Lectures on the ℒ²-Sobolev Theory of the ∂-Neumann problem

Straube, Emil J.

doi:10.4171/076

Cited by 142 publications

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“…Then [1, Proposition 3.1.7] implies that k ∩ D(z 0 , a/2) satisfies property (P) (hence it is B-regular). Therefore, [15,Corollary 4.13] implies that k is B-regular.…”

Section: Proof Of Theoremmentioning

confidence: 91%

“…Hence K 0 has empty fine interior which implies that K 0 satisfies property (P) (see [15,Proposition 4.17] D(z 0 , a) where W is complex tangential direction. Then [1, Proposition 3.1.7] implies that k ∩ D(z 0 , a/2) satisfies property (P) (hence it is B-regular).…”

Section: Proof Of Theoremmentioning

confidence: 93%

“…As mentioned above, compactness of N 1 implies that H ψ is compact on any bounded pseudoconvex domain (see [12,15,Proposition 4.4.1]). The key ingredient of our proof of the converse is the characterization of the compactness of N 1 in terms of ground state energies of certain Schrödinger operators as previously explored in [5,14].…”

Section: Theorem 1 Let Be a Smooth Bounded Pseudoconvex Hartogs Domaimentioning

confidence: 98%

“…Let be a bounded pseudoconvex domain in C n and φ ∈ C( ). If ∂-Neumann operator N 1 is compact on L 2 (0,1) ( ) then the Hankel operator H φ is compact (see [15,Proposition 4.1]). We are interested in the converse of this observation.…”

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confidence: 99%

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On Compactness of Hankel and the $$\overline{\partial }$$ ∂ ¯ -Neumann Operators on Hartogs Domains in $$\mathbb {C}^2$$ C 2

Şahutoğlu¹,

Zeytuncu²

2016

J Geom Anal

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We prove that on smooth bounded pseudoconvex Hartogs domains in C 2 compactness of the ∂-Neumann operator is equivalent to compactness of all Hankel operators with symbols smooth on the closure of the domain. Keywords Hankel operators · ∂-Neumann problem · Hartogs domains Mathematics Subject Classification Primary 32W05 · Secondary 47B35Let be a bounded pseudoconvex domain in C n and L 2 (0,q) ( ) denote the space of square integrable (0, q) forms for 0 ≤ q ≤ n. The complex Laplacian = ∂∂ * + ∂ * ∂ is a densely defined, closed, self-adjoint linear operator on L 2 (0,q) ( ). Hörmander in [7] showed that when is bounded and pseudoconvex, has a bounded solution operator N q , called the ∂-Neumann operator, for all q. Kohn in [9] showed that the Bergman projection, denoted by B below, is connected to the ∂-Neumann operator via the following formulawhere I denotes the identity operator. For more information about the ∂-Neumann problem we refer the reader to two books [4,15].

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“…Then [1, Proposition 3.1.7] implies that k ∩ D(z 0 , a/2) satisfies property (P) (hence it is B-regular). Therefore, [15,Corollary 4.13] implies that k is B-regular.…”

Section: Proof Of Theoremmentioning

confidence: 91%

Section: Proof Of Theoremmentioning

confidence: 93%

Section: Theorem 1 Let Be a Smooth Bounded Pseudoconvex Hartogs Domaimentioning

confidence: 98%

mentioning

confidence: 99%

See 2 more Smart Citations

On Compactness of Hankel and the $$\overline{\partial }$$ ∂ ¯ -Neumann Operators on Hartogs Domains in $$\mathbb {C}^2$$ C 2

Şahutoğlu¹,

Zeytuncu²

2016

J Geom Anal

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“…We refer the reader to the books [CS01,Str10] and references there in, for more information about the ∂-Neumann problem. In the following theorem we list the properties we need about N (see [CS01, Theorem 4.4.1]).…”

Section: Theoremmentioning

confidence: 99%

Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

Čučković¹,

Şahutoğlu²

2017

Math. Scand.

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ABSTRACT. Let Ω ⊂ C 2 be a bounded convex domain with C 1 -smooth boundary and ϕ ∈ C 1 (Ω) such that ϕ is harmonic on the nontrivial disks in the boundary. We estimate the essential norm of the Hankel operator H ϕ in terms of the ∂ derivatives of ϕ "along" the nontrivial disks in the boundary.Let Ω be a domain in C n for n ≥ 1 and bΩ denote the boundary of Ω. Furthermore, let dV denote the volume measure on Ω and A 2 (Ω) be the Bergman space on Ω, the space of square integrable holomorphic functions on Ω with respect to dV. The Bergman projection P is the orthogonal projection fromwhere I denotes the identity operator on L 2 (Ω).In [ČŞ09] we studied compactness of Hankel operators on smooth bounded pseudoconvex domains with the symbols smooth up to the boundary. Our most complete result is attained on smooth bounded convex domains in C 2 . On such domains we characterize compactness of H ϕ in terms of the behavior of ϕ on the analytic disks in bΩ. Throughout this paper D will denote the unit open disk in C. Theorem ([ČŞ09]).Let Ω be a smooth bounded convex domain in C 2 and ϕ ∈ C ∞ (Ω). Then H ϕ is compact if and only if ϕ • F is holomorphic for all holomorphic F : D → bΩ.In this paper we continue our study of compactness of Hankel operators and obtain estimates on their essential norms. The essential norm T e of a bounded linear operator T : X → Y where X and Y are normed linear spaces, is defined as T e = inf T − K : K : X → Y is a compact linear operator .

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Some aspects of Property for

Zhang

2016

Mathematische Nachrichten

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Lectures on the ℒ²-Sobolev Theory of the ∂-Neumann problem

Cited by 142 publications

References 0 publications

On Compactness of Hankel and the $$\overline{\partial }$$ ∂ ¯ -Neumann Operators on Hartogs Domains in $$\mathbb {C}^2$$ C 2

On Compactness of Hankel and the $$\overline{\partial }$$ ∂ ¯ -Neumann Operators on Hartogs Domains in $$\mathbb {C}^2$$ C 2

Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

Some aspects of Property for

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