Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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Unique Continuation Property for C∞ Functions

  • CHUNG, Young-Bok (Department of Mathematics, Chonnam National University)
  • Published : 2003.07.30
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Abstract

We prove a unique continuation theorem for $C^{\infty}$ functions in pseudoconvex domains in ${\mathbb{C}}^{n}$. More specifically, we show that if ${\Omega}$ is a pseudoconvex domain in ${\mathbb{C}}^n$, if f is in $C^{\infty}({\Omega})$ such that for all multi-indexes ${\alpha},{\beta}$ with ${\mid}{\beta}{\mid}{\geq}1$ and for any positive integer k, there exists a positive constant $C_{{\alpha},{\beta},{\kappa}}$ such that $$|{\frac{{\partial}^{{\mid}{\alpha}{\mid}+{\mid}{\beta}{\mid}}f}{{\partial}z^{\alpha}{\partial}{\bar{z}}^{\beta}}{\mid}{\leq}C_{{\alpha},{\beta},{\kappa}}{\mid}f{\mid}^{\kapp}}\;in\;{\Omega}$$, and if there exists $z_0{\in}{\Omega}$ such that f vanishes to infinite order at $z_0$, then f is identically zero. We also have a sharp result for the case of strongly pseudoconvex domains.

Keywords

Acknowledgement

Supported by : Korea Research Foundation

References

  1. The Cauchy transform, potential theory, and conformal mapping Bell, S.
  2. J. Geometric Anal. v.3 Unique continuation theorem for the ${\bar{\partial}}$-operator and applications Bell, S.
  3. J. Diff. Geometry v.32 A $C^{\infty}$ Schwarz reflection principle in one and several complex variables Bell, S.;Lempert, L.
  4. Les fonctions quasi-analytiques Carleman, T.
  5. Indiana Univ. Math. J. v.42 The Bergman kernel function and the Ahlfors mapping in the plane Chung, Y.B.
  6. Partial differential equations in several complex variables;MS/IP Studies in Advanced Mathematics, 19. xii+380 pp Chen, So-Chin;Shaw, Mei-Chi
  7. Ann. Math. v.78 no.1 Harmonic integrals on strongly pseudoconvex manifolds Kohn, J.J.
  8. Ann. Math. v.79 no.3 Harmonic integrals on strongly pseudoconvex manifolds, II Kohn, J.J.
  9. Trans. Amer. Math. Soc. v.181 Global regularity for ${\bar{\partial}}$on weakly pseudoconvex manifolds Kohn, J.J.
  10. Proc. Symposia Pure Math. v.41 A survey of the ${\bar{\partial}}$-Neumann problem Kohn, J.J.
  11. Series adherentes, regularisation des suites, applications Mandelbrojt, S.
  12. Duke Math. J. v.49 no.1 Equation de Lewy-resolubilite globale de l'equation ${\partial}_{b}u$=fsur la frontiere de domaines faiblement pseudo-convexes de $C^2$ (ou $C^n$) Rosay, J.P.
  13. Proc. Amer. Math. Soc. v.59 no.2 On the Lu Qi-Keng conjecture Suita, N.;Yamada, A.
  14. Acta Math. v.145 The local real analyticity of solutions to ${\leqq}_b$and the ${\bar{\partial}}$-Neumann problem Tarkakoff, D.S.