Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAnalysis and geometry on worm domains
113
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.
We prove the existence of a roof function for arclength null quadrature domains having finitely many boundary components. This bridges a gap toward classification of arclength null quadrature domains by removing an a priori assumption from previous classification results.
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.
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.
We obtain uniform estimates for the canonical solution to $barpartial u=f$ on the Cartesian product of smoothly bounded planar domains, when $f$ is continuous up to the boundary. This generalizes Landuccis result for the bidisc toward higher dimensional product domains.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
