ScalingStacks

Remark 2.3. Assuming an $L^{1}$-bound on $\phi$, then we can take a suitable cutoff function $\chi$, and via integration by parts,

$$\int_{B_{1}}\sqrt{-1}\partial\bar{\partial}\phi\wedge\omega_{E}^{n-1}\leq\int_{B_{2}}\chi\sqrt{-1}\partial\bar{\partial}\phi\wedge\omega_{E}^{n-1}=\int_{B_{2}}\phi\sqrt{-1}\partial\bar{\partial}\chi\wedge\omega_{E}^{n-1}\leq\left\lVert\chi\right\rVert_{C^{2}}\left\lVert\phi\right\rVert_{L^{1}}\leq C.$$

This simple idea is a basic version of the Chern-Levine inequality, which is another fundamental reason why psh functions are much more regular than the subharmonic functions in general dimensions.