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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
DOI: https://doi.org/10.1090/S0002-9947-1995-1308003-2
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 ${\left \| {{H_\Omega }f} \right \|_{{L^\infty }(\Omega )}} \leqslant {C_\Omega }{\left \| f \right \|_{{L^\infty }(\Omega )}}$ of the $\partial$-problem on strictly convex domains $\Omega$ in ${{\mathbf {C}}^n}$.


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Keywords: Strictly convex domain, principal curvature, <!– MATH $\bar \partial$ –> <IMG WIDTH="18" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\bar \partial$">-problem
Article copyright: © Copyright 1995 American Mathematical Society