Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExtremal Bases, Geometrically Separated Domains and Applications
91
0
0.0
(
0
)
Authors
Philippe Charpentier
Ask ChatGPT about the research
No Arabic abstract
We introduce the notion of extremal basis of tangent vector fields at a boundary point of finite type of a pseudo-convex domain in $mathbb{C}^n$. Then we define the class of geometrically separated domains at a boundary point, and give a description of their complex geometry. Examples of such domains are given, for instance, by locally lineally convex domains, domains with locally diagonalizable Levi form, and domains for which the Levi form have comparable eigenvalues at a point. Moreover we show that these domains are localizable. Then we define the notion of adapted pluri-subharmonic function to these domains, and we give sufficient conditions for his existence. Then we show that all the sharp estimates for the Bergman ans Szego projections are valid in this case. Finally we apply these results to the examples to get global and local sharp estimates, improving, for examlple, a result of Fefferman, Kohn and Machedon on the Szego projection.
rate research
Read More
We construct holomorphically varying families of Fatou-Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $mathbb C^n$ for $n>1$. This provides a simple proof of the recent result of Yuta Kusakabe to the effect that the complement $mathbb C^nsetminus K$ of any polynomially convex subset $K$ of $mathbb C^n$ is an Oka manifold. The analogous result is obtained with $mathbb C^n$ replaced by any Stein manifold with the density property.
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 present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Caratheodory theorem for univalent holomorphic self-mappings of the open unit disk $mathbb Dsubset mathbb C$. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma.
We study the smoothness of the Siciak-Zaharjuta extremal function associated to a convex body in $mathbb{R}^2$. We also prove a formula relating the complex equilibrium measure of a convex body in $mathbb{R}^n$ to that of its Robin indicatrix. The main tool we use are extremal ellipses.
We consider Sobolev mappings $fin W^{1,q}(Omega,IC)$, $1<q<infty$, between planar domains $Omegasubset IC$. We analyse the Radon-Riesz property for convex functionals of the form [fmapsto int_Omega Phi(|Df(z)|,J(z,f)) ; dz ] and show that under certain criteria, which hold in important cases, weak convergence in $W_{loc}^{1,q}(Omega)$ of (for instance) a minimising sequence can be improved to strong convergence. This finds important applications in the minimisation problems for mappings of finite distortion and the $L^p$ and $Exp$,-Teichmuller theories.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
