Closed Range for ∂ and ∂_b on Bounded Hypersurfaces in Stein Manifolds

2011/06/03 by Harrington, Phillip, Raich, Andrew
#32Q28 #35N15 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Primary 32W05 #Secondary 32W10

paper · doi:10.48550/arxiv.1106.0629

Abstract

We define weak Z(q), a generalization of Z(q) on bounded domains Ω in a Stein manifold Mⁿ that suffices to prove closed range of ∂. Under the hypothesis of weak Z(q), we also show (i) that harmonic (0,q)-forms are trivial and (ii) if ∂Ω satisfies weak Z(q) and weak Z(n-1-q), then \dbar_b has closed range on (0,q)-forms on ∂Ω. We provide examples to show that our condition contains examples that are excluded from (q-1)-pseudoconvexity and the authors' previous notion of weak Z(q).

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