arXiv subjects
Publications and source records attributed to W. Zwonek.
In the paper we discuss the problem of uniqueness of left inverses (solutions of two point Nevanlinna-Pick problem) in bounded convex domains, strongly linearly convex domains, the symmetrized bidisc and the tetrablock.
We present a result on existence of some kind of peak functions for $\C$-convex domains and for the symmetrized polydisc. Then we apply the latter result to show the equivariance of the set of peak points for $A(D)$ under proper holomorphic mappings. Additionally, we present a description of the set of peak points in the class of bounded pseudoconvex Reinhardt domains.
We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz Lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The paper may be seen as a continuation of the results Abouhajar, White and Young. The proofs rely on a necessary form of complex geodesics in general domains which is also proven in the paper.
Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.
A characterization of $L_h^2$-domains of holomorphy in the class of Hartogs domains in $\Bbb C^2$ is given.
We study the problem of the product property for the Lempert function with many poles and consider some properties of this function mostly for plane domains.
We describe all proper holomorphic mappings of the symmetrized polydisc and study its geometric properties. We also apply the obtained results to the study of the spectral unit ball in $\MM_n(\CC^n)$.
In the paper we find effective formulas for the complex geodesics in the symmetrized bidisc.
We give a characterization of $L_h^2$-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.
We construct a bounded plane domain which is Bergman complete but for which the Bergman kernel does not tend to infinity as the point approaches the boundary.
We study the problem of the boundary behaviour of the Bergman kernel and the Bergman completeness in some classes of bounded pseudoconvex domains, which contain also non-hyperconvex domains. Among the classes for which we prove the Bergman completeness and the convergence of the Bergman kernel to infinity while tending to the boundary are all bounded pseudoconvex balanced domains, all bounded Hartogs domains with balanced fibers over regular domains and some bounded Laurent-Hartogs domains.