Theorem 16.27.17337360EF765B

Proof. Choose $A\geq 1$ with $\theta \leq \kappa =A\omega $. Every competitor $v$ for $\operatorname {Cap}_{\theta ,\psi }$ is a competitor for $\operatorname {Cap}_{\kappa ,\psi }$, and its measure for $\kappa $ dominates its measure for $\theta $: since $\kappa +\mathrm{dd}^{\mathrm{c}}v=(\theta +\mathrm{dd}^{\mathrm{c}}v)+(\kappa -\theta )$ with a smooth closed semipositive form $\kappa -\theta $, multilinearity and positivity of non-pluripolar products give $\langle (\kappa +\mathrm{dd}^{\mathrm{c}}v)^ n\rangle \geq \langle (\theta +\mathrm{dd}^{\mathrm{c}}v)^ n\rangle $; hence $\operatorname {Cap}_{\theta ,\psi }\leq \operatorname {Cap}_{\kappa ,\psi }$, and the mass of $\psi $ for $\kappa $ is at least its mass for $\theta $, hence positive. Adding a constant to $\psi $ does not change the capacities, so we may assume $\psi \leq 0$. Let $\Phi =P_\kappa [\psi ]$. By theorem 16.22.11, $\Phi $ is a model potential for $\kappa $ with $\psi \leq \Phi $ and with the same mass for $\kappa $ as $\psi $. Thus $\psi \in \mathcal E(X,\kappa ,\Phi )$, and lemma 16.27.15 and proposition 16.27.12 bound $\operatorname {Cap}_{\kappa ,\psi }$ by a continuous nondecreasing function, vanishing at zero, of $\operatorname {Cap}_\kappa $.

Finally, rescaling the competitors gives

every competitor for $\operatorname {Cap}_\omega $ is one for $\operatorname {Cap}_\kappa $, with a larger measure by the same argument, and if $v$ is a competitor for $\operatorname {Cap}_\kappa $, then $v/A$ is a competitor for $\operatorname {Cap}_\omega $ with measure $A^{-n}(\kappa +\mathrm{dd}^{\mathrm{c}}v)^ n$. This proves the first inequality.

For the second inequality use the model potential $\phi =P_\theta [\psi ]$ for $\theta $, which has the same positive mass $m=\int _ X\langle \theta _\psi ^ n\rangle $ as $\psi $. By (16.27.14), $\operatorname {Cap}_\kappa ^ n\leq (C^ n/m)\operatorname {Cap}_{\theta ,\phi }$. By theorem 16.26.1 with $a=2$, applied to $\psi $ and $\phi $, whose singularity envelopes agree, the potential $\eta =P_\theta (2\psi -\phi )$ has mass $m$ and satisfies $P_\theta [\eta ]=\phi $ and $(\eta +\phi )/2\leq \psi $.

It remains to bound $\operatorname {Cap}_{\theta ,\phi }$ by a function of $\operatorname {Cap}_{\theta ,\psi }$. Put $u=(\eta +\phi )/2$. Since $P_\theta [\eta ]=P_\theta [\phi ]$, the preservation of mixed masses in proposition 16.21.2 shows that $u$ also has mass $m$. Since $u\leq \psi $, this means that $u\in \mathcal E(X,\theta ,\psi )$, and lemma 16.27.15 bounds $\operatorname {Cap}_{\theta ,u}$ by a continuous nondecreasing function, vanishing at zero, of $\operatorname {Cap}_{\theta ,\psi }$. If $v$ is a competitor for $\operatorname {Cap}_{\theta ,\phi }$, then $h=(v+\eta )/2$ satisfies $u-1/2\leq h\leq u$, so it is a competitor for $\operatorname {Cap}_{\theta ,u}$, and $\mu _ h\geq 2^{-n}\mu _ v$. Thus

Combining these inequalities with $\operatorname {Cap}_\omega \leq \operatorname {Cap}_\kappa $ gives $G$; compositions, positive multiples and $n$th roots preserve continuity, monotonicity and the value zero at the origin. The remaining assertions follow by comparing both capacities with $\operatorname {Cap}_\omega $. $\square$