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Lemma 1
There are constants C = C ( n , ω X , χ , γ ) > 0 C=C(n,\omega_{X},\chi,\gamma)>0 β 0 = β 0 ( n , ω X , χ , γ ) > 0 \beta_{0}=\beta_{0}(n,\omega_{X},\chi,\gamma)>0 s > 0 s>0
∫ Ω s exp { β 0 ( − ( φ t + s ) A s 1 / ( n + 1 ) ) n + 1 n } ω X n ≤ C exp ( C E ¯ t ) , \int_{\Omega_{s}}\,{\rm exp}\,\Big\{\beta_{0}\big(\frac{-(\varphi_{t}+s)}{A_{s}^{1/(n+1)}}\big)^{\frac{n+1}{n}}\Big\}\omega_{X}^{n}\leq C\,{\rm exp}\,(C{\overline{E}}_{t}),
where A s := c t n V t ∫ Ω s ( − φ t − s ) e n F t ω X n A_{s}:=\frac{c_{t}^{n}}{V_{t}}\int_{\Omega_{s}}(-\varphi_{t}-s)e^{nF_{t}}\omega_{X}^{n} ( φ t + s ) − (\varphi_{t}+s)_{-}