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 ${{\mathbf {C}}^n}$.
- Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148 T. G. Chen, On Henkin’s solution of the $\partial$-problem on strictly convex domains in ${{\mathbf {C}}^n}$, Ph.D. Thesis, Univ. of California at Berkeley, 1985. S. Gilbarg and N. Trudinger, Elliptic partial equations of second order, Springer, Berlin, 1977.
- Hans Grauert and Ingo Lieb, Das Ramirezsche Integral und die Lösung der Gleichung $\bar \partial f=\alpha $ im Bereich der beschränkten Formen, Rice Univ. Stud. 56 (1970), no. 2, 29–50 (1971) (German). MR 273057
- G. M. Henkin, Integral representation of functions in strongly pseudoconvex regions, and applications to the $\overline \partial $-problem, Mat. Sb. (N.S.) 82 (124) (1970), 300–308 (Russian). MR 0265625 L. Hörmander, ${L^2}$ estimates and existence theorems for the $\partial$ operator, Acta Math. 113 (1965), 82-152. ---, Introduction to complex analysis in several variables, North-Holland, Amsterdam, 1973.
- Norberto Kerzman, Hölder and $L^{p}$ estimates for solutions of $\bar \partial u=f$ in strongly pseudoconvex domains, Comm. Pure Appl. Math. 24 (1971), 301–379. MR 281944, DOI https://doi.org/10.1002/cpa.3160240303
- Steven G. Krantz, Function theory of several complex variables, 2nd ed., The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR 1162310
- Jeffrey Rauch, An inclusion theorem for ovaloids with comparable second fundamental forms, J. Differential Geometry 9 (1974), 501–505. MR 353225
- Richard Sacksteder, On hypersurfaces with no negative sectional curvatures, Amer. J. Math. 82 (1960), 609–630. MR 116292, DOI https://doi.org/10.2307/2372973
- Nessim Sibony, Un exemple de domaine pseudoconvexe régulier où l’équation $\bar \partial u=f$ n’admet pas de solution bornée pour $f$ bornée, Invent. Math. 62 (1980/81), no. 2, 235–242 (French). MR 595587, DOI https://doi.org/10.1007/BF01389159
- F. W. Warner, Extensions of the Rauch comparison theorem to submanifolds, Trans. Amer. Math. Soc. 122 (1966), 341–356. MR 200873, DOI https://doi.org/10.1090/S0002-9947-1966-0200873-6
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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