Choquet-Monge-Ampère classes (Q507193)

Let \((X,\omega)\) be an \(n\)-dimensional compact Kähler manifold. The authors introduce and study classes of quasi-plurisubharmonic functions \(\varphi\) on \(X\) defined in terms of the Monge-Ampère capacity \(C_\omega\) of the sublevel sets \(\{\varphi\leq -t\}\) as \(t\to+\infty\). Namely, these are the Choquet-Monge-Ampère classes \[ {Ch}^p(X,\omega)=\bigg\{\varphi\in \operatorname{PSH}(X,\omega): \int_0^\infty t^{p+n-1}C_\omega(\{\varphi\leq -t\})\,dt<\infty\bigg\}. \] They are characterized by the finiteness of the Choquet energy, \[ \int_X(-\varphi)^p[(-\varphi)\omega+\omega_\varphi]^n<\infty, \] and are compeared with standard finite energy classes.