The fact that the $L^{1}$-norm of $\psi$ with respect to the probability measure $T\wedge\omega_{\varphi}\wedge\omega^{n-p-1}$ is controlled by its $L^{1}$-norm with respect to $T\wedge\omega^{n-p}$ is similar to the phenomenon already encountered in example 1.8: one can write

$$T\wedge\omega_{\varphi}\wedge\omega^{n-p-1}=T\wedge\omega^{n-p}+dd^{c}S,\;S=(\varphi-\inf_{X}\varphi)T\wedge\omega^{n-p-1}\geq 0.$$

This type of estimates is usually referred to as ”Chern-Levine-Nirenberg inequalities”, in reference to [8] where simpler -but fondamental- $L^{\infty}$-estimates were established (with $\psi=constant$). Estimates involving the $L^{1}$-norm of $\psi$ were first proved in the local context by Cegrell [6] and Demailly [12].