Pluricomplex Green and Lempert functions for equally weighted poles

Abstract

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.Comment: 25 page

Convergence and multiplicities for the Lempert function

“…We skip the easy proof, which makes use of Theorem 3.2. We now return to the study of the example presented in [14]. Let us recall the notation.…”

“…Denote by L S the old-style generalized Lempert function defined in [14]. Since it was not monotonic, we also needed the following definition.…”

“…The inequality g(z)≤ (z) always holds, and it is known that it can be strict, see [2], [9] and [14]. If ever turns out to be plurisubharmonic itself, then must be equal to g, [3].…”

“…This idea was put to use in [14] to exhibit an example where a Lempert function with four poles is different from the corresponding Green 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].…”

“…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.…”