The multipole Lempert function is monotone under inclusion of pole sets

Abstract

We prove that the multipole Lempert function is monotone under inclusion of pole sets.Let D be a domain in C n and let A = (a j ) l j=1 , 1 ≤ l ≤ ∞, be a countable (i.e. l = ∞) or a non-empty finite subset of D (i.e. l ∈ N). Moreover, fix a function p :p is called a pole function for A on D and |p| its pole set. In case that B ⊂ A is a non-empty subset we put p B := p on B and p B := 0 on D \ B. p B is a pole function for B.For z ∈ D we set 2000 Mathematics Subject Classification. Primary:32F45.

On the product property for the Lempert function

“…Note that l D (a, ·) := l D ({a}, ·) is the classical Lempert function. The Lempert function is monotone under inclusion of pole sets; moreover (see [6]),…”

Coman conjecture for the bidisc

“…Recall that l D (z; P ; ν) = min{l D (z; A; ν |A ) : ∅ = A ⊂ P } (see [14] for arbitrary D or [19] for D convex). The last equality will be of interest for us since in the case of taut domains (convex and bounded domains are taut) the infimum in the definition of l D (z; P ; ν) will be attained by some analytic disc defining l D (z; A; ν |A ) for some ∅ = A ⊂ P .…”

“…. , N.Recall that l D (z; P ; ν) = min{l D (z; A; ν |A ) : ∅ = A ⊂ P } (see [14] for arbitrary D or [19] for D convex). The last equality will be of interest for us since in the case of taut domains (convex and bounded domains are taut) the infimum in the definition of l D (z; P ; ν) will be attained by some analytic disc defining l D (z; A; ν |A ) for some ∅ = A ⊂ P .The function l D (•; P ; ν) is called the Lempert function with the poles at P and with the weight function ν (or weights ν j ).Analoguously we define the pluricomplex Green function g D (z; P ; ν) with the poles at P and the weight function ν as the supremum of 2010 Mathematics Subject Classification.…”

Convergence and multiplicities for the Lempert function

“…The definition of a generalized Lempert function chosen in [14] had some drawbacks -essentially, it was not monotonic with respect to its system of poles (in an appropriate sense) [14,Proposition 4.3] and did not pass to the limit in some very simple situations [13,Theorem 6.3]. We recall that monotonicity holds when no multiplicities are present, or more generally when a subset of the original set of poles is considered with the same indicators, see [17] and [14,Proposition 3.1] for the convex case, and the more recent [8] for arbitrary domains and weighted Lempert functions.…”