Hearing pseudoconvexity with the Kohn Laplacian

Fu, Siqi

doi:10.1007/s00208-004-0591-5

Cited by 17 publications

(28 citation statements)

References 21 publications

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“…The converse is also true provided D has Lipschitz boundary or more generally, D satisfies interior(D) = D (see e.g. the remark at the end of the paper in [7]). …”

Section: Hausdorff Property For Domains With Connected Complementmentioning

confidence: 99%

On the Hausdorff property of some Dolbeault cohomology groups

Laurent-Thiébaut

Shaw

2012

Math. Z.

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“…The converse is also true provided D has Lipschitz boundary or more generally, D satisfies interior(D) = D (see e.g. the remark at the end of the paper in [7]). …”

Section: Hausdorff Property For Domains With Connected Complementmentioning

confidence: 99%

On the Hausdorff property of some Dolbeault cohomology groups

Laurent-Thiébaut

Shaw

2012

Math. Z.

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“…The converse of Hörmander's theorem also holds, under the assumption that the interior of the closure of Ω is the domain Ω itself. Sheaf theoretic arguments for the Dolbeault cohomology groups ( [24,16,29,2,23]) can be modified to give a proof of this fact (see, e.g., [8,9]). We remark that some regularity of the boundary is necessary in order to characterize pseudoconvexity by the L 2 -Dolbeault cohomology.…”

Section: Introductionmentioning

confidence: 99%

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

Laurent-Thiébaut

Shaw

2017

Math. Z.

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Let Ω = Ω \ D where Ω is a bounded domain with connected complement in C n (or more generally in a Stein manifold) and D is relatively compact open subset of Ω with connected complement in Ω. We obtain characterizations of pseudoconvexity of Ω and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of Ω and D are Lipschitz and C 2 -smooth respectively, then both Ω and D are pseudoconvex if and only if 0 is not in the spectrum of the ∂-Neumann Laplacian on (0, q)-forms for 1 ≤ q ≤ n − 2 when n ≥ 3; or 0 is not a limit point of the spectrum of the ∂-Neumannn Laplacian on (0, 1)-forms when n = 2. (2000): 32C35, 32C37, 32W05. Mathematics Subject Classification

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“…Spectral behavior of the ∂-Neumann Laplacian is more sensitive to the boundary geometry of the domain than the Dirichlet/Neumann Laplacians. (See [Fu05a,Fu05b] and the references therein for related discussions. )…”

Section: Introductionmentioning

confidence: 99%