Maximal properties of the normalized Cauchy transform

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 $p>2$.