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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Geometry of strictly convex domains and an application to the uniform estimate of the $ \overline\partial$-problem


Author: Ten Ging Chen
Journal: Trans. Amer. Math. Soc. 347 (1995), 2127-2137
MSC: Primary 32F20
MathSciNet review: 1308003
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Abstract: In this paper, we construct a nice defining function $ \rho $ for a bounded smooth strictly convex domain $ \Omega $ in $ {R^n}$ with explicit gradient and Hessian estimates near the boundary $ \partial \Omega $ of $ \Omega $. From the approach, we deduce that any two normals through $ \partial \Omega $ do not intersect in any tubular neighborhood of $ \partial \Omega $ with radius which is less than $ \frac{1} {K}$, where $ K$ is the maximum principal curvature of $ \partial \Omega $. Finally, we apply such $ \rho $ to obtain an explicit upper bound of the constant $ {C_\Omega }$ in the Henkin's estimate $ {\left\Vert {{H_\Omega }f} \right\Vert _{{L^\infty }(\Omega )}} \leqslant {C_\Omega }{\left\Vert f \right\Vert _{{L^\infty }(\Omega )}}$ of the $ \partial $-problem on strictly convex domains $ \Omega $ in $ {{\mathbf{C}}^n}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1308003-2
PII: S 0002-9947(1995)1308003-2
Keywords: Strictly convex domain, principal curvature, $ \bar \partial $-problem
Article copyright: © Copyright 1995 American Mathematical Society