Vol. 89, No. 2, 1980

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On holomorphic approximation in weakly pseudoconvex domains

Frank Hayne Beatrous, Jr. and R. Michael Range

Vol. 89 (1980), No. 2, 249–255

A uniform estimate for solutions to the equation u = α in a weakly pseudoconvex domain is obtained, provided that the form α vanishes near the set of degeneracy of the Levi form. Under the additional hypothesis that the closure of the domain is holomorphically convex, analogous estimates are obtained for solutions defined in a full neighborhood of the closure. Applications are given to Mergelyan type approximation problems in a weakly pseudoconvex domain D. In particular, it is shown that any function in A(D) can be uniformly approximated by functions in A(D) which extend holomorphically across all strongly pseudoconvex boundary points. When D is holomorphically convex, it is shown that the Mergelyan problem can be localized to a small neighborhood of the set on which the Levi form degenerates.

Mathematical Subject Classification 2000
Primary: 32E30
Secondary: 32F15
Received: 22 June 1979
Published: 1 August 1980
Frank Hayne Beatrous, Jr.
R. Michael Range