Cauchy singular integral operator with parameters in Log-Hölder spaces

2020/01/11 by Yifei Pan, Yuan Zhang, Pan, Yifei +1
Mathematics · #32A55 #32W05 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2001.03778

openalex publication_date 2020/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is motivated by a claim in the classical textbook of Muskhelishvili concerning the Cauchy singular integral operator S on Hölder functions with parameters. To the contrary of the claim, a counter example was constructed by Tumanov which shows that S with parameters fails to maintain the same Hölder regularity with respect to the parameters. In view of the example, the behavior of the Cauchy singular integral operator with parameters between a type of Log-Hölder spaces is investigated to obtain the sharp norm estimates. At the end of the paper, we discuss its application to the ∂ problem on product domains.

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