Internal characteristics of domains in $\mathbb {C}^{n}$

Vyacheslav Zakharyuta¹

¹ Sabancı University 34956 Tuzla/İstanbul, Turkey

Annales Polonici Mathematici, Tome 111 (2014) no. 3, pp. 215-236

Résumé

This paper is devoted to internal capacity characteristics of a domain $D\subset \mathbb{C}^{n}$, relative to a point $a\in D$, which have their origin in the notion of the conformal radius of a simply connected plane domain relative to a point. Our main goal is to study the internal Chebyshev constants and transfinite diameters for a domain $D\subset \mathbb{C}^{n}$ and its boundary $\partial D$ relative to a point $a\in D$ in the spirit of the author's article [Math. USSR-Sb. 25 (1975), 350–364], where similar characteristics have been investigated for compact sets in $\mathbb{C}^{n}$. The central notion of directional Chebyshev constants is based on the asymptotic behavior of extremal monic “polynomials” and “copolynomials” in directions determined by the arithmetic of the index set $\mathbb{Z}^{n}$. Some results are closely related to results on the $s$th Reiffen pseudometrics and internal directional analytic capacities of higher order (Jarnicki–Pflug, Nivoche) describing the asymptotic behavior of extremal “copolynomials” in varied directions when approaching the point $a$.

Détail

DOI : 10.4064/ap111-3-1

Keywords: paper devoted internal capacity characteristics domain subset mathbb relative point which have their origin notion conformal radius simply connected plane domain relative point main study internal chebyshev constants transfinite diameters domain subset mathbb its boundary partial relative point spirit authors article math ussr sb where similar characteristics have investigated compact sets mathbb central notion directional chebyshev constants based asymptotic behavior extremal monic polynomials copolynomials directions determined arithmetic index set mathbb results closely related results sth reiffen pseudometrics internal directional analytic capacities higher order jarnicki pflug nivoche describing asymptotic behavior extremal copolynomials varied directions approaching point