In our context, it holds that

(9.133)

$$\mathscr{F}(0)=-\mathfrak{F}_{0}\Big(\TF(-Q_{\beta})\Big).$$

Since $\mathfrak{F}_{0}:\mathscr{S}_{0}(\mathbb{R}^{3})\to\mathscr{S}_{0}(\mathbb{R}^{3})$ is a smooth map on the space of trace-free symmetric $(3\times 3)$-matrices, and $\mathfrak{F}_{0}(0)=0$, so we have

(9.134)

$$\|\mathscr{F}(0)\|_{\mathfrak{B}}\leq C\|\TF(Q_{\beta})\|_{\mathfrak{B}}.$$

The proof immediately follows from the estimate in Corollary 6.5. ∎