Appendix A. Proof of Lemma 2.4
In this proof, we use Lemma 2.3-(i) and (ii) together with the following inequalities when integrating in \(x \in {\mathbb {T}}^3\),
$$\begin{aligned} \Vert u \Vert _{L^\infty ({\mathbb {T}}^3_x)} \lesssim \Vert u \Vert _{H^2 ({\mathbb {T}}^3_x)}, \quad \Vert u \Vert _{L^6 ({\mathbb {T}}^3_x)} \lesssim \Vert u \Vert _{H^1 ({\mathbb {T}}^3_x)}, \quad \Vert u \Vert _{L^3 ({\mathbb {T}}^3_x)} \lesssim \Vert u \Vert _{H^1 ({\mathbb {T}}^3_x)}.\nonumber \\ \end{aligned}$$
(A.1)
Proof of (i). We write
$$\begin{aligned} \langle Q(f,g),h \rangle _{{{\mathcal {H}}}^3_x L^2_v(m)} = \langle Q(f,g),h \rangle _{L^2_{x,v}(m)} + \sum _{1 \le |\beta | \le 3} \langle \partial ^\beta _x Q(f,g), \partial ^\beta _x h \rangle _{L^2_{x,v}(m \langle v \rangle ^{-2|\beta |s})}, \end{aligned}$$
and
$$\begin{aligned} \partial ^\beta _x Q(f,g) = \sum _{\beta _1 + \beta _2 = \beta } C_{\beta _1,\beta _2} \, Q ( \partial ^{\beta _1}_x f , \partial ^{\beta _2}_x g). \end{aligned}$$
In the following steps we will always consider \(\ell \in (\gamma +1+3/2,k-6s]\) which is possible since \(k>\gamma /2+3+8s\), \(\gamma \le 1\) and \(s \ge 0\).
Step 1. Using Lemma 2.3-(i) applied with \(\varsigma _1=\varsigma _2=s\), \(N_1=\gamma /2+2s\), \(N_2=\gamma /2\) and (A.1) we have
$$\begin{aligned} \begin{aligned}&\langle Q(f,g),h \rangle _{L^2_{x,v}(m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2+2s}m)} \, \Vert h\Vert _{H^s_v(\langle v \rangle ^{\gamma /2}m)} \\&\qquad + \Vert f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2}m)} \, \Vert g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert h\Vert _{L^2_v (\langle v \rangle ^{\gamma /2}m)}\bigg ) \\&\quad \lesssim \Vert f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2+2s}m)} \, \Vert h\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2}m)} \\&\qquad + \Vert f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2}m)}\, \Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Step 2.Case\(|\beta |=1\). Arguing as in the previous step,
$$\begin{aligned} \begin{aligned}&\langle Q(f,\partial ^\beta _xg),\partial ^\beta _x h \rangle _{L^2_{x,v}(\langle v \rangle ^{-2s} m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _xg\Vert _{H^s_v(\langle v \rangle ^{\gamma /2}m)} \, \Vert \nabla _xh\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-2s}m)} \\&\qquad + \Vert f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-2s}m)} \, \Vert \nabla _xg\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _xh\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-2s}m)}\bigg ) \\&\quad \lesssim \Vert f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \,\Vert \nabla _xg\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2}m)} \, \Vert \nabla _xh\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-2s}m)} \\&\qquad + \Vert f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-2s}m)}\, \Vert \nabla _xg\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-2s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Moreover,
$$\begin{aligned} \begin{aligned}&\langle Q(\partial ^\beta _x f, g) , \partial ^\beta _x h \rangle _{L^2_{x,v}(\langle v \rangle ^{-2s} m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert \nabla _x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2} m)} \, \Vert \nabla _x h\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-2s}m)} \, \Vert g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x h\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-2s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla _x f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )}\, \Vert g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2} m)} \, \Vert \nabla _x h\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-2s}m)} \,\Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-2s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Step 3.Case\(|\beta |=2\). When \(\beta _2=\beta \), we have
$$\begin{aligned} \begin{aligned}&\langle Q(f,\partial ^\beta _xg),\partial ^\beta _x h \rangle _{L^2_{x,v}(\langle v \rangle ^{-4s} m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_xg\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-2s}m)} \, \Vert \nabla ^2_xh\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s}m)}\\&\qquad + \Vert f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert \nabla _x^2g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_xh\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)}\bigg ) \\&\quad \lesssim \Vert f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_xg\Vert _{L^2_{x}H^s_{v}(\langle v \rangle ^{\gamma /2-2s}m)} \, \Vert \nabla ^2_xh\Vert _{L^2_x H^s_v (\langle v \rangle ^{\gamma /2-4s}m)}\\&\qquad + \Vert f\Vert _{H^2_x L^2_v (\langle v \rangle ^{\gamma /2-4s}m)}\, \Vert \nabla ^2_xg\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
When \(\beta _1=\beta \), we have
$$\begin{aligned} \begin{aligned}&\langle Q(\partial ^\beta _x f, g) , \partial ^\beta _x h \rangle _{L^2_{x,v}(\langle v \rangle ^{-4s} m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert \nabla ^2_x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x h\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x h\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla ^2_x f\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )}\, \Vert g\Vert _{H^{2,s}_{x,v}(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x h\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \,\Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Finally, when \(|\beta _1|=|\beta _2|=1\), we obtain
$$\begin{aligned} \begin{aligned}&\langle Q( \partial ^{\beta _1}_x f, \partial ^{\beta _2}_x g ), \partial ^{\beta }_x h \rangle _{L^2_x L^2_v(\langle v \rangle ^{-4s}m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3}\bigg (\Vert \nabla _x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x h\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert \nabla _xg\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x h\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla _x f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )}\, \Vert \nabla _xg\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x h\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad +\Vert \nabla _x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert \nabla _xg\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Step 4. Case\(|\beta |=3\). When \(\beta _2=\beta \) we obtain
$$\begin{aligned} \begin{aligned}&\langle Q(f,\partial ^\beta _xg),\partial ^\beta _x h \rangle _{L^2_{x,v}(\langle v \rangle ^{-6s} m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x h\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla _x^3g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_xh\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)}\bigg ) \\&\quad \lesssim \Vert f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x h\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert f\Vert _{H^2_xL^2_{v} (\langle v \rangle ^{\gamma /2-6s}m)}\, \Vert \nabla ^3_xg\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )} \,\Vert \nabla ^3_x h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
If \(|\beta _1|=1\) and \(|\beta _2|=2\) then
$$\begin{aligned} \begin{aligned}&\langle Q( \partial ^{\beta _1}_x f, \partial ^{\beta _2}_x g ), \partial ^{\beta }_x h \rangle _{L^2_x L^2_v( \langle v \rangle ^{-6s}m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3}\bigg (\Vert \nabla _x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x h\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla ^2_xg\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x h\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla _x f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )}\, \Vert \nabla ^2_xg\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x h\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{H^2_xL^2_{v} (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla ^2_xg\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)}\\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
When \(|\beta _1|=2\) and \(|\beta _2|=1\) then we get
$$\begin{aligned} \begin{aligned}&\langle Q( \partial ^{\beta _1}_x f, \partial ^{\beta _2}_x g ), \partial ^{\beta }_x h \rangle _{L^2_x L^2_v(m \langle v \rangle ^{-6s})} \\&\quad \lesssim \int _{{\mathbb {T}}^3}\bigg (\Vert \nabla ^2_x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x h\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla _xg\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x h\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla ^2_x f\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )}\, \Vert \nabla _xg\Vert _{H^{2,s}_{x,v}(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x h\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla _xg\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Finally, when \(\beta _1=\beta \), it follows
$$\begin{aligned} \begin{aligned}&\langle Q( \partial ^{\beta }_x f, g ), \partial ^{\beta }_x h \rangle _{L^2_x L^2_v(m \langle v \rangle ^{-6s})} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert \nabla ^3_x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x h\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^3_x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x h\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla ^3_x f\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )}\, \Vert g\Vert _{H^{2,s}_{x,v}(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x h\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^3_x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x h\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\bar{\mathbf{Y}}}} \, \Vert h\Vert _{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert h\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Proof of (ii). As in the proof of (i), we write
$$\begin{aligned} \langle Q(f,g),g \rangle _{{{\mathcal {H}}}^3_x L^2_v(m)} = \langle Q(f,g),g \rangle _{L^2_{x,v}(m)} + \sum _{1 \le |\beta | \le 3} \langle \partial ^\beta _x Q(f,g), \partial ^\beta _x g \rangle _{L^2_{x,v}(m \langle v \rangle ^{-2|\beta |s})}, \end{aligned}$$
and
$$\begin{aligned} \partial ^\beta _x Q(f,g) = \sum _{\beta _1 + \beta _2 = \beta } C_{\beta _1,\beta _2} \, Q ( \partial ^{\beta _1}_x f , \partial ^{\beta _2}_x g). \end{aligned}$$
In the following steps, we will always consider \(\ell \in (4-\gamma +3/2,k-6s]\). Notice that since \(\gamma \le 1\) and \(s \le 1/2\), the condition \(k>4-\gamma +3/2+6s\) implies \(k>\gamma /2+3+8s\) so that we can apply results from Lemma 2.3.
Step 1. Using Lemma 2.3-(ii) and (A.1), we have
$$\begin{aligned} \begin{aligned}&\langle Q(f,g),g \rangle _{L^2_{x,v}(m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \left( \Vert f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert ^2_{H^{s,*}_v(m)} + \Vert f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2}m)} \, \Vert g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{L^2_v (\langle v \rangle ^{\gamma /2}m)}\right) \\&\quad \lesssim \Vert f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert ^2_{L^2_{x}H^{s,*}_v(m)} + \Vert f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2}m)}\, \Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}^*} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Step 2.Case\(|\beta |=1\). Arguing as in the previous step,
$$\begin{aligned} \begin{aligned}&\langle Q(f,\partial ^\beta _xg),\partial ^\beta _x g \rangle _{L^2_{x,v}(\langle v \rangle ^{-2s} m)} \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _xg\Vert ^2_{H^{s,*}_v(\langle v \rangle ^{-2s} m)} \\&\qquad + \Vert f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-2s}m)} \, \Vert \nabla _xg\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _xg\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-2s}m)}\bigg ) \\&\quad \lesssim \Vert f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _xg\Vert ^2_{L^2_xH^{s,*}_v(\langle v \rangle ^{-2s} m)} \\&\qquad +\Vert f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-2s}m)}\, \Vert \nabla _xg\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-2s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}^*} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Moreover, we also have using Lemma 2.3-(i),
$$\begin{aligned} \begin{aligned}&\langle Q(\partial ^\beta _x f, g) , \partial ^\beta _x g \rangle _{L^2_{x,v}(\langle v \rangle ^{-2s} m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert \nabla _x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2} m)} \, \Vert \nabla _x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-2s}m)} \, \Vert g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x g\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-2s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla _x f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )}\, \Vert g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2} m)} \, \Vert \nabla _x g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-2s}m)} \, \Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-2s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Step 3.Case\(|\beta |=2\). When \(\beta _2=\beta \), we have
$$\begin{aligned} \begin{aligned}&\langle Q(f,\partial ^\beta _xg),\partial ^\beta _x g \rangle _{L^2_{x,v}(\langle v \rangle ^{-4s} m)} \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_xg\Vert ^2_{H^{s,*}_v(\langle v \rangle ^{-4s} m)} \\&\qquad + \Vert f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert \nabla _x^2g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_xg\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)}\bigg ) \\&\quad \lesssim \Vert f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_xg\Vert ^2_{L^2_xH^{s,*}_v(\langle v \rangle ^{-4s} m)} \\&\qquad + \Vert f\Vert _{H^2_{x}L^2_v (\langle v \rangle ^{\gamma /2-4s}m)}\,\Vert \nabla ^2_xg\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}^*} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
When \(\beta _1=\beta \), we have
$$\begin{aligned} \begin{aligned}&\langle Q(\partial ^\beta _x f, g) , \partial ^\beta _x g \rangle _{L^2_{x,v}(\langle v \rangle ^{-4s} m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert \nabla ^2_x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla ^2_x f\Vert _{L^6_xL^2_v(\langle v \rangle ^\ell )}\, \Vert g\Vert _{L^3_x H^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x g\Vert _{L^2_x H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \,\Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \\&\quad \lesssim \Vert \nabla ^2_x f\Vert _{H^1_xL^2_v(\langle v \rangle ^\ell )}\, \Vert g\Vert _{H^{1,s}_{x,v}(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Finally, when \(|\beta _1|=|\beta _2|=1\), we obtain
$$\begin{aligned} \begin{aligned}&\langle Q( \partial ^{\beta _1}_x f, \partial ^{\beta _2}_x g ), \partial ^{\beta }_x g \rangle _{L^2_x L^2_v(\langle v \rangle ^{-4s}m)} \\&\quad \lesssim \int _{{\mathbb {T}}^3}\bigg (\Vert \nabla _x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert \nabla _xg\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-4s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla _x f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )}\, \Vert \nabla _xg\Vert _{L^2_x H^s_v(\langle v \rangle ^{\gamma /2-2s} m)} \, \Vert \nabla ^2_x g\Vert _{L^2_x H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \, \Vert \nabla _xg\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-4s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Step 4. Case \(|\beta |=3\). When \(\beta _2=\beta \) we obtain
$$\begin{aligned} \begin{aligned}&\langle Q(f,\partial ^\beta _xg),\partial ^\beta _x g \rangle _{L^2_{x,v}(\langle v \rangle ^{-6s} m)} \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_xg\Vert ^2_{H^{s,*}_v(\langle v \rangle ^{-6s}m)} \\&\qquad + \Vert f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla _x^3g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_xg\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)}\bigg ) \\&\quad \lesssim \Vert f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_xg\Vert ^2_{L^2_xH^{s,*}_v(\langle v \rangle ^{-6s}m)} \\&\qquad + \Vert f\Vert _{H^2_{x} L^2_{v} (\langle v \rangle ^{\gamma /2-6s}m)}\, \Vert \nabla ^3_xg\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}^*} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
If \(|\beta _1|=1\) and \(|\beta _2|=2\) then
$$\begin{aligned} \begin{aligned}&\langle Q( \partial ^{\beta _1}_x f, \partial ^{\beta _2}_x g ), \partial ^{\beta }_x g \rangle _{L^2_x L^2_v(m \langle v \rangle ^{-6s})} \\&\quad \lesssim \int _{{\mathbb {T}}^3}\bigg (\Vert \nabla _x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla ^2_xg\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla _x f\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )}\, \Vert \nabla ^2_xg\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla _x f\Vert _{H^2_xL^2_{v} (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla ^2_xg\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)}\\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
When \(|\beta _1|=2\) and \(|\beta _2|=1\), we get
$$\begin{aligned} \begin{aligned}&\langle Q( \partial ^{\beta _1}_x f, \partial ^{\beta _2}_x g ), \partial ^{\beta }_x g \rangle _{L^2_x L^2_v(m \langle v \rangle ^{-6s})} \\&\quad \lesssim \int _{{\mathbb {T}}^3}\bigg (\Vert \nabla ^2_x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla _x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla _xg\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla ^2_x f\Vert _{L^6_xL^2_v(\langle v \rangle ^\ell )}\, \Vert \nabla _xg\Vert _{L^3_xH^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla _xg\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \\&\quad \lesssim \Vert \nabla ^2_x f\Vert _{H^1_xL^2_v(\langle v \rangle ^\ell )}\, \Vert \nabla _xg\Vert _{H^{1,s}_{x,v}(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^2_x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert \nabla _xg\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
Finally, when \(\beta _1=\beta \), it follows
$$\begin{aligned} \begin{aligned}&\langle Q( \partial ^{\beta }_x f, g ), \partial ^{\beta }_x g \rangle _{L^2_x L^2_v(m \langle v \rangle ^{-6s})} \\&\quad \lesssim \int _{{\mathbb {T}}^3} \bigg (\Vert \nabla ^3_x f\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-4s} m)} \, \Vert \nabla ^3_x g\Vert _{H^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^3_x f\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert g\Vert _{L^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^2_x g\Vert _{L^2_v (\langle v \rangle ^{\gamma /2-6s}m)} \bigg ) \\&\quad \lesssim \Vert \nabla ^3_x f\Vert _{L^2_{x,v}(\langle v \rangle ^\ell )}\, \Vert g\Vert _{H^{2,s}_{x,v}(\langle v \rangle ^{\gamma /2-4s} m))} \, \Vert \nabla ^3_x g\Vert _{L^2_xH^s_v(\langle v \rangle ^{\gamma /2-6s} m)} \\&\qquad + \Vert \nabla ^3_x f\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \, \Vert g\Vert _{H^2_xL^2_v(\langle v \rangle ^\ell )} \, \Vert \nabla ^3_x g\Vert _{L^2_{x,v} (\langle v \rangle ^{\gamma /2-6s}m)} \\&\quad \lesssim \Vert f\Vert _{{\mathbf {X}}} \, \Vert g\Vert ^2_{{\mathbf {Y}}} + \Vert f\Vert _{{\mathbf {Y}}} \, \Vert g\Vert _{{\mathbf {X}}} \, \Vert g\Vert _{{\mathbf {Y}}}. \end{aligned} \end{aligned}$$
We conclude noticing that \(\Vert g\Vert ^2_{{\mathbf {Y}}} \lesssim \Vert g\Vert ^2_{{\mathbf {Y}}^*}\) from Lemma 2.1.
Proof of (iii). The result is immediate from (ii) and the fact that \(\Vert f\Vert ^2_{{\mathbf {Y}}} \lesssim \Vert f\Vert ^2_{{\mathbf {Y}}^*}\).
Appendix B. Cancellation Lemma and Carleman Representation
We state here two classical tools in the analysis of Boltzmann operator, the cancellation lemma and the Carleman representation. The cancellation lemma comes from [1, Lemma 1], we here state it for the kernel \(B(v-v_*, \sigma ) = b(\cos \theta ) \, |v-v_*|^\gamma \) but it can be generalized to other kernels very easily (for example, we us it with \(B_\delta (v-v_*, \sigma ) = b_\delta (\cos \theta ) \, |v-v_*|^\gamma \) in Sects. 2.2 and 4.3 of with \({\widetilde{B}}_1 (v-v_*,\sigma ) = \chi (|v'-v|) \, b(\cos \theta ) \, |v-v_*|^\gamma \) in Sect. 2.4).
Lemma B.1
(Cancellation lemma). Let f be a measurable function defined on \({\mathbb {R}}^3\). For almost every \(v \in {\mathbb {R}}^3\), we have:
$$\begin{aligned} \int _{{\mathbb {R}}^3 \times {{\mathbb {S}}}^2} B(v-v_*,\sigma ) (f'_*-f_*) \, {\mathrm{d}v}_* \, \mathrm{d}\sigma = (f *S)(v) \end{aligned}$$
where
$$\begin{aligned} S(z) := 2 \pi \, \int _0^{\pi /2} \sin \theta \, b(\cos \theta ) \left( \frac{|z|^\gamma }{\cos ^{\gamma +3} (\theta /2)} - |z|^\gamma \right) \, \mathrm{d}\theta . \end{aligned}$$
For the Carleman representation, we refer to [2] for more details on the version that we state here.
Lemma B.2
(Carleman representation). Let F be a measurable function defined on \(({\mathbb {R}}^3)^4\). For any vector \(\vartheta \in {\mathbb {R}}^3\), we denote by \(E_{0,\vartheta }\) the (hyper)vector plane orthogonal to \(\vartheta \). Then, when all sides are well defined, we have the following equality :
$$\begin{aligned}&\int _{{\mathbb {R}}^3 \times {\mathbb {S}}^2} b(\cos \theta ) |v-v_*|^\gamma F(v,v_*,v',v'_*) \, {\mathrm{d}v}_* \, \mathrm{d}\sigma \\&\quad = \int _{{\mathbb {R}}^3_\vartheta } \mathrm{d}\vartheta \int _{E_{0,\vartheta }} \mathrm{d}\alpha \, \tilde{b}(\alpha ,\vartheta )\,\mathbb {1}_{|\alpha | \ge |\vartheta |} \frac{|\alpha +\vartheta |^{\gamma +1+2s} }{|\vartheta |^{3+2s}} \,F(v,v+\alpha -\vartheta ,v-\vartheta ,v+\alpha ) \end{aligned}$$
where \(\tilde{b}(\alpha ,\vartheta )\) is bounded from above and below by positive constants and \(\tilde{b}(\alpha ,\vartheta )=\tilde{b}(\pm \alpha ,\pm \vartheta )\).
Appendix C. Pseudodifferential Tools
1.1 C.1. Pseudodifferential calculus
We first recall the definitions of the quantizations we shall use in the following. Let us consider a temperate symbol \({\sigma } \in {{\mathcal {S}}}\), we define its standard quantization \({\sigma }(v,D_v)\) for \(f\in L^2({\mathbb {R}}^d)\) by
$$\begin{aligned} {\sigma }(v,D_v) f (v) := \frac{1}{(2\pi )^d} \int e^{iv \cdot \eta } {\sigma }(v,\eta ) \hat{f}(\eta ) \, \mathrm{d}\eta . \end{aligned}$$
The Weyl quantization is defined by
$$\begin{aligned} {\sigma }^w f (v) := \frac{1}{(2\pi )^d} \iint e^{i(v-w) \cdot \eta } {\sigma }\left( \frac{v+w}{2},\eta \right) f(w) \, \mathrm{d}\eta \, \mathrm{d}w. \end{aligned}$$
We recall that for two symbols \({\sigma }\) and \(\tau \) we have
$$\begin{aligned} {{\sigma }^w \tau ^w = ({\sigma } \sharp \tau )^w, \quad {\sigma } \sharp \tau = {\sigma }\tau + \int _0^1 (\partial _\eta {\sigma } \sharp _\theta \partial _v \tau - \partial _v {\sigma } \sharp _\theta \partial _\eta \tau ) \, \mathrm{d}\theta } \end{aligned}$$
(C.1)
where for \(V=(v,\eta )\) we have \(\sharp = \sharp _1\) and for \(\theta \in (0,1]\),
$$\begin{aligned} {\sigma } \sharp _\theta \tau (V):= \frac{1}{2i} \iint e^{-2i [V-V_1,V-V_2]/\theta } {\sigma }(V_1) \tau (V_2)\, \mathrm{d}V_1 \, \mathrm{d}V_2 /(\pi \theta )^d \end{aligned}$$
with \([V_1,V_2] = v_2\cdot \eta _1 - v_1\cdot \eta _2\) the canonical symplectic form on \({\mathbb {R}}^{2d}\). We shall also use the Wick quantization, which has very nice properties concerning positivity of operators (see [30,31,32] for more details on the subject). For this, we first introduce the Gaussian in phase variables
$$\begin{aligned} N(v,\eta ) := (2\pi )^{-d} \text {e}^{-(|v|^2+|\eta |^2)/2}. \end{aligned}$$
(C.2)
The Wick quantization is then defined by
$$\begin{aligned} {\sigma }^\mathrm{Wick}f(v) := ({\sigma } \star N )^w f(v), \end{aligned}$$
(C.3)
where \(\star \) denotes the usual convolution in \((v,\eta )\) variables. Recall that one of the main property of Wick quantization is its positivity:
$$\begin{aligned} \forall \, (v,\eta ) \in {\mathbb {R}}^6, \; {\sigma }(v,\eta )\ge 0 \Rightarrow {\sigma }^{\mathrm{Wick}}\ge 0, \end{aligned}$$
(C.4)
and that the following relation holds (see e.g. [30, Proposition 3.4]):
$$\begin{aligned}{}[ g^{\mathrm{Wick}},iv\cdot \xi ]=\{ g, v\cdot \xi \}^{\mathrm{Wick}}. \end{aligned}$$
(C.5)
The previous definitions extend to symbols in \({{\mathcal {S}}}'\) by duality.
1.2 C.2. The weak semiclassical class \(S_K(g)\)
Let \(\Gamma := |{\mathrm{d}v}|^2 + |\mathrm{d}\eta |^2\) be the flat metric on \({\mathbb {R}}^6_{v,\eta }\). The first point is to verify that the introduced symbols and weights are indeed in a suitable symbolic calculus with large parameter K uniformly in the parameter \(\xi \). For this, we first recall that a weight \(1 \le g\) is said to be temperate with respect to \(\Gamma \) if there exist \(N \ge 1\) and \(C_{N}\) such that for all \((v, \eta )\), \((v',\eta ') \in {\mathbb {R}}^6\)
$$\begin{aligned} g(v',\eta ') \le C_N \, g(v,\eta ) ( 1 + |v'-v| + |\eta '-\eta |)^N. \end{aligned}$$
We now introduce adapted classes of symbols.
Definition C.1
Let g be a temperate weight. We denote by S(g) the symbol class of all smooth functions \({\sigma }(v,\eta )\) (possibly depending on parameters K and \(\xi \)) such that
$$\begin{aligned} \left| \partial _v^\alpha \partial _\eta ^\beta {\sigma }(v,\eta ) \right| \le C_{\alpha ,\beta } g(v,\eta ) \end{aligned}$$
where for any multiindex \(\alpha \) and \(\beta \), \(C_{\alpha ,\beta }\) is uniform in K and \(\xi \). We denote also \(S_K(g)\) the symbol class of all smooth functions \({\sigma }(v,\eta )\) (possibly depending on K and \(\xi \) again) such that
$$\begin{aligned} \left| {\sigma }\right| \le C_{0,0} g \quad \text {and} \quad \forall \, |\beta | \ge 1, \quad \left| \partial _v^\alpha \partial _\eta ^\beta {\sigma } \right| \le C_{\alpha ,\beta } K^{-1/2} g \end{aligned}$$
uniformly in K and \(\xi \). Note that \(S_K(g) \subset S(g)\) and that these definitions are with respect to the flat metric.
Eventually, we shall say that a symbol \({\sigma }\) is elliptic positive in S(g) or \(S_K(g)\) if in addition \({\sigma } \ge 1\) and if there exists a constant C uniform in parameters such that we have \(C^{-1} g \le {\sigma } \le C g\).
Before focusing on the class \(S_K(g)\), we first recall one of the main results concerning the class without parameter (and without weight) S(1):
Lemma C.2
(Calderon Vaillancourt Theorem). Let \({\sigma } \in S(1)\), then \({\sigma }^w\) is a bounded operator with norm depending only on a finite number of semi-norms of \({\sigma }\) in S(1).
The classes \(S_K\) and S have standard internal properties:
Lemma C.3
For K sufficiently large, we have the following:
- (i):
Let g be a temperate weight and consider \({\sigma }\) an elliptic positive symbol in \(S_K(g)\) then for all \(\nu \in {\mathbb {R}}\), \({\sigma }^\nu \in S_K(g^\nu )\);
- (ii):
Let g, h be temperate weights and consider \({\sigma }\) in \(S_K(g)\), \(\tau \) in \(S_K(h)\), then \({\sigma }\tau \) is in \( S_K(g h)\).
Proof
For point a), just notice that if \({\sigma }\) is an elliptic positive symbol in \(S_K(g)\), then \({\sigma } \simeq g\) so that \({\sigma }^\nu \simeq g^\nu \). We also have directly for \(\beta \) a multiindex of length 1
$$\begin{aligned} \left| \partial _\eta ^\beta {\sigma }^\nu \right| = |\nu | {\sigma }^{\nu -1} \left| \partial _\eta ^\beta {\sigma }\right| \le C g^{\nu -1} K^{-1/2} g = C K^{-1/2} g^\nu \end{aligned}$$
using \({\sigma } \simeq g\). Estimates on higher order derivatives are straightforward.
For point b), the computation is also straightforward using the Leibniz rule. \(\square \)
Now we can quantize the previously introduced symbols. The main semiclassical idea behind the introduction of the class \(S_K\) for K large is that invertibility and powers of operators associated to symbols are direct consequences of similar properties of symbols, essentially independently of the quantization.
We first check that the class \(S_K\) is essentially stable by change of quantization.
Lemma C.4
Let g be a temperate weight and consider \(\tilde{{\sigma }}\) a positive elliptic symbol in \(S_K(g)\). We denote \({{\sigma }}\) the Weyl symbol of the operator \(\tilde{{\sigma }}(v,D_v)\) so that \({\sigma }^w = \tilde{{\sigma }}(v,D_v)\) and recall that the Weyl symbol of \({\sigma }^\mathrm{Wick}\) is \({\sigma } \star N\). Then \({\sigma }\) and \( {\sigma }\star N\) are both in \(S_K(g)\). If in addition \(\tilde{{\sigma }}\) is elliptic positive, then \({{\text {Re}}\,}{\sigma }\) and \({{\text {Re}}\,}{\sigma }\star N\) are elliptic positive.
Proof
We first prove the result for \({\sigma }\) supposing that \(\tilde{{\sigma }}\) is elliptic positive. From for e.g. [32] and an adaptation of Lemma 4.4 in [2], we know that
$$\begin{aligned} {\sigma } - \tilde{{\sigma }} \in K^{-1/2} S(g). \end{aligned}$$
(C.6)
Since \(K^{-1/2} S(g) \subset S_K(g)\), this gives that \({{\sigma }} \in S_K(g)\). If in addition \(\tilde{{\sigma }}\) is elliptic positive, then let us prove that \({{\text {Re}}\,}{\sigma }\) also is. There exist constants C, \(C'\) uniform in K large such that
$$\begin{aligned} C^{-1} g - C'K^{-1/2} g \le {{\text {Re}}\,}{{\sigma }} \le C g + C'K^{-1/2} g \end{aligned}$$
if \(C^{-1} g \le {\sigma } \le C g\). Taking K sufficiently large then gives the result.
We now deal with \({\sigma }\star N\), supposing that \({\sigma }\) is in \(S_K(g)\). For \(V=(v,\eta )\) we have
$$\begin{aligned} {\sigma }\star N (V) = \iint {\sigma }(V-W) N(W) \mathrm{d}W \end{aligned}$$
and using the temperance property of g, we get uniformly in all other possible parameters (including K)
$$\begin{aligned} \left| {\sigma }\star N(V)\right| \le \iint C g(V) (1+|W|)^N N(W) \, \mathrm{d}W \le C' g(V). \end{aligned}$$
For the derivatives, we get similarly for multiindex \(\alpha \) and \(\beta \) with \(|\beta | \ge 1\)
$$\begin{aligned} \begin{aligned} \left| \partial ^\alpha _v \partial ^\beta _\eta {\sigma }\star N (V)\right|&\le \iint \left| \partial ^\alpha _v \partial ^\beta _\eta {\sigma }(V-W)\right| N(W) \, \mathrm{d}W\\&\le C K^{-1/2} \iint g(V-W) N(W) \, \mathrm{d}W \\&\le C' K^{-1/2} \iint g(V) (1+|W|)^N N(W) \, \mathrm{d}W \\&\le C'' K^{-1/2} g(V). \end{aligned} \end{aligned}$$
(C.7)
Suppose now that in addition \(\tilde{{\sigma }}\) is elliptic positive, then \({{\text {Re}}\,}{\sigma }\) is elliptic positive and \(C^{-1} g(V) \le {{\text {Re}}\,}{\sigma }(V) \le C g(V)\) for a constant \(C>0\). Since \({{\text {Re}}\,}{\sigma }\star N\) is positive, this implies with the temperance of g that
$$\begin{aligned} c' g(V)\le & {} \iint C^{-1} C_N^{-1} g(V) (1+|W|)^{-N} N(W) \mathrm{d}W \nonumber \\\le & {} {{\text {Re}}\,}{\sigma }\star N(V) \le \iint C C_N g(V) (1+|W|)^N N(W) \mathrm{d}W = C' g(V)\nonumber \\ \end{aligned}$$
(C.8)
for some positive constants \(c'\) and \(C'\), so that \({{\text {Re}}\,}{\sigma }\star N\) is indeed elliptic positive. \(\square \)
Remark C.5
Note that using exactly the same argument as in the proof before, we also get that if \(\tau \) is a given elliptic positive symbol in \(S_K(g)\), with g a temperate weight, then \(\tau \star N\) is also an elliptic positive symbol in \(S_K(g)\).
The next technical lemma is also proven in [2]:
Lemma C.6
(Lemma 4.2 in [2]). Let g be a temperate weight and \({\sigma } \in S_K(g)\). Then for K sufficiently large (depending on a finite number of semi-norms of \({\sigma }\)), the operator \({\sigma }^w\) is invertible and there exists \(H_L\) and \(H_R\) bounded invertible operators that are close to identity as well as their inverse such that
$$\begin{aligned} ({\sigma }^w)^{-1} = H_L ({\sigma }^{-1})^w = ({\sigma }^{-1})^w H_R. \end{aligned}$$
The norms of operators \(H_L\) and \(H_R\) and their inverse can be bounded uniformly in parameters (including K).
Note that by “close to identity uniformly in parameters”, we mean that
$$\begin{aligned} \left\| H_L f\right\| \simeq \left\| H_R f\right\| \simeq \left\| f\right\| \end{aligned}$$
with constants uniform in parameters (including K sufficiently large).
Proof
The proof follows exactly the lines of the one given in [2, Lemma 4.2. i)]. \(\square \)
We now give the main Proposition that will be used in the proof of the technical Lemmas in Sect. 3.2.3.
Proposition C.7
Let g be a temperate weight and consider \({\sigma }\) an elliptic positive symbol in \(S_K(g)\). Then for K sufficiently large, we have the following
$$\begin{aligned} \left\| ({\sigma }^w)^{1/2} f\right\| \simeq \left\| ({\sigma }^{1/2})^w f\right\| \qquad \text { and } \qquad \left\| ({\sigma }^w)^{-1} f\right\| \simeq \left\| ({\sigma }^{-1})^w f\right\| . \end{aligned}$$
(C.9)
In addition, suppose that \(\tau \) is another elliptic positive symbol in \(S_K(g)\) then
$$\begin{aligned} \left\| {\sigma }^w f\right\| \simeq \left\| \tau ^w f\right\| . \end{aligned}$$
(C.10)
In particular, we have
$$\begin{aligned} \left\| {\sigma }^w f\right\| ^2 \simeq \left\| {\sigma }^\mathrm{Wick}f\right\| ^2 \simeq \left( ({\sigma }^2)^\mathrm{Wick}f,f\right) \end{aligned}$$
(C.11)
and
$$\begin{aligned} \left( {\sigma }^w f,f\right) \simeq \left( {\sigma }^\mathrm{Wick}f,f\right) \end{aligned}$$
(C.12)
uniformly in parameters (in particular K).
Proof
We first prove (C.9). For the second almost equality, we just have to notice that from Lemma C.6, we have
$$\begin{aligned} \left\| ({\sigma }^w)^{-1} f\right\| = \left\| H_L({\sigma }^{-1})^w f\right\| \simeq \left\| ({\sigma }^{-1})^w f\right\| \end{aligned}$$
since \(H_L\) is close to identity (uniformly in parameters). For the first part of (C.9), we write that
$$\begin{aligned} \begin{aligned} \left\| {\sigma }^w f\right\| ^2&= ( ({\sigma } \sharp {\sigma })^w f,f) = ( ({\sigma }^2)^w f,f) + ( r^w f,f) \end{aligned} \end{aligned}$$
(C.13)
where \(r = {\sigma }\sharp {\sigma } - {\sigma }^2 \in K^{-1/2} S(g^2)\) by standard symbolic calculus. More precisely, we can write from (C.1)
$$\begin{aligned} r = \int _0^1 (\partial _v {\sigma } \sharp _\theta \partial _\eta {\sigma } -\partial _\eta {\sigma } \sharp _\theta \partial _v {\sigma }) \, \mathrm{d}\theta \end{aligned}$$
and using that \(\partial _v {\sigma } \in S(g)\) and \(\partial _\eta {\sigma } \in K^{-1/2} S(g)\) gives the result by stability of the flat symbol class S(g). We therefore get that
$$\begin{aligned} \begin{aligned} |( r^w f,f)|&= \big | \big ( ({\sigma }^w)^{-1}r^w ({\sigma }^w)^{-1} {\sigma }^w f, {\sigma }^w f ) \big | \\&= \big | \big ( H_L ({\sigma }^{-1})^w r^w ({\sigma }^{-1})^w H_R {\sigma }^w f, {\sigma }^w f ) \big |. \end{aligned} \end{aligned}$$
Now \({\sigma }^{-1} \sharp r \sharp {\sigma }^{-1} \in K^{-1/2} S(1)\) since \({\sigma }^{-1} \in S(g)\), so that \(({\sigma }^{-1})^w r^w ({\sigma }^{-1})^w\) is a bounded operator with norm controlled by a constant times \(K^{-1/2}\). Since \(H_L\) and \(H_R\) are bounded operators independently of K, there exists a constant such that
$$\begin{aligned} |( r^w f,f)| \le C K^{-1/2}\left\| {\sigma }^w f\right\| ^2. \end{aligned}$$
This estimate and (C.13), gives that for K sufficiently large,
$$\begin{aligned} \begin{aligned} \frac{1}{2} \left\| {\sigma }^w f\right\| ^2 \le ( ({\sigma }^2)^w f,f) \le 2\left\| {\sigma }^w f\right\| ^2. \end{aligned} \end{aligned}$$
(C.14)
Taking \({\sigma }^{1/2} \in S_K(g^{1/2})\) (by Lemma C.3) instead of \({\sigma }\), we obtain
$$\begin{aligned} \begin{aligned} \left\| ({\sigma }^{1/2})^w f\right\| ^2 \simeq ( {\sigma }^w f,f)&= \left\| ({\sigma }^w)^{1/2}\right\| ^2 \end{aligned} \end{aligned}$$
and the proof of (C.9) is complete.
Concerning (C.10), we just have to prove one inequality since the result is symmetric in \(\tau \) and \({\sigma }\). For K sufficiently large, we have
$$\begin{aligned} \left\| \tau ^w f\right\|= & {} \left\| \tau ^w ({\sigma }^w)^{-1} {\sigma }^w f\right\| = \left\| \tau ^w ({\sigma }^{-1})^w H_R {\sigma }^w f\right\| \\= & {} \left\| ( \tau \sharp ({\sigma }^{-1}))^w H_R {\sigma }^w f\right\| \le C \left\| {\sigma }^w f\right\| \end{aligned}$$
since \(\tau \sharp ({\sigma }^{-1}) \in S(1)\), so that \(( \tau \sharp ({\sigma }^{-1}))^w\) is bounded (with bound independent of K). By symmetry, this proves (C.10).
We then prove (C.11). We first recall that \({\sigma }^\mathrm{Wick}= ({\sigma }\star N)^w\) and that \({\sigma }\star N\) is elliptic positive in \(S_K(g)\) by Lemma C.4. From (C.10), this directly yields
$$\begin{aligned} \left\| {\sigma }^w f\right\| \simeq \left\| ({\sigma } \star N)^w f\right\| = \left\| {\sigma }^\mathrm{Wick}f\right\| . \end{aligned}$$
By direct computation \(({\sigma }^2\star N)^{1/2}\) is also in \(S_K(g)\) by point b) of Lemma C.3 with \(\nu =2\) and \(\nu =1/2\), respectively, and Lemma C.4. Using again (C.10) and (C.9), yields that
$$\begin{aligned} \left\| {\sigma }^w f\right\|\simeq & {} \left\| (({\sigma }^2\star N)^{1/2})^w f\right\| \\\simeq & {} \left\| (({\sigma }^2\star N)^w)^{1/2} f\right\| = (({\sigma }^2\star N)^w f,f) = (({\sigma }^2)^\mathrm{Wick}f,f). \end{aligned}$$
The proof of the last point (C.12) follows exactly the same lines and we skip it. \(\square \)
