Ideals of holomorphic functions with 𝐶^{∞} boundary values on a pseudoconvex domain

Bierstone, Edward; Milman, Pierre D.

doi:10.1090/S0002-9947-1987-0906818-9

Ideals of holomorphic functions with $C^ \infty$ boundary values on a pseudoconvex domain
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by Edward Bierstone and Pierre D. Milman

Trans. Amer. Math. Soc. 304 (1987), 323-342

DOI: https://doi.org/10.1090/S0002-9947-1987-0906818-9

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Abstract:

We give natural sufficient conditions for the solution of several problems concerning division in the space ${\mathcal {A}^\infty }(\Omega )$ of holomorphic functions with ${\mathcal {C}^\infty }$ boundary values on a pseudoconvex domain $\Omega$ with smooth boundary. The sufficient conditions come from upper semicontinuity with respect to the analytic Zariski topology of a local invariant of coherent analytic sheaves (the "invariant diagram of initial exponents"), and apply to division in the space of ${\mathcal {C}^\infty }$ Whitney functions on an arbitrary closed set. Our theorem on division in ${\mathcal {A}^\infty }(\Omega )$ follows using Kohn’s theorem on global regularity in the $\bar \partial$-Neumann problem.

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