(Chern-Levine type estimate) Let $u$ be a psh function on the annulus region $U=\{|\log|z_{i}||<s,\forall i\}\subset(\mathbb{C}^{*})^{n}$, with $\left\lVert u\right\rVert_{L^{\infty}}\lesssim 1$. Then

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  On the shrinked set $E=\{|\log|z_{i}||<s/2\}$ the measure

  $$\int_{E}(\sqrt{-1}\partial\bar{\partial}u)^{n}\leq Cs^{-n}.$$

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  Let $u+v$ is another psh function, with $\left\lVert v\right\rVert_{L^{\infty}}\ll 1$. Let $f$ be any compactly supported function on the square $\{|x_{i}|<1\}\subset\mathbb{R}^{n}$. Then

  $$\int f\{(\sqrt{-1}\partial\bar{\partial}(u+v))^{n}-(\sqrt{-1}\partial\bar{\partial}u)^{n}\}\leq Cs^{-n}\left\lVert f\right\rVert_{C^{2}}\left\lVert v\right\rVert_{L^{\infty}}.$$