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

Abstract

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 Akudemie der Wneenechaften llrstiiut fur M d h e n d i k DDR -1086 B e r h Mdremtmfie 39

Holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds

“…The main ingredient of the proof of Theorem 2.4 is uniform estimates for solutions of certain ∂-equations on M . In fact, similar estimates are valid on coverings of so-called nondegenerate pseudoconvex polyhedrons on Stein manifolds (see [SH80] and [Heu83] for their definition). This class contains, in particular, piecewise strictly pseudoconvex domains and non-degenerate analytic polyhedrons on Stein manifolds.…”

“…(2) Using the main result of [Heu83] and the estimates from [SH80] one can show that the result of Proposition 5.1 is also valid for coverings of non-degenerate pseudoconvex polyhedrons on Stein manifolds (see [Heu83] and [SH80] for the definition).…”

On the approximation property for Banach spaces predual to H∞-spaces

“…Using the Remmert embedding theorem (see, e.g., [10]) we may assume without loss of generality that M is a closed complex submanifold of some C p . Then according to [14,Lemma 1], there exist a strongly pseudoconvex domain W ⊂ C p with C 2 boundary such that W ∩ M = Ω and a holomorphic map π from a neighbourhood U ofW onto U ∩ M such that π(W ) = Ω and π| U ∩M is the identity map. From here by the covering homotopy theorem, see, e.g., [16], we obtain that for every unbranched covering r : Ω → Ω there exist an unbranched covering q : W → W and holomorphic maps π : W → Ω and i : Ω → W such that π • i = Id and r • π = π • q.…”

An Approximation Theorem and OKA's Principle for Holomorphic Vector Bundles which are Continuous on the Boundary of Strictly Pseudoconvex Domains

“…Further, Theorem 1 holds for strictly pseudoconvex domains, defined as above, on STEIW manifolds. It is possible to reduce this problem t o the case of domains in C" using the simple method of such reduction from [4].…”