Growth Estimates for Cauchy Integrals of Measures and Rectifiability期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Growth Estimates for Cauchy Integrals of Measures and Rectifiability

Authors:

Institution:

1.Institució Catalana de Recerca i Estudis Avan?ats (ICREA) and Departament de Matemàtiques,Universitat Autònoma de Barcelona,Barcelona,Spain

Abstract:

We show that if μ is a finite Borel measure on the complex plane such that

${\mathcal{C}}_{*} \mu (z) = {\mathop {\sup }\limits_{\varepsilon > 0} }{\left| {{\mathcal{C}}_{\varepsilon } \mu (z)} \right|} = {\mathop {\sup }\limits_{\varepsilon > 0} }{\left| {{\int_{|\xi - z| > \varepsilon } {\frac{1}{{\xi - z}}d\mu (\xi )} }} \right|} < \infty$

for μ-a.e. ${z \in \mathbb{C}}$ , then μ must be the addition of some point masses, plus some measure absolutely continuous with respect to arc length on countably many rectifiable curves, plus another measure with zero linear density. We also prove that the same conclusion holds if instead of the condition ${\mathcal{C}_{*}\mu(z) < \infty}$ μ-a.e. one assumes ${\mathcal{C}_{*}\mu(z) = o(\mu(B(z,\varepsilon))/\varepsilon)}$ as ${\varepsilon \, \rightarrow \, 0+ \mu}$ -a.e. Partially supported by grants MTM2004-00519 and Acción Integrada HF2004-0208 (Spain), and 2001-SGR-00431 (Generalitat de Catalunya). Received: July 2005 Accepted: October 2005

Keywords:

and phrases:" target="_blank"> and phrases: Cauchy transform maximal Cauchy transform principal values rectifiability curvature of measures

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