ and $f\mu$, respectively. The normalized Cauchy transform is defined as $C_{\mu }: f\mapsto \frac {Kf\mu }{K\mu }$. We prove that $C_{\mu }$ is bounded as an operator in $L^{p}(\mu )$ for $1<p\leq 2$ but is unbounded (in general) for $p>2$. The associated maximal non-tangential operator is bounded for $1<p<2$ and has weak type $(2,2)$ but is unbounded for $p>2$.## References

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## Bibliographic Information

**Alexei Poltoratski**- Affiliation: Department of Mathemathcs, Texas A&M University, College Station, Texas 77843
- MR Author ID: 292108
- Email: alexeip@math.tamu.edu
- Received by editor(s): June 12, 2000
- Published electronically: August 27, 2002
- Additional Notes: The author is supported in part by N.S.F. grant DMS 9970151
- © Copyright 2002 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**16**(2003), 1-17 - MSC (2000): Primary 30E20
- DOI: https://doi.org/10.1090/S0894-0347-02-00403-4
- MathSciNet review: 1937196