$L^{2}$ Serre duality on domains in complex manifolds and applications

Abstract

An L 2 version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the ∂-operator is established. This duality is used to study the solution of the ∂-equation with prescribed support. Applications are given to ∂-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions.2000 Mathematics Subject Classification. 32C37, 35N15, 32W05.

Hearing pseudoconvexity in Lipschitz domains with holes via $${\bar{\partial }}$$ ∂ ¯

“…The equivalence between (i) and (ii) is a direct consequence of the Serre duality (see [4] or Theorem 2.2). From Proposition 4.7 in [20], we know that H n,n c,L 2 (X \ D) is Hausdorff if and only if H n,n−1 W 1 (D) = 0.…”

“…The necessary condition is a direct consequence of Corollary 3.14 and Theorem 3.13. The sufficient condition follows from Theorem 5 in [4].…”

On the Hausdorff property of some Dolbeault cohomology groups

“…The space L 2 (D) being an Hilbert space is self-dual and moreover the weak maximal realization of a differential operator and its strong minimal realization are dual to each other (see [2]). …”

An extension theorem of holomorphic functions on hyperconvex domains

“…Corollary 3.2 implies that H 2 n,n−q (Ω, e −ψ+δφ , ω) and H 2 n,n−q+1 (Ω, e −ψ+δφ , ω) are {0}. Therefore, the Serre duality in [4] implies that ∂ : L 2 0,q−1 (Ω, e ψ−δφ , ω) → L 2 0,q (Ω, e ψ−δφ , ω) has a closed range and For the convenience of readers, we repeat Lemma 5 in [9]. Let Ω be a smoothly bounded pseudoconvex domain in C n with a plurisubharmonic defining function ϕ ∈ C ∞ (Ω), i.e.…”