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Existence and weak-type inequalities for Cauchy integrals of general measures on rectifiable curves and sets


Authors: Pertti Mattila and Mark S. Melnikov
Journal: Proc. Amer. Math. Soc. 120 (1994), 143-149
MSC: Primary 30E20; Secondary 42B20
DOI: https://doi.org/10.1090/S0002-9939-1994-1160305-3
MathSciNet review: 1160305
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Abstract | References | Similar Articles | Additional Information

Abstract: If $ \mu $ is a finite complex Borel measure and $ \Gamma $ a Lipschitz graph in the complex plane, then for $ \lambda > 0$

$\displaystyle \left\vert {\left\{ {z \in \Gamma :\mathop {\sup }\limits_{\varep... ...\right\vert \leqslant c(\Gamma ){\lambda ^{ - 1}}\vert\vert\mu \vert{\vert _1}.$

It follows that for any finite Borel measure $ \mu $ and any rectifiable curve $ \Gamma $ the finite principal value

$\displaystyle \mathop {\lim }\limits_{\varepsilon \downarrow 0} \int_{\vert\zeta - z\vert \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}d\mu \zeta } $

exists for almost all (with respect to length) $ z \in \Gamma $.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1160305-3
Article copyright: © Copyright 1994 American Mathematical Society