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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Real analytic boundary regularity of the Cauchy kernel on convex domains

Author: So-Chin Chen
Journal: Proc. Amer. Math. Soc. 108 (1990), 423-432
MSC: Primary 32A25; Secondary 32A40, 32F15
MathSciNet review: 1000149
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Abstract: It is well-known that in one complex variable the Cauchy integral preserves real analyticity near the boundary. In this paper we show that the same conclusion also holds on convex domains with real analytic boundary in higher dimension, where the Cauchy kernel is given by the Cauchy-Fantappiè form of order zero generated by the (l.0)-form $ C\left( {\xi ,z} \right)$,

$\displaystyle C\left( {\xi ,z} \right) = \left( {\sum\limits_{j = 1}^n {\frac{{... ...xi _j}}}} \left( \xi \right)\left( {{\xi _j} - {z_j}} \right)} \right)^{ - 1}},$

where $ r\left( \xi \right)$ is the defining function of the domain.

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

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  • [2] R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923
  • [3] David S. Tartakoff, On the global real analyticity of solutions to 𝑐𝑚_{𝑏} on compact manifolds, Comm. Partial Differential Equations 1 (1976), no. 4, 283–311. MR 0410809

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Keywords: convex domain, Cauchy-Fantappiè form
Article copyright: © Copyright 1990 American Mathematical Society