PomeFE03F7C266DDLemma 5.35.4.FE03F7C266DD Let $u$ be quasi-plurisubharmonic, let $m>0$, and let $a_ m$ be a function which locally satisfies $a_ m=m^{-1}\log \sum _ k|g_ k|^2+O(1)$, where the $g_ k$ generate $\mathcal I(mu)$. For $t>0$ and $s>1$,
Moreover, $u\preceq a_ m$ locally, and hence $\mathcal I(tu)\subseteq \mathcal I(ta_ m)$ for every $t>0$.
Proof. On a sufficiently small coordinate neighborhood, choose generators $g_1,\ldots ,g_ N$ of $\mathcal I(mu)$ and put $B=(\sum _ k|g_ k|^2)\mathrm{e}^{-mu}$. After shrinking the neighborhood, $B$ is integrable and $a_ m=m^{-1}\log \sum _ k|g_ k|^2+O(1)$ with a bounded remainder. If $f\in \mathcal I(sta_ m)$, shrink once more so that $f$ is bounded and $A=|f|^2\mathrm{e}^{-sta_ m}$ is integrable. Then, almost everywhere,
Hölder's inequality with exponents $s$ and $s/(s-1)$ applies because $st/(m(s-1))\leq 1$, and it shows that the left side is locally integrable, also in the case of equality in the bound for $m$. Finally, the local Bergman lower estimate of proposition 5.35.1 gives $u\preceq a_ m$ locally; bounded smooth corrections affect neither this comparison nor any of the ideals. $\square$
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