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Uniform Estimates for the ∂-Equation on Pseudoconvex Polyhedra on Stein Manifolds

✍ Scribed by Dieter Heunemann

Publisher
John Wiley and Sons
Year
1983
Tongue
English
Weight
285 KB
Volume
114
Category
Article
ISSN
0025-584X

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✦ Synopsis

Uniform estimates for the &equation on strictly pseudoconvex smooth domains in Cn were obtained in 1969/70 by G R A U E R T ~E B [2], HENKIN Ell], KERZW [5], [0], ~E B 171, and OVRELID [8]. RANGE and Sm generalized these results to p i e c e h e smooth strictly paeudoconvex domains. HENKIN proved uniform estimates a180 for non-degenerated WEIL polyhedra [ 121, [ 131. HENKIN and SERGEJEV [ 101 obtained such estimates for a certain class of so-called pseudoconvex polyhedra which contains the eaaes mentioned above. In all these papers global integral formulas for solving the &equation are used. HENKIN and LEITERER [a] generalized these formulas to STEIN manifolds. This makes it possible to obtain uniform estimates for WE= polyhedra on STEIN manifolds. Earlier it was shown by KERZW [6], [6] that on strictly pseudoconvex smooth domains on STEIN manifolds uniform and HOLDER estimate for the &equation can be obtained by localization.

In the present paper we get uniform estimates for the &equation on pseudoconvex polyhedra on STEIN manifolds by direct reduction to the ease of subdomains in C". We prove that every paeudoconvex polyhedron on a STEIN manifold M is a holomorphic retraction of some pseudoconvex polyhedron D' c C". For the exact formulation see Lemma 1, Lemma 2, Lemma 3.

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