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Lemma 8.4.1B276B9DFAC29

Lemma 8.4.1.B276B9DFAC29 For $\epsilon >0$, let $\chi (t)=-(1/12)\log (2\epsilon -t)$ on $|t|\leq \epsilon $. Suppose $q=\chi '(t)\geq 8$. For any $a,b\in \mathbb C$,

\[ \chi ''(t)|a-b|^2-(9q^2+2q)|a|^2-q|a| \geq q^2|a|^2-1-84q^2|b|^2. \]

In particular, let $v,\widetilde\varphi $ be smooth real functions with $|v-\widetilde\varphi |\leq \epsilon $, and take derivatives in an $\omega $-unitary frame. For a positive diagonal linearized operator with coefficients $B_ p$, setting $t=v-\widetilde\varphi $, $a=v_ p$, $b=\widetilde\varphi _ p$ and summing gives an error bounded by $(1+84q^2\sup |\partial \widetilde\varphi |_\omega ^2)\sum _ pB_ p$. The condition $q\geq 8$ holds throughout this interval if $\epsilon \leq 1/288$.

Proof. Young's inequality gives $|a-b|^2\geq (7/8)|a|^2-7|b|^2$. Since $\chi ''=12q^2$, the coefficient of $|a|^2$ after subtracting $9q^2$ is $3q^2/2$. Moreover, $2q\leq q^2/4$ and $q|a|\leq q^2|a|^2/4+1$. The remaining coefficient is at least $q^2$, and the error involving $b$ is $84q^2|b|^2$. Finally $q\geq 1/(36\epsilon )$, which proves the last assertion. The weighted version follows by summing the scalar inequality with positive coefficients. $\square$

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