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

2017/09/02 by Deng, Guantie, Liu, Rong
#30E20 #30E25 #30H10 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.00563

Abstract

We provide the details of the first proof in~\citeCJS89, which proved that Cauchy transform of L²~functions on Lipschitz curves is bounded. We then prove that every L²~function on Lipschitz curves is the sum of non-tangential boundary limit of functions in H²(Ω_±), 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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