Notes of Boundedness on Cauchy Integrals on Lipschitz Curves ($p=2$)

TL;DR

This paper details the proof that the Cauchy transform of L^2 functions on Lipschitz curves is bounded and explores boundary limits and Hardy space representations, especially for the real axis case.

Contribution

It provides a detailed proof of boundedness of the Cauchy transform on Lipschitz curves and characterizes L^2 functions via Hardy space boundary limits.

Findings

Cauchy transform of L^2 functions on Lipschitz curves is bounded.

L^2 functions can be represented as boundary limits of Hardy space functions.

Enhanced boundary behavior for the Cauchy transform on the real axis.

Abstract

We provide the details of the first proof in~\cite{CJS89}, which proved that Cauchy transform of $L^2$~functions on Lipschitz curves is bounded. We then prove that every $L^2$~function on Lipschitz curves is the sum of non-tangential boundary limit of functions in $H^2(\Omega_\pm)$, the Hardy spaces on domains over and under the Lipschitz curve. We also obtain a more accurate boundary of Cauchy transform under the condition that the Lipschitz curve is the real axis.