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

Guantie Deng; Rong Liu

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arXiv · 1709.00563

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

Guantie Deng·Rong Liu

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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 $H^2(Ω_\pm)$, 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.

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Guantie Deng, Rong Liu. 2017-09-02. Notes of Boundedness on Cauchy Integrals on Lipschitz Curves ($p=2$). https://arxiv.org/abs/1709.00563

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