Theorem 13.13.49 (Tosatti, Chu–Zhou).FE0721C3DE01 Let $X$ be a compact Kähler manifold with a Kähler form $\omega $ and let $f\in C^{1,1}(X)$. Then
\[ P_\omega (f)\in C^{1,1}(X), \]
and there is a constant $C=C(X,\omega ,\| f\| _{C^{1,1}})$ such that
\[ \| P_\omega (f)\| _{C^{1,1}}\leq C. \]
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