ScalingStacks

Proof. (Sketch) We may assume $K$ is not pluripolar, for otherwise $\int_{K}\omega_{\phi}^{n}=0$ and $\text{Cap}_{\omega}(K)=0$. We introduce the Siciak extremal function

$$V_{K,\omega}=\sup\{u\in PSH(X,\omega)|u\leq 0\text{ on }K\},$$

whose upper semicontinuous regularisation $V_{K,\omega}^{*}\in PSH(X,\omega)$. By the Alexander-Taylor comparison principle (cf. [22, Prop. 6.1]),

$$\exp(-\sup_{X}V_{K,\omega})\leq e\exp\left(-(\frac{\text{Vol}(X)}{\text{Cap}_{\omega}(K)})^{1/n}\right).$$

By the Skoda integrability assumption (2), and the fact that $V_{K,\omega}=V_{k,\omega}^{*}$ a.e with respect to $\omega^{n}$ (so by absolute continuity also for $\omega_{\phi}^{n}$),

$$\int_{X}e^{\alpha(\sup_{X}V_{K,\omega}-V_{K,\omega})}\omega_{\phi}^{n}=\int_{X}e^{\alpha(\sup_{X}V_{K,\omega}-V_{K,\omega}^{*})}\omega_{\phi}^{n}\leq A\text{Vol}(X),$$

hence

$$\int_{K}e^{-\alpha V_{K,\omega}}\omega_{\phi}^{n}\leq\int_{X}e^{-\alpha V_{K,\omega}}\omega_{\phi}^{n}\leq Ae^{\alpha}\text{Vol}(X)\exp\left(-\alpha(\frac{\text{Vol}(X)}{\text{Cap}_{\omega}(K)})^{1/n}\right).$$

The volume-capacity estimate (3) follows because $V_{K,\omega}\leq 0$ on $K$. ∎