Lemma 23.1.19.AB7F46061184 There is a constant $C$ depending only on $E$ with the following property. For $0<\kappa <1/(200C_0)$, $0<\eta <1/10$, $0<t\leq 1$ and $1/2\leq \mu \leq 1$, define $\sigma >0$ and the functions $h,\zeta $ of $w\in E$ by
\[ \sigma ^2=\frac{2\kappa }{\eta }\log \frac1{\eta t},\qquad h=\eta k_ t,\qquad \zeta =\frac{\sigma }{\mu }\, t\partial _{\bar w}\log k_ t. \]
Then
ED58C2434E12
23.1.20
\begin{equation} \kappa tk_ t- \frac{\mu ^2(1-\eta )h}{1-\eta +h} \left|\zeta -\frac1{1-\eta }\right|^2 \leq C\kappa \exp \left(-\frac1{C\sigma ^2t}\right). \end{equation}
Proof.
If $\sigma ^2t\geq 1/100$, the bound $tk_ t\leq C_0$ gives (23.1.20) after enlarging $C$. Assume therefore that $\sigma ^2t<1/100$. If $|\zeta -(1-\eta )^{-1}|\geq 1/4$, the subtracted expression is at least $h/[100(1+h)]$. Since $\log (1/(\eta t))>2$, the definition of $\sigma $ gives
\[ \kappa t(1+h)\leq \frac{\eta }{400}+C_0\kappa \eta <\frac{\eta }{100}. \]
Multiplication by $k_ t/(1+h)$ shows that the left side of (23.1.20) is nonpositive.
Otherwise $|\zeta |\geq 3/4$, so
\[ |t\partial _{\bar w}\log k_ t|=\frac{\mu |\zeta |}{\sigma } \geq \frac3{8\sigma }. \]
Discarding the nonnegative square and applying lemma 23.1.17 gives (23.1.20) in this case as well.
$\square$
Pome
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