Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An approximation theorem for $ \overline \partial $-closed forms of type $ (n,\,n-1)$


Author: Barnet M. Weinstock
Journal: Proc. Amer. Math. Soc. 26 (1970), 625-628
MSC: Primary 32.70
DOI: https://doi.org/10.1090/S0002-9939-1970-0265638-4
MathSciNet review: 0265638
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ D$ be a bounded open set in $ {C^n}$ with smooth boundary. Then every closed form of type $ (n,n - 1)$ which is $ {C^\infty }$ on $ \bar D$ can be approximated uniformly on $ \bar D$ by $ (n,n - 1)$ forms which are closed in a neighborhood of $ \bar D$. If $ {C^n} - D$ is connected these forms can be chosen to be closed in $ {C^n}$. This is applied to prove that a continuous function on the connected boundary of a bounded domain in $ {C^n}$ admits a holomorphic extension to the interior if and only if it is a weak solution of the tangential Cauchy-Riemann equations.


References [Enhancements On Off] (What's this?)

  • [1] F. Browder, Functional analysis and partial differential equations. II, Math. Ann. 145 (1961/62), 81-226. MR 25 #318. MR 0136857 (25:318)
  • [2] J. Dieudonné and L. Schwartz, La dualité dans les espaces $ (\mathfrak{F})$ et $ (\mathfrak{L}\mathfrak{F})$, Ann. Inst. Fourier Grenoble 1 (1949), 61-101. MR 12, 417. MR 0038553 (12:417d)
  • [3] L. Hörmander, An introduction to complex analysis in several variables, Van Nostrand, Princeton, N. J., 1966. MR 34 #2933. MR 0203075 (34:2933)
  • [4] J. L. Kelley and I. Namioka, Linear topological spaces, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1963. MR 29 #3851. MR 0166578 (29:3851)
  • [5] L. Schwartz, Théorie des distributions. Tome I, Actualités Sci. Indust., no. 1091, Hermann, Paris, 1950. MR 12, 31. MR 0035918 (12:31d)
  • [6] B. Weinstock, Continuous boundary values of analytic functions of several complex variables, Proc. Amer. Math. Soc. 21 (1969), 463-466. MR 38 #6106. MR 0237825 (38:6106)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32.70

Retrieve articles in all journals with MSC: 32.70


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0265638-4
Keywords: Complex differential form, $ {C^\infty }$ topology, theory of distributions, boundary values of analytic functions, tangential Cauchy-Riemann equations
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society