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Register a new userIntrinsic geometry and boundary structure of plane domains
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For a non-empty compact set $E$ in a proper subdomain $Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $Omega$ by $d(E)$ and $d(E,partialOmega),$ respectively. The quantity $d(E)/d(E,partialOmega)$ is invariant under similarities and plays an important role in Geometric Function Theory. In the present paper, when $Omega$ has the hyperbolic distance $h_Omega(z,w),$ we consider the infimum $kappa(Omega)$ of the quantity $h_Omega(E)/log(1+d(E)/d(E,partialOmega))$ over compact subsets $E$ of $Omega$ with at least two points, where $h_Omega(E)$ stands for the hyperbolic diameter of the set $E.$ We denote the upper half-plane by $mathbb{H}$. Our main results claim that $kappa(Omega)$ is positive if and only if the boundary of $Omega$ is uniformly perfect and that the inequality $kappa(Omega)leqkappa(mathbb{H})$ holds for all $Omega,$ where equality holds precisely when $Omega$ is convex.
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In this paper we consider Hankel operators on domains with bounded intrinsic geometry. For these domains we characterize the $L^2$-symbols where the associated Hankel operator is compact (respectively bounded) on the space of square integrable holomorphic functions.
We describe recent work on the Bergman kernel of the (non-smooth) worm domain in several complex variables. An asymptotic expansion is obtained for the Bergman kernel. Mapping properties of the Bergman projection are studied. Irregularity properties of the kernal at the boundary are established. This is an expository paper, and considerable background is provided. Discussion of the smooth worm is also included.
In this paper, we generalize a recent work of Liu et al. from the open unit ball $mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Caratheodory metric near the boundary of $C^2$ domains and the well-known Grahams estimate on the boundary behavior of the Caratheodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.
The subject of this paper is Beurlings celebrated extension of the Riemann mapping theorem cite{Beu53}. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem contains a number of gaps which seem inherent in Beurlings geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurlings geometric method with a number of new analytic tools, notably $H^p$-space techniques and methods from the theory of Riemann-Hilbert-Poincare problems. One additional advantage of this approach is that it leads to an extension of the Beurling-Riemann mapping theorem for analytic maps with prescribed branching. Moreover, it allows a complete description of the boundary regularity of solutions in the (generalized) Beurling-Riemann mapping theorem extending earlier results that have been obtained by PDE techniques. We finally consider the question of uniqueness in the extended Beurling-Riemann mapping theorem.
We study two geometric properties of reproducing kernels in model spaces $K_theta$where $theta$ is an inner function in the disc: overcompleteness and existence of uniformly minimalsystems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern--Clark point. It is shown that uniformly minimal non-Riesz$ $ sequences of reproducing kernelsexist near each Ahern--Clark point which is not an analyticity point for $theta$, whileovercompleteness may occur only near the Ahern--Clark points of infinite orderand is equivalent to a zero localization property. In this context the notion ofquasi-analyticity appears naturally, and as a by-product of our results we give conditions in thespirit of Ahern--Clark for the restriction of a model space to a radius to be a class ofquasi-analyticity.
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