The Lelong number, the Monge-Ampère mass, and the Schwarz symmetrization of plurisubharmonic functions (Q2210777)

Let \(\Omega\subset\mathbb C^n\simeq\mathbb R^{2n}\) be a bounded domain and let \(u:\Omega\longrightarrow\mathbb R\). The Schwarz symmetrization of \(u\) is a radial function \(\widehat u(x)=f(|x|)\) with \(f\) non-decreasing and such that \(|\{u<t\}|=|\{\widehat u<t\}|\), \(t\in\mathbb R\). Let \(B\subset\mathbb C^n\) be the unit ball. The main results of the paper are the following two theorems: -- Let \(u\) be an \(S^1\)-invariant plurisubharmonic function on \(B\), which can be extended invariantly to a slightly larger ball \(B_{1+\delta}\). Then \(\nu_u(0)=\max_{x\in\overline B}\nu_u(x)\) and \(\iota_u(0)=\max_{x\in\overline B}\iota_u(x)\), where \(\nu_u(x)\) (resp. \(\iota_u(x)\)) denotes the Lelong number (resp. the integrability index) of \(u\) at \(x\). In particular, \(\iota_u(0)=\frac{\nu_{\widehat u}(0)}n=\lim_{t\to-\infty}\frac{2t}{\log|\{u<t\}|}\). -- Let \(u\) be an \((S^1)^{\times n}\)-invariant plurisubharmonic function on \(B\) with a single pole at the origin, which can be extended invariantly to a slightly larger ball \(B_{1+\delta}\). Then \(\tau_{\widehat u}(0)\leq\tau_u(0)\).