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 μ \mu be a positive finite measure on the unit circle of the complex plane and f ∈ L 1 ( μ ) f\in L^{1}(\mu ) . Denote by K μ K\mu and K f μ Kf\mu the Cauchy integrals of the measures μ \mu and f μ f\mu , respectively. The normalized Cauchy transform is defined as C μ : f ↦ K f μ K μ C_{\mu }: f\mapsto \frac {Kf\mu }{K\mu } . We prove that C μ C_{\mu } is bounded as an operator in L p ( μ ) L^{p}(\mu ) for 1 > p ≤ 2 1>p\leq 2 but is unbounded (in general) for p > 2 p>2 . The associated maximal non-tangential operator is bounded for 1 > p > 2 1>p>2 and has weak type ( 2 , 2 ) (2,2) but is unbounded for p > 2 p>2 .