ScalingStacks

Proof. (Thm 2.7) Combining the first two ingredients, the function $f(t)=(\frac{\int_{\phi\leq-t}\omega_{\phi}^{n}}{\text{Vol}(X)})^{1/2n}$ satisfies

$$tf(t+\tau)\leq Bf(\tau)^{2},\quad 0\leq t\leq 1,\quad\tau\geq 0,$$

We conclude that for $t>t_{0}+4Bf(t_{0})$ the sublevel set $\{\phi\leq-t\}$ has zero $\omega_{\phi}$-measure, and therefore zero capacity by Lemma 2.8, so $\phi$ has the lower estimate as claimed in the first statement.

For the second statement, by (2) we have an a priori exponential decay

$$f(t)\leq A^{1/2n}e^{-\alpha t/2n},\quad t\geq 0,$$

which allows us to find an appropriate $t_{0}$. ∎