Non-analytic Bergman and Szegö kernels for weakly pseudoconvex tube domains in $\mathbb C^2$

For any weakly pseudoconvex tube domain in with real analytic boundary, there exist points on the boundary off the diagonal where the Bergman kernel and the Szeg? kernel fail to be real analytic. Received April 6, 1999; in final form August 28, 1999 / Published online December 8, 2000     

On the asymptotics of the on-diagonal Szegö kernel of certain Reinhardt domains

We compute the leading and subleading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.     

Estimates of the Bergman kernel for some pseudoconvex domains

Denote by K_ω(z, ζ) the Bergman kernel of a pseudoconvex domain Ω. For some classes of domains Ω, a relationship is found between the rate of increase of K_ω(z, z) as z tends to ∂Ω, and a purely geometric property of Ω. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 222–245.     

A characterization of products of balls by their isotropy groups

In this paper we will characterize products of balls – especially the ball and the polydisc – in by properties of the isotropy group of a single point. It will be shown that such a characterization is possible in the class of Siegel domains of the second kind, a class that extends the class of bounded homogeneous domains, and that such a characterization is no longer possible in the class of bounded domains with noncompact automorphism groups. The main result is that a Siegel domain of the second kind is biholomorphically equivalent to a product of balls, iff there is a point such that the isotropy group of p contains a torus of dimension n. As an application it will be proved that the only domains biholomorphically equivalent to a Siegel domain of the second kind and to a Reinhardt domain are exactly the domains biholomorphically equivalent to a product of b alls. Received: 27 February 1998 / In final form: 6 August 1998     

Comparing the Bergman and Szegö projections on domains with subelliptic boundary Laplacian

We prove that the difference between the Bergman and Szegö projections on a bounded, pseudoconvex domain (with C ^∞ boundary) is smoothing whenever the boundary Laplacian is subelliptic. An equivalent statement is that the Bergman projection can be represented as a composition of the Szegö and harmonic Bergman projections (along with the restriction and Poisson extension operators) modulo an error that is smoothing. We give several applications to the study of optimal mapping properties for these projections and their difference.     

On Bergman completeness and Bergman stability

It is proved that any bounded pseudoconvex domain in is complete w.r.t. the Bergman metric if its boundary can be described locally as the graph of a continuous function in suitable coordinates for . Further arguments are given concerning the stability problems of the Bergman kernel on non-smooth pseudoconvex domains. Received November 30, 1998 / Revised December 21, 1999 / Published online September 5, 2000     

Extension of bounded holomorphic functions¶in convex domains

The E. Amar and G. Henkin theorem on the bounded extendability of bounded holomorphic functions from certain closed complex submanifolds of strictly pseudoconvex domains to the whole domain is generalized to the case of finite type convex domains and their intersections with affine linear hyperplanes. Suitable integral operators of Berndtsson–Andersson type are constructed and estimated for this purpose. Received: 7 July 2000     

Commutative Algebras of Toeplitz Operators on the Reinhardt Domains

Let D be a bounded logarithmically convex complete Reinhardt domain in centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the C ^*-algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending only on is commutative. We show that the natural action of the n-dimensional torus defines (on a certain open full measure subset of D) a foliation which carries a transverse Riemannian structure having distinguished geometric features. Its leaves are equidistant with respect to the Bergman metric, and the orthogonal complement to the tangent bundle of such leaves is integrable to a totally geodesic foliation. Furthermore, these two foliations are proved to be Lagrangian. We specify then the obtained results for the unit ball.     

The Bergman Kernel Function and Full Group of Holomorphic Automorphism on a Reinhardt Domain

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On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let be a domain with smooth boundary and let α be a C ²- diffeomorphism on satisfying the Carleman condition .We denote by the C*-algebra generated by the Bergman projection of G, all multiplication operators aI and the operator where is the Jacobian of α. A symbol algebra of is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator is isomorphic and isometric to . Submitted: February 11, 2001?Revised: January 27, 2002     

On the Notion of the Bochner–Martinelli Integral for Domains with Rectifiable Boundary

In this paper we discuss the notion of the Bochner–Martinelli kernel for domains with rectifiable boundary in , by expressing the kernel in terms of the exterior normal due to Federer (see [17,18]). We shall use the above mentioned kernel in order to prove both Sokhotski–Plemelj and Plemelj–Privalov theorems for the corresponding Bochner–Martinelli integral, as well as a criterion of the holomorphic extendibility in terms of the representation with Bochner–Martinelli kernel of a continuous function of two complex variables. Explicit formula for the square of the Bochner–Martinelli integral is rediscovered for more general surfaces of integration extending the formula established first by Vasilevski and Shapiro in 1989. The proofs of all these facts are based on an intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis. Submitted: September 6, 2006. Accepted: November 1, 2006.     

Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains

We study regularity of Bergman and Szeg? projections on Sobolev type weighted-sup spaces. The paper covers the case of strongly pseudoconvex domains with C⁴ boundary and, partially, domains of finite type in the sense of D’Angelo. Received: 6 October 2005     

Analytic discs, plurisubharmonic hulls, and non-compactness of the -Neumann operator

We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain Ω in is an obstruction to compactness of the -Neumann operator on Ω, provided that at some point of M, the Levi form of bΩ has the maximal possible rank n−1−dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. Research supported in part by NSF grant number DMS-0100517.     

Serre Problem for Unbounded Pseudoconvex Reinhardt Domains in ℂ<Superscript>2</Superscript>

We give a characterization of non-hyperbolic pseudoconvex Reinhardt domains in ℂ² for which the answer to the Serre problem is positive. Moreover, all non-hyperbolic pseudoconvex Reinhardt domains in ℂ² with non-compact automorphism group are explicitly described.     

Geometry of Reinhardt domains of finite type in ℂ2

The asymptotic behavior of the holomorphic sectional curvature of the Bergman metric on a pseudoconvex Reinhardt domain of finite type in ?² is obtained by rescaling locally the domain to a model domain that is either a Thullen domain Ω _m = {(z ₁,z ₂); ¦z ₁¦² + ¦z ₂¦^2m < 1 or a tube domainT _m = {(z ₁,z ₂); Im_z1 + (Im_z2)^2m < 1}. The Bergman metric for the tube domainT _m is explicitly calculated by using Fourier-Laplace transformation. It turns out that the holomorphic sectional curvature of the Bergman metric on the tube domainT _m at (0, 0) is bounded above by a negative constant. These results are used to construct a complete Kähler metric with holomorphic sectional curvature bounded above by a negative constant for a pseudoconvex Reinhardt domain of finite type in ?².     

On hyperbolicity of pseudoconvex Reinhardt domains

We give a characterization of Kobayashi hyperbolicity of pseudoconvex Reinhardt domains. All such domains turn out to be biholomorphic to a bounded Reinhardt domain. In particular, any Kobayashi hyperbolic pseudoconvex Reinhardt domain is Kobayashi complete.     

Hilbert–Schmidt Hankel Operators over Complete Reinhardt Domains

Let ${\mathcal{R}}$ be an arbitrary bounded complete Reinhardt domain in ${\mathbb{C}^n}$ . We show that for ${n \geq 2}$ , if a Hankel operator with an anti-holomorphic symbol is Hilbert–Schmidt on the Bergman space ${A^2(\mathcal{R})}$ , then it must equal zero. This fact has previously been proved only for strongly pseudoconvex domains and for a certain class of pseudoconvex domains.     

Transformation formulas for the Bergman kernels and projections of Reinhardt domains

Formulas that relate the Bergman kernel and projection of a bounded Reinhardt domain whose closure does not intersect the coordinate planes to those of its covering tube domain are obtained via the Poisson summation formula.