PomeRemark 16.6.31.0F2E59A9BE30 One could also define the following interpolation between the $\mathcal{I}$-envelope and the $P$-envelope: Suppose $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, $j\in \{ 0,\ldots ,n\} $. Then we let
\[ \begin{split} P_{\theta ,j}[\varphi ]\coloneqq \operatorname *{sup}_{}{}\! {}^{*}\left\{ \psi \in \mathrm{PSH}(X,\theta ): \psi \leq 0,\varphi \preceq \psi , \int _ X \theta _{\varphi }^ j\wedge \theta _{P_{\theta }[\varphi ]_{\mathcal{I}}}^{n-j}\right.\\ \left.=\int _ X \theta _{\psi }^ j\wedge \theta _{P_{\theta }[\psi ]_{\mathcal{I}}}^{n-j} \right\} . \end{split} \]
Based on the techniques developed in section 16.7, one could show that $P_{\theta ,j}[\bullet ]$ is a projection operator. The cases $j=0$ and $j=n$ recover the $\mathcal{I}$- and $P$-envelopes respectively; the intermediate operators $P_{\theta ,j}$ have not, to our knowledge, been systematically studied in the literature.
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