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Analyticity in the boundary of a pseudoconvex domain

Author: Alan V. Noell
Journal: Proc. Amer. Math. Soc. 94 (1985), 450-454
MSC: Primary 32F15; Secondary 32E25
MathSciNet review: 787892
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Abstract: Let $ D$ be a bounded pseudoconvex domain with $ {C^\infty }$ boundary in $ {{\mathbf{C}}^n},{A^\infty }(D)$ the algebra of functions holomorphic in $ D$ and $ {C^\infty }$ up to the boundary, and $ M$ a compact real-analytic manifold in the boundary which is integral for the complex structure of the boundary and which has no complex tangent vectors. A necessary and sufficient condition that each element of $ {A^\infty }(D)$ be real-analytic on $ M$ is that the germ of the complexification of $ M$ be in the boundary. Examples indicate that the quasi-analyticity of $ {A^\infty }(D)$ along $ M$ is possible even in the absence of complex manifolds in the boundary.

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  • [1] A Browder, Introduction to function algebras, Benjamin, New York, 1969. MR 0246125 (39:7431)
  • [2] K. Diederich and J. E. Fornaess, Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), 129-141. MR 55 #10728. MR 0437806 (55:10728)
  • [3] M. Hakim and N. Sibony, Spectre de $ A(\bar \Omega )$ pour des domaines bornés faiblement pseudoconvexes réguliers, J. Funct. Anal. 37 (1980), 127-135. MR 81 #46072. MR 578928 (81g:46072)
  • [4] S. Mandelbrojt, Séries adhérentes, regularisation des suites, applications, Gauthier-Villars, Paris, 1952. MR 0051893 (14:542f)
  • [5] A. Noell, Peak points in boundaries not of finite type, Pacific J. Math. (to appear). MR 840849 (87i:32023)
  • [6] -, Interpolation in weakly pseudoconvex domains in $ {{\mathbf{C}}^2}$, Math. Ann. (to appear).
  • [7] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR 0365062 (51:1315)
  • [8] N. Sibony, Prolongement analytique des fonctions holomorphes bornées, C. R. Acad. Sci. Paris Ser. A 275 (1972), 973-976. MR 47 #7062. MR 0318515 (47:7062)
  • [9] E. Stein, Singular integrals and estimates for the Cauchy-Riemann equations, Bull. Amer. Math. Soc. (N.S.) 79 (1973), 440-445. MR 47 #3851. MR 0315302 (47:3851)
  • [10] R. O. Wells, Holomorphic approximation on real-analytic submanifolds of a complex manifold, Proc. Amer. Math. Soc. 17 (1966), 1272-1275. MR 34 #832. MR 0200946 (34:832)

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Keywords: Pseudoconvex domain, integral manifold, complexification
Article copyright: © Copyright 1985 American Mathematical Society

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