PomeLemma 5.42.1.D38324EEBF88 There are constants $A,K>0$ and $t_*>0$, depending only on $(X,\omega )$ and $C_0$, such that $R_ tu$ is smooth for $0<t<t_*$,
\[ -C_0\leq R_ tu\leq 0,\qquad R_ tu+Kt^2\downarrow u\quad \text{as }t\downarrow 0. \]
For $0<\delta <t_*$ and $c>0$, the bounded function
\[ \Phi _{c,\delta } =\inf _{0<t\leq \delta } \left\{ R_ tu+K(t^2-\delta ^2)+K(t-\delta ) -c\log (t/\delta )\right\} \]
is quasi-plurisubharmonic and satisfies
\[ \omega +\mathrm{dd}^{\mathrm{c}}\Phi _{c,\delta }\geq -(Ac+2K\delta )\omega . \]
At every point, the infimum is attained at some $t>0$.
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