Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Geometry of strictly convex domains and an application to the uniform estimate of the $\overline \partial$-problem
HTML articles powered by AMS MathViewer

by Ten Ging Chen PDF
Trans. Amer. Math. Soc. 347 (1995), 2127-2137 Request permission


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 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 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 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 10.1090/S0002-9947-1966-0200873-6
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 32F20
  • Retrieve articles in all journals with MSC: 32F20
Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 2127-2137
  • MSC: Primary 32F20
  • DOI:
  • MathSciNet review: 1308003