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Maximal properties of the normalized Cauchy transform

Author: Alexei Poltoratski
Journal: J. Amer. Math. Soc. 16 (2003), 1-17
MSC (2000): Primary 30E20
Published electronically: August 27, 2002
MathSciNet review: 1937196
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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 $p>2$.

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Additional Information

Alexei Poltoratski
Affiliation: Department of Mathemathcs, Texas A&M University, College Station, Texas 77843

Keywords: Cauchy integrals, boundary convergence, non-tangential maximal function
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
Article copyright: © Copyright 2002 American Mathematical Society

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