Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Čučković, Željko; Şahutoğlu, Sönmez

doi:10.1016/j.jfa.2009.02.018

Cited by 21 publications

(18 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our first result gives a necessary condition for compactness of Hankel operators; the case q = 1 and n = 2 is in [6] (for symbols in C(Ω)), the case q = 1 but general n is in [7] (for symbols in C ∞ (Ω)).…”

Section: Introduction and Resultsmentioning

confidence: 94%

See 1 more Smart Citation

Convex domains, Hankel operators, and maximal estimates

Çeli̇k

Şahutoğlu²,

Straube

2019

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Let 1 ≤ q ≤ (n − 1). We first show that a necessary condition for a Hankel operator on (0, q − 1)-forms on a convex domain to be compact is that its symbol is holomorphic along q-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension (n − 1) (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level q if and only if they hold at all levels. q s=1 λ j s (z) ≤ λ 1 (z) + · · · + λ n−1 (z) for any q-tuple (j 1 , . . . , j q ) andDate: February 24, 2019. 2010 Mathematics Subject Classification. Primary 32W05; Secondary 47B35. Even if we were to drop the requirement that the basis be orthonormal, the next condition would remain independent of the basis chosen, in view of a theorem of Ostrowski which relates eigenvalues of matrices of the form M and S T MS; see for example [12], Theorem 4.5.9.

show abstract

Section: Introduction and Resultsmentioning

confidence: 94%

“…We also note that (n − 1)-dimensional polydiscs in bΩ are open subsets of a complex hyperplane. Indeed, the argument in [11], section 2 (see also [7], Lemma 2) shows that if the supporting real hyperplane to Ω at a point is {x n = 0} in suitable coordinates, then the embedding ψ maps into the complex hyperplane {z n = 0}.…”

Section: Corollarymentioning

confidence: 99%

Convex domains, Hankel operators, and maximal estimates

Çeli̇k

Şahutoğlu²,

Straube

2019

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…But the domain Ω is convex which implies that the disk F(D) is an affine disk (see [FS98,ČŞ09]). Using Lemma 2 we can thus assume that there exists…”

Section: Theorem Let ω Be a Bounded Pseudoconvex Domain Inmentioning

confidence: 99%

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

Čučković¹,

Şahutoğlu²

2017

Math. Scand.

Self Cite

View full text Add to dashboard Cite

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 .

show abstract

“…It is clear that H φ is a bounded operator; however, its compactness depends on both the function theoretic properties of the symbol φ as well as the geometry of the boundary of the domain (see [6]). …”

mentioning

confidence: 99%

“…This will be enough to conclude that N 1 is compact on L 2 (0,1) ( ) The proof of the fact that 0 is B-regular is essentially contained in [5, Lemma 10.1] together with the following fact: Let be a smooth bounded pseudoconvex domain in C 2 . If H z and H w are compact on A 2 ( ) then there is no analytic disc in b (see[6, Corollary 1]).…”

mentioning

confidence: 99%

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

Self Cite

View full text Add to dashboard Cite

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].

show abstract

Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Cited by 21 publications

References 21 publications

Convex domains, Hankel operators, and maximal estimates

Convex domains, Hankel operators, and maximal estimates

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

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

Contact Info

Product

Resources

About