PomeProposition 5.33.2.32843D040902 If $u\preceq v$, then $\mathcal I(ku)\subseteq \mathcal I(kv)$ for every real $k>0$. If $u\sim v$, then these ideals agree for every $k>0$.
Proposition 5.33.232843D040902
Proof. Let $g$ be a local holomorphic function on $X$. If $u\leq v+C$, then $\mathrm{e}^{-kv}\leq \mathrm{e}^{kC}\mathrm{e}^{-ku}$. Multiplying by $|g|^2$ shows that local integrability of $|g|^2\mathrm{e}^{-ku}$ implies local integrability of $|g|^2\mathrm{e}^{-kv}$. If $u\sim v$, the same argument applies in both directions. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.