Product properties in weighted pluripotential theory (Q1002455)

Given a compact subset \(E\) of \(\mathbb C^n\) and a nonnegative, upper semicontinuous function \(\omega\) on \(E\) such that the set \(\{z\in E:\;\omega(z)>0\}\) is non-pluripolar, one defines the weighted pluricomplex Green function by \[ V_{E,Q}(z):=\sup\{u(z):\;u\in\mathcal L,\;u\leq Q=-\log\omega\;\text{on}\;E\}, \] where \(\mathcal L=\mathcal L(\mathbb C^n):=\{u\in PSH(\mathbb C^n):\;u(z)\leq\frac12\log(1+|z|^2)+c_u\}\) is the Lelong class of plurisubharmonic functions with minimal growth. Similarly to the unweighted case (\(\omega(z)\equiv1\), see [\textit{M. Klimek}, Pluripotential Theory. Oxford etc.: Clarendon Press. (1991; Zbl 0742.31001)]), starting from \(V_{E,Q}\) one can introduce the notions of weighted directional Chebyshev constant \({\tau}^{\omega}(E,\theta)\), weighted transfinite diameter \(d^{\omega}(E)\) as well as weighted logarithmic capacity \(c^{\omega}(E)\). By a known result of \textit{J. Siciak} [Ann. Pol. Math. 39, 175--211 (1981; Zbl 0477.32018)], if \(E\subset\mathbb C^n\) and \(F\subset\mathbb C^m\) are compact, then \(V_{E\times F}(z,w)=\max\{V_E(z),V_F(w)\}\), where \(V_K:=V_{K,0}\). This product property of the pluricomplex Green function has been the departure point to numerous investigations in the unweighted case. In particular, \textit{T. Bloom} and \textit{J.-P. Calvi} [Ann. Pol. Math. 72, No. 3, 285--305 (1999; Zbl 0954.32020)] showed that \(d(E\times F)=d^{\frac{n}{n+m}}(E)\cdot d^{\frac{m}{n+m}}(F)\). In this paper, the authors give a simple proof of the above result via a product property of weighted directional Chebyshev constants. Then, they generalize the Bloom-Calvi result to the weighted case but only with inequalities.