Proof. From (3.7), we have

$$ \begin{align*}\big\| \sqrt{R_0}(L - L_0)\sqrt{R_0} \,\big\|_{\mathfrak I_2}\leq \sqrt2 \|\Lambda\|_{\mathfrak I_2} \lesssim \|q\|_{H^s_\kappa} \lesssim \delta \end{align*} $$

uniformly for $|\kappa |\geq 1$ . Thus, for $\delta>0$ sufficiently small, the series (3.4) converges in operator norm uniformly for $|\kappa |\geq 1$ . It is elementary to then verify that the sum acts as a (two-sided) inverse to $L(\kappa )$ .

This argument also yields that $R-R_0 \in \mathfrak I_2$ . In particular, it admits an integral kernel in $L^2({{\mathbb {R}}}^2)$ . To prove (3.8) is continuous, we only need to verify that the series defining $R-R_0$ converges in the sense of Hilbert–Schmidt operators from $H^{-\frac 34 - \frac s2}_\kappa $ to $H^{\frac 34 + \frac s2}_\kappa $ . This follows readily from (2.20).

The continuity of $G-G_0$ as a function of $(x,y)$ follows from the Hilbert–Schmidt bound on (3.8) because $\frac 34 + \frac s2>\frac 12$ .

For regular q, the identities (3.9) and (3.10) precisely express the fact that G is an integral kernel for $R(\kappa )$ . The issue of how to make sense of them for irregular q is settled by (3.8).

Proof. It suffices to consider the case $\kappa \geq 1$ , as the case $\kappa \leq -1$ is similar; moreover, by (3.15), it suffices to consider $g_{12}(\kappa )$ . Recalling (3.20), we obtain

(3.26) $$ \begin{align} \|g_{12}^{[1]}(\kappa)\|_{H^{s+1}_\kappa} = \|q\|_{H^s_\kappa}. \end{align} $$

To bound the remaining terms in the series, we employ duality and Lemma 2.5:

$$ \begin{align*} \big|\langle f, g_{12}^{[\geq 3]}(\kappa)\rangle\big|&\leq \|(\kappa + \partial)^{-(1+s)}\bar f(\kappa - \partial)^{-(1+s)}\|_{\mathrm{op}}\|(\kappa - \partial)^s q (\kappa+\partial)^{-(\frac34+\frac s2)}\|_{\mathfrak I_2}^2\\ &\qquad\qquad \times\sum_{\ell=1}^\infty\|(\kappa - \partial)^{-\frac14+\frac{s}2}q(\kappa + \partial)^{-\frac14+\frac{s}2}\|_{\mathfrak I_2}^{2\ell-1}\kappa^{-(\ell-1)(1+2s)}\\ &\lesssim |\kappa|^{-(2s+1)} \|f\|_{H^{-(1+s)}_\kappa} \|q\|_{H_\kappa^s}^3, \end{align*} $$

provided $\delta>0$ is sufficiently small. This proves (3.25) and completes the proof of (3.24).

We wish to apply the inverse function theorem to obtain the diffeomorphism property. At the linearized level, we already have

$$\begin{align*}\tfrac{\delta g_{12}}{\delta q}(\kappa)\big|_{q=0} = - (2\kappa - \partial)^{-1} \quad\text{and}\quad \tfrac{\delta g_{12}}{\delta r}(\kappa)\big|_{q=0} = 0 \end{align*}$$

which is an isomorphism, as noted already in (3.26). At the nonlinear level, we apply the resolvent identity, which shows that for any test function $f\in \mathcal {S}$ , we have

$$ \begin{align*} {\left.\frac d{d\varepsilon}\right|}_{\varepsilon = 0} G(x,z;q + \varepsilon f) &= -\int G(x,y;q)\begin{bmatrix}0&f(y)\\\mp\bar f(y)&0\end{bmatrix}G(y,z;q)\,dy. \end{align*} $$

Repeating the analysis used to prove (3.25), we find

$$ \begin{align*} \big\| \tfrac{\delta g_{12}}{\delta r}(\kappa) \big\|_{H^s_\kappa\rightarrow H^{s+1}_\kappa} + \big\| \tfrac{\delta g_{12}}{\delta q}(\kappa)+ (2\kappa - \partial)^{-1} \big\|_{H^s_\kappa\rightarrow H^{s+1}_\kappa} \lesssim \delta^2|\kappa|^{-(2s+1)} \lesssim \delta^2 \end{align*} $$

and so deduce that the diffeomorphism property holds for $\delta>0$ sufficiently small, which can be chosen independent of $|\kappa |\geq 1$ .

Next, we seek to show $g_{12} \in \mathcal {S}$ whenever $q\in B_\delta \cap \mathcal {S}$ , beginning with a consideration of derivatives. For any $h\in {{\mathbb {R}}}$ , we have

$$\begin{align*}g_{12}(x+h;q) = g_{12}(x;q(\cdot + h)). \end{align*}$$

In particular, differentiating n times with respect to h and evaluating at $h=0$ , we may use duality to bound

$$ \begin{align*} \|\partial_x^n g_{12}(\kappa)\|_{H^{s+1}} &\leq \sum\limits_{\ell=0}^\infty\sum\limits_{\substack{\sigma\in \mathbb{N}^{2\ell+1}\\|\sigma| = n}}\binom n\sigma |\kappa|^{-(2s+1)\ell}\prod\limits_{m=1}^{2\ell+1} \|\partial_x^{\sigma_m}q\|_{H^s}\\ &\leq \sum\limits_{\ell=0}^\infty C^{2\ell+1}(2\ell+1)^n|\kappa|^{-(2s+1)\ell} \|q\|_{H^s}^{2\ell}\|\partial_x^nq\|_{H^s}\lesssim_{n}\|\partial_x^nq\|_{H^s}, \end{align*} $$

where the constant $C = C(s)> 0$ may be chosen independent of $\kappa $ . To handle spatial weights, we observe that

$$\begin{align*}x^n(\kappa - \partial)^{-1} = \sum\limits_{m=0}^n(-1)^m \frac{n!}{(n-m)!} (\kappa - \partial)^{-m-1}x^{n-m}. \end{align*}$$

In particular, by duality, we may bound

$$ \begin{align*} \|x^ng_{12}(\kappa)\|_{H^{s+1}} &\leq \sum\limits_{\ell=0}^\infty \sum\limits_{m=0}^nC^{2\ell+1}\frac{n!}{(n-m)!} |\kappa|^{-m-(2s+1)\ell}\|x^{n-m}q\|_{H^s}\|q\|_{H^s}^{2\ell}\\ &\lesssim_{n} \|\langle x\rangle^n q\|_{H^s}. \end{align*} $$

Combining these, we see that if $q\in B_\delta \cap \mathcal {S}$ , then $g_{12}(\kappa )\in \mathcal {S}$ .

Proof. Once again, it suffices to consider the case $\kappa \geq 1$ . Using (2.5) and (3.22), we obtain

$$ \begin{align*} \|\gamma^{[2]}\|_{H_\kappa^{s+1}}\lesssim \kappa^{-(s+\frac12)}\| q\|_{H^s_\kappa}^2. \end{align*} $$

To handle $\gamma ^{[\geq 4]}$ , we use the series representation (3.19) and the same duality argument used to prove (3.25). The estimate (3.28) then follows from (3.27) via (2.4).

Setting $\varkappa =\kappa $ in (3.14), we find that

$$ \begin{align*}\partial_x \big\{ 2 g_{12}(x;\kappa)g_{21}(x;\kappa) - \tfrac12\gamma(x;\kappa)^2 - \gamma(x;\kappa) \big\} = 0. \end{align*} $$

From (3.24) and (3.27), we see that the term in braces vanishes as $|x|\to \infty $ . Thus, the quadratic identity (3.31) follows by integration.

By using this quadratic identity, we may write

(3.32) $$ \begin{align} \gamma^{[\geq 4]} = - \tfrac12 \gamma^2 + 2g_{12}^{[\geq 3]}\cdot g_{21} + 2g_{12}^{[1]}\cdot g_{21}^{[\geq 3]}. \end{align} $$

By Proposition 3.2 and (3.27), we have

$$ \begin{align*} \|g_{12}^{[\geq 3]}\|_{L^2}+\|g_{21}^{[\geq 3]}\|_{L^2}&\lesssim |\kappa|^{-(1+s)}\big[\|g_{12}^{[\geq 3]}\|_{H^{s+1}_\kappa}+ \|g_{21}^{[\geq 3]}\|_{H^{s+1}_\kappa}\big] \lesssim |\kappa|^{-(2+3s)}\|q\|_{H^s_\kappa}^3\\ \|g_{12}\|_{L^2}+\|g_{21}\|_{L^2}&\lesssim |\kappa|^{-(1+s)} \big( \|g_{12}\|_{H_\kappa^{s+1}}+\|g_{21}\|_{H_\kappa^{s+1}}\big)\lesssim |\kappa|^{-(1+s)}\|q\|_{H^s_\kappa} \\ \|\gamma\|_{L^2}&\lesssim |\kappa|^{-(1+s)}\|\gamma\|_{H^{s+1}_\kappa}\lesssim|\kappa|^{-(\frac 32 +2s)}\|q\|_{H^s_\kappa}^2. \end{align*} $$

Thus

$$ \begin{align*} \|\gamma^{[\geq 4]}\|_{L^1} &\lesssim \|\gamma\|_{L^2}^2 + \|g_{12}^{[\geq 3]}\|_{L^2}\|g_{21}\|_{L^2} + \|g_{12}^{[1]}\|_{L^2}\|g_{21}^{[\geq 3]}\|_{L^2}\lesssim |\kappa|^{-(3+4s)}\|q\|_{H^s_\kappa}^4, \end{align*} $$

which yields the estimate (3.30). The estimate (3.29) then follows from applying the Cauchy-Schwarz inequality to (3.22).

If $q\in B_\delta \cap \mathcal {S}$ , then from Proposition 3.2 and the quadratic identity (3.31), we see that $\gamma + \frac 12\gamma ^2\in \mathcal {S}$ . As $H^{s+1}$ is an algebra, we may then bound

$$ \begin{align*} \|\partial_x^n \gamma\|_{H^{s+1}}\left(1 - \|\gamma\|_{H^{s+1}}\right) &\leq \left\|\partial_x^n\left(\gamma + \tfrac12\gamma^2\right)\right\|_{H^{s+1}} + \left(2^{n-1} - 1\right)\|\gamma\|_{H^{n+s}}^2,\\ \|x^n \gamma\|_{H^{s+1}}\left(1 - \|\gamma\|_{H^{s+1}}\right) &\leq \left\|x^n\left(\gamma + \tfrac12\gamma^2\right)\right\|_{H^{s+1}}, \end{align*} $$

so using the estimate (3.27), we see that $\gamma (\kappa )\in \mathcal {S}$ , provided $0<\delta \ll 1$ is sufficiently small.

Proof. Applying the product estimate (2.15) with the quadratic identity (3.31) and the symmetry relation (3.15), we may bound

$$ \begin{align*} |\varkappa|\|\gamma\|_{X^\sigma_\kappa} + \|\gamma\|_{X^{\sigma+1}_\kappa}&\lesssim |\varkappa|^{-(s+\frac12)}\|\gamma\|_{L^\infty_t H^{s+1}_\varkappa}\left(|\varkappa|\|\gamma\|_{X^\sigma_\kappa} + \|\gamma\|_{X^{\sigma+1}_\kappa}\right)\\ &\quad +|\varkappa|^{-(s+\frac12)} \max_{\pm\varkappa}\|g_{12}\|_{L^\infty_t H^{s+1}_\varkappa}\max_{\pm\varkappa}\left(|\varkappa|\|g_{12}\|_{X^\sigma_\kappa} + \|g_{12}\|_{X^{\sigma+1}_\kappa}\right). \end{align*} $$

In view of (3.27), taking $0<\delta \ll 1$ sufficiently small (independently of $\varkappa $ ) and using (3.24), we get

(3.36) $$ \begin{align} |\varkappa|\|\gamma\|_{X^\sigma_\kappa} + \|\gamma\|_{X^{\sigma+1}_\kappa}\lesssim |\varkappa|^{-(s+\frac12)}\|q\|_{L^\infty_t H^s_\varkappa} \max_{\pm\varkappa}\left(|\varkappa|\|g_{12}\|_{X^\sigma_\kappa} + \|g_{12}\|_{X^{\sigma+1}_\kappa}\right). \end{align} $$

As a consequence, the estimate (3.35) follows from the estimate (3.33).

To prove the estimate (3.33), we first apply the estimate (2.14) to obtain

(3.37) $$ \begin{align} |\varkappa|\|g_{12}\|_{X^\sigma_\kappa} + \|g_{12}\|_{X^{\sigma+1}_\kappa} \lesssim \|q\|_{X^\sigma_\kappa} + \|q\|_{L^\infty_t H^s} + |\varkappa|\|g_{12}^{[\geq 3]}\|_{X^\sigma_\kappa} + \|g_{12}^{[\geq 3]}\|_{X^{\sigma+1}_\kappa}. \end{align} $$

From the identity (3.12) for $g_{12}$ , we see that $g_{12}^{[\geq 3]} = -(2\varkappa - \partial )^{-1} (q\gamma )$ . Thus, employing (2.14), we find

$$\begin{align*}|\varkappa|\|g_{12}^{[\geq 3]}\|_{X^\sigma_\kappa} + \|g_{12}^{[\geq 3]}\|_{X^{\sigma+1}_\kappa} \lesssim \|q\gamma\|_{X^\sigma_\kappa} + \|q\gamma\|_{L^\infty_t H^s}. \end{align*}$$

To continue, we use (2.16) together with (3.27) and (3.36) for $\gamma $ to obtain

$$\begin{align*}\|q\gamma\|_{X^\sigma_\kappa}\lesssim |\varkappa|^{-(2s+1)}\|q\|_{L^\infty_t H^s_\varkappa}^2 \Big[\|q\|_{X^\sigma_\kappa} +\max_{\pm \varkappa}\big(|\varkappa|\|g_{12}\|_{X^\sigma_\kappa} + \|g_{12}\|_{X^{\sigma+1}_\kappa}\big)\Big]. \end{align*}$$

Using (2.6) and (3.27), we may bound

$$ \begin{align*} \|q\gamma\|_{L^\infty_t H^s}&\lesssim |\varkappa|^{-(2s+1)} \|q\|_{L^\infty_t H^s}\|q\|_{L^\infty_t H^s_\varkappa}^2. \end{align*} $$

As a consequence,

$$ \begin{align*} &\max_{\pm \varkappa}\big(|\varkappa|\|g_{12}^{[\geq 3]}\|_{X^\sigma_\kappa} + \|g_{12}^{[\geq 3]}\|_{X^{\sigma+1}_\kappa}\big) \\ &\quad\lesssim |\varkappa|^{-(2s+1)} \|q\|_{L^\infty_t H^s_\varkappa}^2\Big[\|q\|_{X^\sigma_\kappa} + \|q\|_{L^\infty_t H^s}+\max_{\pm \varkappa}\big(|\varkappa|\|g_{12}\|_{X^\sigma_\kappa} + \|g_{12}\|_{X^{\sigma+1}_\kappa}\big)\Big]. \end{align*} $$

Combining this with (3.37) and choosing $0<\delta \ll 1$ sufficiently small (independently of $\kappa ,\varkappa $ ), we obtain (3.33) and so also (3.34).

Proof. From (3.20) and (3.38), we see that

$$ \begin{align*}|\varkappa|\big\|\big(\tfrac{g_{12}}{2 + \gamma}\big)^{[1]}\big\|_{H^s} + \big\| \big(\tfrac{g_{12}}{2 + \gamma}\big)^{[1]}\big\|_{ H^{s+1}} \approx \big\| (2\varkappa-\partial) \big(\tfrac{g_{12}(\varkappa)}{2 + \gamma(\varkappa)}\big)^{[1]}\big\|_{ H^s} \approx \|q\|_{H^s}. \end{align*} $$

Thus, (3.41) will follow once we prove (3.42). Moreover, using also (3.12), we find

$$ \begin{align*}(2\varkappa-\partial) \big(\tfrac{g_{12}}{2 + \gamma}\big)^{[\geq 3]} = - \tfrac{\gamma}{2(2+\gamma)} q + \tfrac{g_{12}}{(2 + \gamma)^2} \gamma' \end{align*} $$

and thence

$$ \begin{align*} \text{LHS}(3.42) &\lesssim \big\| \tfrac{\gamma}{2+\gamma} q \big\|_{H^s} + \big\| \tfrac{g_{12}}{(2 + \gamma)^2} \gamma' \big\|_{H^{s}} \\ &\lesssim |\varkappa|^{-(s+\frac12)} \|q\|_{H^s} \big\| \tfrac{\gamma}{2+\gamma} \big\|_{ H^{s+1}_\varkappa} + |\varkappa|^{-(2s+1)} \|q\|_{H^s}\|q\|_{H^s_\varkappa} \big\| \tfrac{g_{12}}{(2 + \gamma)^2} \big\|_{ H^{s+1}_\varkappa}, \end{align*} $$

where the second step was an application of (2.6) and (3.27). To handle the remaining rational functions, we expand as series and employ the algebra property (2.5), together with (3.24) and (3.27). This yields (3.41) for $\delta>0$ sufficiently small.

Next, we prove (3.44), since (3.43) follows from this, (3.38), and (2.14).

In order to prove (3.44), we first employ (3.39). The requisite estimate for the first term was given already in Lemma 3.4. The second summand can be treated by combining that lemma with the algebra property (2.17).

It remains to prove (3.45). Recalling the expansion (3.40), the last two terms are easily controlled using (3.34), (3.35), (3.43), and Lemma 2.4. To control the first two terms, we use (3.12) and (3.32).

Proof. First, we observe that the series (4.1) converges absolutely and uniformly for $|\kappa |\geq 1$ and $q\in B_\delta $ , provided $0<\delta \ll 1$ . This follows from the estimate (3.7). In the same way, convergence holds for the term-wise derivative of the series (4.1) with respect to $\kappa $ . The terms appearing are exactly those from (3.18) and (3.19), and so we may deduce that

$$ \begin{align*}\frac{\partial A}{\partial \kappa} = \int \gamma(x;\kappa)\,dx. \end{align*} $$

This proves the first assertion in (4.5) as well as justifying 1.10. The second assertion of (4.5) then follows, since (3.7) guarantees that $A(\kappa )\to 0$ uniformly on $B_\delta $ as $|\kappa |\to \infty $ .

The conjugation symmetry (4.3) follows immediately from (3.15) and (4.5).

Differentiating the series (4.1) with respect to r yields the series (3.19) for $g_{12}$ with an additional minus sign, thus giving the second assertion in (4.4). The first assertion follows in a parallel manner, or by invoking conjugation symmetry. The third part of (4.4) follows from the first two parts via (3.11).

We now turn our attention to (4.6). First, we must clarify what we mean by $\int \rho $ . When $q\in \mathcal {S}$ , then $\rho $ also belongs to Schwartz class (for $\delta $ small enough), and so the integral can be taken in the classical sense. For $q\in H^s$ , however, we interpret this integral via the duality between $H^s$ and $H^{-s}$ , noting that

$$ \begin{align*}q,r \in H^{s}({{\mathbb{R}}}) \quad\text{and}\quad \tfrac{g_{21}(\kappa)}{2+\gamma(\kappa)} , \tfrac{g_{12}(\kappa)}{2+\gamma(\kappa)} \in H^{1+s}({{\mathbb{R}}}) \hookrightarrow H^{-s}({{\mathbb{R}}}) \end{align*} $$

(see Corollary 3.5). By density and continuity, it suffices to verify (4.6) for $q\in \mathcal {S}$ .

Differentiating (3.12), (3.13), and (3.31) with respect to $\kappa $ and then combining these with the original versions shows

$$ \begin{align*} \partial_x\Big(g_{12}\tfrac{\partial g_{21}}{\partial\kappa} - \tfrac{\partial g_{12}}{\partial\kappa} g_{21}\Big) & =-\gamma(2+\gamma) + (1+\gamma)\tfrac{\partial\ }{\partial\kappa}\big(qg_{21}-rg_{12}\big) - \big(qg_{21}-rg_{12}\big) \tfrac{\partial\gamma}{\partial\kappa}. \end{align*} $$

Using also (3.11), we obtain

$$ \begin{align*} - \big( g_{12}\tfrac{\partial g_{21}}{\partial\kappa} - \tfrac{\partial g_{12}}{\partial\kappa} g_{21}\big) \gamma' = -\gamma(2+\gamma)\tfrac{\partial\ }{\partial\kappa}\big(qg_{21}-rg_{12}\big)+ \big(qg_{21}-rg_{12}\big)(1+\gamma)\tfrac{\partial\gamma}{\partial\kappa}. \end{align*} $$

These identities then combine to show

$$ \begin{align*} \partial_x\frac{g_{12}\tfrac{\partial g_{21}}{\partial\kappa} - \tfrac{\partial g_{12}}{\partial\kappa} g_{21}}{2+\gamma} &= - \gamma + \frac{\partial\ }{\partial\kappa}\frac{qg_{21}-rg_{12}}{2+\gamma} , \end{align*} $$

which can then be integrated in x to yield

$$ \begin{align*} \frac{\partial\ }{\partial\kappa} \int \frac{qg_{21}-rg_{12}}{2+\gamma}\,dx = \int \gamma\,dx = \frac{\partial A}{\partial\kappa}. \end{align*} $$

The veracity of (4.6) then follows by observing that both sides of (4.6) vanish in the limit $|\kappa |\to \infty $ .

Proof. While the first few terms can readily be discovered by brute force, we follow a systematic method based on the biHamiltonian relations

(4.8) $$ \begin{align} \nonumber -2\kappa \tfrac{\delta A}{\delta q} &= \partial\tfrac{\delta A}{\delta q} - r\big[\gamma+1\big] = \partial\tfrac{\delta A}{\delta q} + 2r\partial^{-1}\big(r\tfrac{\delta A}{\delta r} - q\tfrac{\delta A}{\delta q}\big) - r,\\ 2\kappa \tfrac{\delta A}{\delta r} &= \partial\tfrac{\delta A}{\delta r} + q\big[\gamma+1\big] = \partial\tfrac{\delta A}{\delta r} - 2q\partial^{-1}\big(r\tfrac{\delta A}{\delta r} - q\tfrac{\delta A}{\delta q}\big) + q, \end{align} $$

which, in view of (4.4), are merely a recapitulation of (3.12) and (3.13).

By iterating (4.8), we find

(4.9) $$ \begin{align} \nonumber \tfrac{\delta A}{\delta r} &= - g_{12} = \tfrac{q}{2\kappa} + \tfrac{q'}{(2\kappa)^2} + \tfrac{q"-2q^2r}{(2\kappa)^3} + \tfrac{q"'-6qq'r}{(2\kappa)^4} + O(\kappa^{-5}), \\ \tfrac{\delta A}{\delta q} &= g_{21} = \tfrac{r}{2\kappa} - \tfrac{r'}{(2\kappa)^2} + \tfrac{r"-2qr^2}{(2\kappa)^3} - \tfrac{r"'-6qrr'}{(2\kappa)^4} + O(\kappa^{-5}), \end{align} $$

which can then be integrated to recover the series for A; indeed,

$$ \begin{align*}A(q) = \int_0^1 \partial_\theta A(\theta q)\,d\theta = \int_0^1 \langle \bar q, \tfrac{\delta A}{\delta q}(\theta q)\rangle + \langle \bar r, \tfrac{\delta A}{\delta r}(\theta q)\rangle \,d\theta. \end{align*} $$

In following this algorithm, we have found it convenient to successively update the asymptotic expansion of $\gamma $ using (3.31), rather than compute $\partial ^{-1}(r\tfrac {\delta A}{\delta r} - q\tfrac {\delta A}{\delta q})$ by laboriously finding complete derivatives. We record here the key result:

$$ \begin{align*} \tfrac12\gamma = -\tfrac{qr}{(2\kappa)^2} - \tfrac{q'r-qr'}{(2\kappa)^3} & - \tfrac{q"r - q'r' + qr"-3q^2r^2}{(2\kappa)^4} \\ &-\tfrac{q"'r - q"r' + q'r" - qr"' - 6qq'r^2 + 6q^2rr'}{(2\kappa)^5} + O(\kappa^{-6}). \end{align*} $$

This technique is easily automated on a computer algebra system, which we have done as a check on our hand computations.

Proof. If $\kappa = \varkappa $ , there is nothing to prove. Suppose now that $\kappa \neq \varkappa $ . From (4.4) and then (3.14), we deduce that

$$ \begin{align*} \{A(\kappa),A(\varkappa)\} &= \tfrac1{i} \int g_{12}(\kappa)g_{21}(\varkappa) - g_{21}(\kappa)g_{12}(\varkappa) =0.\\[-46pt] \end{align*} $$

Proof. We first decompose $\rho (\varkappa )= \rho ^{[2]} (\varkappa )+ \rho ^{[\geq 4]}(\varkappa )$ with

(4.13) $$ \begin{align} \rho^{[2]} (\varkappa)&= \tfrac12\big(q\cdot\tfrac r{2\varkappa + \partial} + \tfrac q{2\varkappa - \partial}\cdot r\big), \end{align} $$

(4.14) $$ \begin{align} \rho^{[\geq 4]} (\varkappa) &= q\cdot \big(\tfrac{g_{21}(\varkappa)}{2+\gamma(\varkappa)}\big)^{[\geq 3]} - r\cdot (\tfrac{g_{12}(\varkappa)}{2 + \gamma(\varkappa)}\big)^{[\geq 3]}. \end{align} $$

Inspired by (4.2), we compute

(4.15) $$ \begin{align} \pm \int \tfrac12\big[\rho^{[2]}(x;\varkappa)-\rho^{[2]}(x;-\varkappa)\big]\,dx = 2\varkappa \int\frac{|\hat q(\xi)|^2\,d\xi}{4\varkappa^2+\xi^2} \end{align} $$

and so, invoking (2.7), deduce that

(4.16) $$ \begin{align} \pm \int_\kappa^\infty \! \int \tfrac12\big[\rho^{[2]}(x;\varkappa)-\rho^{[2]}(x;-\varkappa)\big] \varkappa^{2s}\,dx\,d\varkappa \approx \|q\|_{H^s_\kappa}^2. \end{align} $$

On the other hand, interpolating the bounds in (3.42), we find

(4.17) $$ \begin{align} \big\| \big(\tfrac{g_{21}(\varkappa)}{2+\gamma(\varkappa)}\big)^{[\geq 3]} \big\|_{H^{-s}} \lesssim |\varkappa|^{-2(1+2s)} \delta \|q\|_{H^s_\varkappa}^2 \end{align} $$

and consequently,

(4.18) $$ \begin{align} \int_\kappa^\infty \bigg| \int \tfrac12\big[\rho^{[\geq 4]}(x;\varkappa)-\rho^{[\geq 4]}(x;-\varkappa)\big]\varkappa^{2s}\,dx \bigg| d\varkappa \lesssim \kappa^{-(1+2s)} \delta^2 \|q\|_{H^s_\kappa}^2. \end{align} $$

Thus, (4.11) follows by choosing $\delta>0$ sufficiently small.

To deduce (4.12), we exploit continuity in time.

Proof. By Plancherel (cf. [Reference Killip and Vişan37, Section 4]), it is easy to show that a bounded set $Q\subset H^s$ is equicontinuous if and only if

$$\begin{align*}\lim\limits_{\kappa\rightarrow+\infty}\sup\limits_{q\in Q}\|q\|_{H^s_\kappa} = 0. \end{align*}$$

The result then follows directly from the estimate (4.12).

Proof. Both identities are elementary computations using (4.4) and (4.8).

Proof. The identities (4.20) simply recapitulate (1.2) and (4.4).

Combining (1.2), the resolvent identity, and Proposition 4.7, we have

$$ \begin{align*} i \frac{d\ }{dt} G(x,z;\varkappa) &= - \int G(x,y;\varkappa)\begin{bmatrix}0&\tfrac{\delta A}{\delta r}(y)\\\tfrac{\delta A}{\delta q}(y)&0\end{bmatrix}G(y,z;\varkappa)\,dy\\ &= \tfrac1{2(\kappa - \varkappa)}\left(\begin{bmatrix}0&\tfrac{\delta A}{\delta r}(x)\\\tfrac{\delta A}{\delta q}(x)&0\end{bmatrix}G(x,z;\varkappa) + G(x,z;\varkappa)\begin{bmatrix}0&\tfrac{\delta A}{\delta r}(z)\\\tfrac{\delta A}{\delta q}(z)&0\end{bmatrix}\right)\\ &\qquad + \tfrac1{4(\kappa - \varkappa)}[\gamma(x;\kappa)+1] \begin{bmatrix}1&0\\0&-1\end{bmatrix}G(x,z;\varkappa) \\ &\qquad - \tfrac1{4(\kappa - \varkappa)}G(x,z;\varkappa)[\gamma(z;\kappa)+1]\begin{bmatrix}1&0\\0&-1\end{bmatrix}. \end{align*} $$

This quantity is actually a continuous function of x and z (as can be seen from the middle expression), and so we may restrict to $z=x$ . Thus, by (4.4),

$$ \begin{align*} i \tfrac{d\ }{dt} G(x,x;\varkappa) &= \tfrac{-1}{2(\kappa - \varkappa)}\begin{bmatrix}g_{12}(\kappa)g_{21}(\varkappa)-g_{12}(\varkappa)g_{21}(\kappa)\!\! & g_{12}(\kappa)[\gamma(\varkappa)+1] \\ -g_{21}(\kappa)[\gamma(\varkappa)+1] & \!\!g_{12}(\kappa)g_{21}(\varkappa)-g_{12}(\varkappa)g_{21}(\kappa) \end{bmatrix} \\ &\quad {} + \tfrac{\gamma(\kappa)+1}{2(\kappa - \varkappa)} \begin{bmatrix}0&g_{12}(\varkappa)\\-g_{21}(\varkappa)&0\end{bmatrix}. \end{align*} $$

This then yields (4.21) directly and (4.22) by invoking (3.14).

The claim (4.23) follows from a lengthy computation using (4.21), (4.22), (3.14), and (3.31).

Proof. From (4.23), we readily find

(4.24) $$ \begin{align} \int j^{[2]}(\varkappa,\kappa)\,dx = \int \frac{2i\varkappa-\xi}{(2\kappa-i\xi)^2} \frac{\hat q(\xi) \hat r(-\xi) }{4\varkappa^2+\xi^2} \,d\xi, \end{align} $$

from which the expansion is readily verified. The analogous formula for $j_\gamma $ is

(4.25) $$ \begin{align} \int j_\gamma^{[2]}(\varkappa,\kappa)\,dx = 2 \int \frac{4\varkappa\xi+i(\xi^2 -4 \varkappa^2)}{(2\kappa-i\xi)^2} \frac{\hat q(\xi) \hat r(-\xi) }{(4\varkappa^2+\xi^2)^2} \,d\xi. \end{align} $$

The fact that this coincides with the $\varkappa $ -derivative of (4.24) is not a coincidence; it reflects the first identity in (4.5).

It is easy to verify that $\partial _\varkappa [\varkappa {\mkern 2mu} C_{\ell }(\varkappa )]$ is never sign definite because it contains the factor $\xi ^2 - 4\varkappa ^2$ .

Proof. As in the proof of Proposition 4.4, we write $\rho (\varkappa ) = \rho ^{[2]} (\varkappa )+ \rho ^{[\geq 4]}(\varkappa )$ . From (4.13), we bound

$$\begin{align*}\left|\int \rho^{[2]}(\varkappa)\,\Psi_h\,dx\right| \lesssim \|\Psi_h q\|_{H^{-\frac12}_\varkappa}\|q\|_{H^{-\frac12}_\varkappa}\lesssim \|q\|_{H^{-\frac12}_\varkappa}^2. \end{align*}$$

Using (4.14) and (4.17), we may bound

$$\begin{align*}\left|\int \rho^{[\geq 4]}(\varkappa)\,\Psi_h\,dx\right| \lesssim \varkappa^{-2(2s+1)}\delta^2\|q\|_{H^s}^2. \end{align*}$$

This completes the proof of the lemma.

Proof. Consider the imaginary part of (5.1). Applying the estimates (5.3) and (5.8) to the LHS and the estimate (5.7) to the RHS, we obtain

$$ \begin{align*} \|(\psi_h^6 q)'\|_{L^2_t H^{-1}_\varkappa}^2 &\lesssim \|q\|_{L^\infty_t H^{-\frac12}_\varkappa}^2 +\varkappa^{-2(2s+1)}\delta^2\left(\|q\|_{X^{s+\frac12}}^2 + \|q\|_{L^\infty_t H^s}^2\right), \end{align*} $$

where the implicit constant is independent of $h,\varkappa $ . We then choose $-\frac 12<s'<s$ and apply the a priori estimate (4.12) to obtain

$$ \begin{align*} \|(\psi_h^6 q)'\|_{L^2_t H^{-1}_\varkappa}^2 &\lesssim \varkappa^{-(2s'+1)} \|q(0)\|_{H^{s'}_\varkappa}^2+ \varkappa^{-2(2s+1)}\delta^2\left(\|q\|_{X^{s+\frac12}}^2 + \|q(0)\|_{H^s}^2\right). \end{align*} $$

Taking $\kappa \geq 1$ and using (2.7), we obtain

(5.14) $$ \begin{align} \nonumber\|(\psi_h^6 q)'\|_{L^2_t H^{s-\frac12}_\kappa}^2 &\approx\int_\kappa^\infty \varkappa^{2s+1}\|(\psi_h^6 q)'\|_{L^2_t H^{-1}_\varkappa}^2\,\tfrac{d\varkappa}\varkappa\\ &\lesssim \|q(0)\|_{H^s_\kappa}^2 + \kappa^{-(2s+1)}\delta^2\left(\|q\|_{X^{s+\frac12}}^2 + \|q(0)\|_{H^s}^2\right). \end{align} $$

To complete the proof, we take $\kappa = 1$ to deduce

$$ \begin{align*} \|\psi_h^6q\|_{L^2_t H^{s+\frac12}}^2 &\lesssim \|P_{>1}(\psi_h^6 q)'\|_{L^2_t H^{s-\frac12}}^2 + \|P_{\leq 1}(\psi_h^6 q)\|_{L^\infty_t H^s}^2 \\ &\lesssim \delta^2\|q\|_{X^{s+\frac12}}^2 + \|q(0)\|_{H^s}^2. \end{align*} $$

Taking the supremum over $h\in {{\mathbb {R}}}$ and choosing $0<\delta \ll 1$ sufficiently small, we obtain the estimate (5.12). The claim (5.13) then follows from (5.12) and (5.14).

Proof. Consider the real part of (5.1). Applying (5.4) and (5.9) to the LHS and applying (5.7) to the RHS, we deduce that

$$ \begin{align*} \varkappa\|(\psi_h^6 q)'\|_{L^2_t H^{-1}_\varkappa}^2 &\lesssim \varepsilon\|(\psi_h^6 q)'\|^2_{L^2_t H^{-\frac12}_\varkappa} + (1 + \tfrac1\varepsilon)\|q\|_{L^\infty_t H^{-\frac12}_\varkappa}^2\\ & \quad + \big[\varkappa^{-1}+\varkappa^{-2(2s+1)}\log^4\!|2\varkappa|\big]\delta^2 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}}^2 \end{align*} $$

for any $0<\varepsilon <1$ . Here, the implicit constant is independent of $h,\varkappa , \varepsilon $ . Applying the a priori estimate (4.12), for any $-\frac 12<s'<s$ , we obtain

$$ \begin{align*} \varkappa\|(\psi_h^6 q)'\|_{L^2_t H^{-1}_\varkappa}^2 &\lesssim \varepsilon\|(\psi_h^6 q)'\|_{L^2_t H^{-\frac12}_\varkappa}^2+ (1 + \tfrac1\varepsilon)\varkappa^{-(2s'+1)}\|q(0)\|_{H^{s'}_\varkappa}^2\\ & \quad + \big[\varkappa^{-1}+\varkappa^{-2(2s+1)}\log^4\!|2\varkappa|\big]\delta^2\Big(\|q\|_{X^{s+1}}^2 + \|q(0)\|_{H^s}^2\Big). \end{align*} $$

Using the estimate (2.7), we obtain

$$ \begin{align*} \|(\psi_h^6 q)'\|_{L^2_t H^s_\kappa}^2&\approx \int_{\kappa}^\infty\varkappa^{2s+2}\|(\psi_h^6 q)'\|_{H^{-1}_\varkappa}^2\,\tfrac{d\varkappa} \varkappa \\ &\lesssim \varepsilon \|(\psi_h q)'\|_{L^2_t H^s_\kappa}^2 + (1 + \tfrac1\varepsilon)\|q(0)\|_{H^s_\kappa}^2\\ & \quad + \big[\kappa^{2s}+\kappa^{-(2s+1)}\log^4\!|2\kappa|\big]\delta^2\Big(\|q\|_{X^{s+1}}^2 + \|q(0)\|_{H^s}^2\Big). \end{align*} $$

Choosing $0<\varepsilon <1$ sufficiently small to defeat the implicit constant, we get

(5.17) $$ \begin{align} \|(\psi_h^6 q)'\|_{L^2_t H^s_\kappa}^2 &\lesssim \|q(0)\|_{H^s_\kappa}^2+ \big[\kappa^{2s}+\kappa^{-(2s+1)}\log^4\!|2\kappa|\big]\delta^2\Big(\|q\|_{X^{s+1}}^2 + \|q(0)\|_{H^s}^2\Big). \end{align} $$

To complete the proof, we apply the estimate (5.17) with $\kappa = 1$ to bound

$$ \begin{align*} \|\psi_h^6 q\|_{L^2_t H^{s+1}}^2 &\lesssim \|P_{>1}(\psi_h^6 q)'\|_{L^2_t H^s}^2 + \|P_{\leq 1}(\psi_h^6 q)\|_{L^\infty_t H^s}^2 \\ &\lesssim \delta^2\|q\|_{X^{s+1}}^2 + \|q(0)\|_{H^s}^2. \end{align*} $$

Taking the supremum over $h\in {{\mathbb {R}}}$ and choosing $0<\delta \ll 1$ sufficiently small, we obtain (5.15). The estimate (5.16) then follows from (5.15), (5.17), and the observation that $\kappa ^{2s}\|q(0)\|_{H^s}^2\lesssim \|q(0)\|_{H^s_\kappa }^2$ .

Proof. Let us write $I=[\kappa ^{\frac 23},\frac 12\kappa ]\cup [2\kappa ,\infty )$ , which is the region of $\varkappa $ over which the estimate (5.10) will be proved.

Taking the imaginary part of (5.1) and applying (5.5), (5.10), and (5.7), we find

$$ \begin{align*} \|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-1}_\varkappa}^2 &\lesssim \|q\|_{L^\infty_t H^{-\frac12}_\varkappa}^2 + \big[\tfrac\kappa{\kappa + \varkappa}\kappa^{-\frac43(2s+1)} + \varkappa^{-2(2s+1)}\big]\delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2, \end{align*} $$

uniformly for $\varkappa \in I$ . Choosing $-\frac 12<s'<s$ and employing the a priori estimate (4.12), we deduce that

$$ \begin{align*} \|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-1}_\varkappa}^2 &\lesssim \varkappa^{-(2s'+ 1)}\|q(0)\|_{H^{s'}_\varkappa}^2 + \big[\tfrac\kappa{\kappa + \varkappa}\kappa^{-\frac43(2s+1)} + \varkappa^{-2(2s+1)}\big]\delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2, \end{align*} $$

uniformly for $\varkappa \in I$ . Next, we wish to integrate out $\varkappa $ .

By Lemma 2.1, we have

$$ \begin{align*} \|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{s-\frac12}_{\kappa^{2/3}}}^2 \approx\int_{\kappa^{\frac23}}^\infty \varkappa^{2s+1}\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-1}_\varkappa}^2\tfrac{d\varkappa}\varkappa, \end{align*} $$

from which it follows that

(5.19) $$ \begin{align} \|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{s-\frac12}_{\kappa^{2/3}}}^2 &\approx\int_I \varkappa^{2s+1}\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-1}_\varkappa}^2\tfrac{d\varkappa}\varkappa , \end{align} $$

because the integrand on the interval $[\kappa /2,2\kappa ]$ is comparable to that on $[2\kappa ,4\kappa ]$ .

Proceeding in this way, we find that

$$ \begin{align*} \|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{s-\frac12}_{\kappa^{2/3}}}^2 \lesssim \|q(0)\|_{H^s}^2 + \kappa^{-\frac13(1+2s)}\delta^2\Big(\|q\|_{X^{s+\frac12}_\kappa}^2 + \|q(0)\|_{H^s}^2\Big). \end{align*} $$

To complete the proof, we decompose

$$ \begin{align*} \|\tfrac{\psi_h^6 q}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{s+\frac32}}^2 &\lesssim \|\tfrac1{\sqrt{4\kappa^2 - \partial^2}}P_{\leq\kappa^{\frac23}}(\psi_h^6 q)\|_{L^2_t H^{s+\frac32}}^2 + \|\tfrac1{\sqrt{4\kappa^2 - \partial^2}}P_{>\kappa^{\frac23}}(\psi_h^6 q)\|_{L^2_t H^{s+\frac32}}^2\\ &\lesssim \|q(0)\|_{H^s}^2 + \|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{s-\frac12}_{\kappa^{2/3}}}^2\\ &\lesssim \|q(0)\|_{H^s}^2 + \delta^2\|q\|_{X^{s+\frac12}_\kappa}^2. \end{align*} $$

Taking the supremum over $h\in {{\mathbb {R}}}$ , we obtain the estimate (5.18) whenever $0<\delta \ll 1$ is sufficiently small, depending only on s.

Proof. The claim (5.20) follows immediately from (5.18) and Bernstein inequalities. To obtain (5.21), we decompose into Littlewood–Paley pieces, use (5.20) and Lemma 2.8, and then sum.

Proof. Consider the real part of (5.1). Applying the estimates (5.6), (5.11), and (5.7), we deduce that

$$ \begin{align*} \varkappa\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-1}_\varkappa}^2 &\lesssim \varepsilon\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|^2_{L^2_t H^{-\frac12}_\varkappa} + (1 + \tfrac1\varepsilon)\|q\|_{L^\infty_t H^{-\frac12}_\varkappa}^2\\ &\quad\ + \big[\tfrac\kappa{\kappa + \varkappa}\kappa^{-(2s+1)} + \varkappa^{-2(2s+1)}\log|2\varkappa|\big]\delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2, \end{align*} $$

uniformly for $0<\varepsilon <1$ and $\varkappa \in I:= [\kappa ^{\frac 12},\frac 12\kappa ]\cup [2\kappa ,\infty )$ .

Choosing $-\frac 12<s'<s$ and applying the a priori estimate (4.12), this becomes

$$ \begin{align*} \varkappa\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-1}_\varkappa}^2 &\lesssim \varepsilon\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-\frac12}_\varkappa}^2+ \varkappa^{-(2s'+1)}(1 + \tfrac1\varepsilon)\|q(0)\|_{H^{s'}_\varkappa}^2\\ &\quad\ + \big[\tfrac\kappa{\kappa + \varkappa}\kappa^{-(2s+1)} + \varkappa^{-2(2s+1)}\log|2\varkappa|\big] \delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2. \end{align*} $$

Next, we wish to integrate over $\varkappa \in I$ . Using Lemma 2.1 as in the proof of Proposition 5.6, we obtain the following analogues of (5.19):

$$ \begin{align*} \|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^s_{\kappa^{1/2}}}^2 &\approx\int_I \varkappa^{2s+2}\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-1}_\varkappa}^2\tfrac{d\varkappa}\varkappa \approx\int_I \varkappa^{2s+1}\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{-\frac12}_\varkappa}^2\tfrac{d\varkappa}\varkappa. \end{align*} $$

Proceeding in this way, and choosing $0<\varepsilon <1$ sufficiently small, we obtain

$$\begin{align*}\big\|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\big\|_{L^2_t H^s_{\kappa^{1/2}}}^2\lesssim \|q(0)\|_{H^s}^2 + \delta^2\|q\|_{X^{s+1}_\kappa}^2. \end{align*}$$

To complete the proof, we decompose

$$ \begin{align*} \|\tfrac{\psi_h^6 q}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{s+2}}^2 &\lesssim \|\tfrac1{\sqrt{4\kappa^2 - \partial^2}}P_{\leq\kappa^{\frac12}}(\psi_h^6 q)\|_{L^2_t H^{s+2}}^2 + \|\tfrac1{\sqrt{4\kappa^2 - \partial^2}}P_{>\kappa^{\frac12}}(\psi_h^6 q)\|_{L^2_t H^{s+2}}^2\\ &\lesssim \|q(0)\|_{H^s}^2 + \|\tfrac{(\psi_h^6 q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{L^2_t H^{s}_{\kappa^{1/2}}}^2\\ &\lesssim \|q(0)\|_{H^s}^2 + \delta^2\|q\|_{X^{s+1}_\kappa}^2. \end{align*} $$

Taking the supremum over $h\in {{\mathbb {R}}}$ , we obtain the estimate (5.18) whenever $0<\delta \ll 1$ is sufficiently small, depending only on s.

Proof of Lemma 5.1.

We introduce the paraproduct $\mathbf R[q,r]$ with symbol

$$\begin{align*}R(\xi,\eta) = \tfrac1{2(2\varkappa - i\xi)} + \tfrac1{2(2\varkappa + i\eta)} \end{align*}$$

so that by (4.13), we may write

$$\begin{align*}\rho^{[2]}(x;\varkappa)=\mathbf R[q,r](x) = \frac1{2\pi}\int R(\xi,\eta)\hat q(\xi)\hat r(\eta) e^{ix(\xi+\eta)}\,d\xi\,d\eta. \end{align*}$$

We then observe that the quadratic part of the current $j(\varkappa ,\kappa )$ defined in (4.23) may be written as

$$\begin{align*}j^{[2]}(\varkappa,\kappa) = i\mathbf R[\tfrac q{2\kappa - \partial},\tfrac r{2\kappa + \partial}]. \end{align*}$$

Expanding in powers of $\kappa $ , we readily obtain the expressions

$$ \begin{align*} j_{\mathrm{NLS}}^{[2]}(\varkappa) &= i\Big(\mathbf R[q,r'] - \mathbf R[q',r]\Big),\\ j_{\mathrm{mKdV}}^{[2]} (\varkappa) &= \big(\mathbf R[q,r]\big)" - 3 \mathbf R[q',r'], \end{align*} $$

for the NLS and mKdV flows, as well as the expressions

$$ \begin{align*} j_{\mathrm{NLS}}^{\mathrm{diff}}{}^{[2]}(\varkappa,\kappa) &= i\Big(\mathbf R[q,r'] - \mathbf R[q',r]\Big) - 16\kappa^4i\Big(\mathbf R[\tfrac q{4\kappa^2 - \partial^2},\tfrac {r'}{4\kappa^2 - \partial^2}] - \mathbf R[\tfrac{q'}{4\kappa^2 - \partial^2},\tfrac r{4\kappa^2 - \partial^2}]\Big),\\ j_{\mathrm{mKdV}}^{\mathrm{diff}}{}^{[2]} (\varkappa,\kappa)&= \big(\mathbf R[q,r]\big)" - 3 \mathbf R[q',r'] - 16\kappa^4\big(\mathbf R[\tfrac{q}{4\kappa^2 - \partial^2},\tfrac{r}{4\kappa^2 - \partial^2}]\big)" + 48\kappa^4 \mathbf R[\tfrac{q'}{4\kappa^2 - \partial^2},\tfrac{r'}{4\kappa^2 - \partial^2}]\\ &\quad + 4\kappa^2 \mathbf R[\tfrac{q"}{4\kappa^2 - \partial^2},\tfrac{r"}{4\kappa^2 - \partial^2}] \end{align*} $$

for the corresponding difference flows. (Alternatively, we may use the definition of the currents from Corollary 4.14 to compute the quadratic components directly.)

If we could simply replace $q,r$ by $\psi _h^6q,\psi _h^6r$ in these expressions, rather than integrating them against $\psi _h^{12}$ , then we would obtain the leading order terms in (5.3)–(5.6). Thus, the focal point of our analysis will be bounding the various commutator terms that arise.

Proof of (5.3). Using the above expression, we may write

$$ \begin{align*} \mathrm{LHS}(5.3) &= \pm 2\|(\psi_h^{6}q)'\|_{H^{-1}_\varkappa}^2 + \operatorname{Re}\int \Big(\mathbf R[q,r']\,\psi_h^{12} - \mathbf R[\psi_h^{6}q,(\psi_h^{6} r)']\Big)\,dx\\ &\quad - \operatorname{Re}\int\Big(\mathbf R[q',r]\,\psi_h^{12} - \mathbf R[(\psi_h^{6} q)',\psi_h^{6}r]\Big)\,dx. \end{align*} $$

By symmetry, it suffices to bound

$$ \begin{align*} &\int \Big(\mathbf R[q,r']\,\psi_h^{12} - \mathbf R[\psi_h^{6}q,(\psi_h^{6}r)']\Big)\,dx\\ &\qquad = \int [\psi_h^{6},\tfrac1{2(2\varkappa - \partial)}] q\cdot \psi_h^{6} r'\,dx - \int \tfrac1{2(2\varkappa - \partial)}(\psi_h^{6}q)\cdot (\psi_h^{6})'r\,dx\\ &\qquad\quad + \int \psi_h^{6} q\cdot [\psi_h^{6},\tfrac\partial{2(2\varkappa + \partial)}]r\,dx, \end{align*} $$

which may be bounded by

$$ \begin{align*} &\left|\int \Big(\mathbf R[q,r']\,\psi_h^{12} - \mathbf R[\psi_h^{6}q,(\psi_h^{6}r)']\Big)\,dx\right|\\ &\qquad \leq \|[\psi_h^{6},\tfrac1{2(2\varkappa - \partial)}] q\|_{H^{\frac32}_\varkappa}\|\psi_h^{6} r'\|_{H^{-\frac32}_\varkappa} + \|\tfrac1{2(2\varkappa - \partial)}(\psi_h^{6} q)\|_{H^{\frac12}_\varkappa}\|(\psi_h^{6})'r\|_{H^{-\frac12}_\varkappa}\\ &\qquad\quad + \|\psi_h^{6} q\|_{H^{-\frac12}_\varkappa}\|[\psi_h^{6},\tfrac\partial{2(2\varkappa + \partial)}]r\|_{H^{\frac12}_\varkappa}\\ &\qquad\lesssim \|q\|_{H^{-\frac12}_\varkappa}^2, \end{align*} $$

as required.

Proof of (5.5). We observe that the difference $ j_{\mathrm {NLS}}^{[2]} - j_{\mathrm {NLS}}^{\mathrm {diff}}{}^{[2]} $ has an identical expression to $j_{\mathrm {NLS}}^{[2]}$ with q replaced by $\tfrac {4\kappa ^2 q}{4\kappa ^2 - \partial ^2}$ . The estimate (5.5) then follows from the estimate (5.3) and the estimates

$$\begin{align*}\|\tfrac{4\kappa^2}{4\kappa^2 - \partial^2}q\|_{H^{-\frac12}_\varkappa}\lesssim \|q\|_{H^{-\frac12}_\varkappa} \quad\text{and}\quad \|\partial[\psi_h^{6},\tfrac{4\kappa^2}{4\kappa^2 - \partial^2}]q\|_{H^{-\frac12}_\varkappa}\lesssim \|q\|_{H^{-\frac12}_\varkappa}. \end{align*}$$

Proof of (5.4). Integrating by parts, we find

$$ \begin{align*} \mathrm{LHS}(5.4) &= \mp 6\varkappa\|(\psi_h^{6} q)'\|_{H^{-1}_\varkappa}^2 + \operatorname{Re}\int \mathbf R[q,r]\,(\psi_h^{12})"\,dx\\ &\quad - 3\operatorname{Re}\int \Big(\mathbf R[q',r']\,\psi_h^{12}- \mathbf R[(\psi_h^{6}q)',(\psi_h^{6}r)']\Big)\,dx. \end{align*} $$

As in the proof of Lemma 5.2, the second term may be readily bounded by

$$\begin{align*}\left|\int \mathbf R[q,r]\,(\psi_h^{12})"\,dx\right|\lesssim \|q\|_{H^{-\frac12}_\varkappa}^2. \end{align*}$$

For the remaining term, we write

$$ \begin{align*} &\int \Big(\mathbf R[q',r']\,\psi_h^{12} - \mathbf R[(\psi_h^{6}q)',(\psi_h^{6}r)']\Big)\,dx\\ &\qquad = \int [\psi_h^{6},\tfrac\partial{2(2\varkappa - \partial)}]q\cdot (\psi_h^{6} r)'\,dx - \int \tfrac\partial{2(2\varkappa - \partial)}(\psi_h^{6}q)\cdot (\psi_h^{6})' r\,dx \\ &\qquad \quad - \int [\psi_h^{6},\tfrac\partial{2(2\varkappa - \partial)}]q\cdot (\psi_h^{6})' r\,dx + \int (\psi_h^{6} q)'\cdot [\psi_h^{6},\tfrac\partial{2(2\varkappa + \partial)}]r\,dx\\ &\qquad \quad - \int (\psi_h^{6})' q\cdot \tfrac\partial{2(2\varkappa + \partial)}(\psi_h^{6} r)\,dx - \int (\psi_h^{6})' q\cdot[\psi_h^{6},\tfrac\partial{2(2\varkappa + \partial)}] r\,dx. \end{align*} $$

The first three summands, here, may be bounded in magnitude via

$$ \begin{align*} \|[\psi_h^{6},\tfrac\partial{2(2\varkappa - \partial)}]q\|_{H^{\frac12}_\varkappa}\|(\psi_h^{6} r)'\|_{H^{-\frac12}_\varkappa} + \|\tfrac\partial{2(2\varkappa - \partial)}&(\psi_h^{6} q)\|_{H^{\frac12}_\varkappa}\|(\psi_h^{6})' r\|_{H^{-\frac12}_\varkappa} \lesssim \|(\psi_h^{6} q)'\|_{H^{-\frac12}_\varkappa} \|q\|_{H^{-\frac12}_\varkappa} \end{align*} $$

and

$$ \begin{align*} \|[\psi_h^{6},\tfrac\partial{2(2\varkappa - \partial)}]q\|_{H^{\frac12}_\varkappa}\|(\psi_h^{6})' r\|_{H^{-\frac12}_\varkappa} &\lesssim \|q\|_{H^{-\frac12}_\varkappa}^2, \end{align*} $$

both of which are acceptable. The remaining three summands can then be estimated in a parallel fashion; indeed, this is tantamount to replacing $\varkappa $ by $-\varkappa $ .

Proof of (5.6). Integrating by parts several times, we obtain the identity

$$ \begin{align*} \mathrm{LHS}(5.6) &= \mp 2\varkappa\int \tfrac{(20\kappa^2 + 3\xi^2)\xi^4}{(4\varkappa^2 + \xi^2)(4\kappa^2 + \xi^2)^2}|\widehat{\psi_h^{6}q}|^2\,d\xi\\ &\quad - 3\operatorname{Re}\int \Big(\mathbf R[\tfrac{q"'}{4\kappa^2 - \partial^2},\tfrac{r"'}{4\kappa^2 - \partial^2}]\,\psi_h^{12} - \mathbf R[\tfrac{(\psi_h^{6} q)"'}{4\kappa^2 - \partial^2},\tfrac{(\psi_h^{6}r)"'}{4\kappa^2 - \partial^2}]\Big)\,dx\\ &\quad - 20\kappa^2\operatorname{Re}\int \Big(\mathbf R[\tfrac{q"}{4\kappa^2 - \partial^2},\tfrac{r"}{4\kappa^2 - \partial^2}]\,\psi_h^{12} - \mathbf R[\tfrac{(\psi_h^{6}q)"}{4\kappa^2 - \partial^2},\tfrac{(\psi_h^{6}r)"}{4\kappa^2 - \partial^2}]\Big)\,dx\\ &\quad + \operatorname{Re}\int \mathbf R[\tfrac{q"}{4\kappa^2 - \partial^2},\tfrac{r"}{4\kappa^2 - \partial^2}]\,(\psi_h^{12})"\,dx\\ &\quad + 20\kappa^2\operatorname{Re}\int \mathbf R[\tfrac{q'}{4\kappa^2 - \partial^2},\tfrac{r'}{4\kappa^2 - \partial^2}]\,(\psi_h^{12})"\,dx\\ &\quad - 4\kappa^2\operatorname{Re}\int \mathbf R[\tfrac{q}{4\kappa^2 - \partial^2},\tfrac{r}{4\kappa^2 - \partial^2}]\,(\psi_h^{12})""\,dx. \end{align*} $$

The final three terms are lower order errors that may be bounded by

$$ \begin{align*} &\left|\int \mathbf R[\tfrac{q"}{4\kappa^2 - \partial^2},\tfrac{r"}{4\kappa^2 - \partial^2}]\,(\psi_h^{12})"\,dx\right| \lesssim \|\tfrac{q"}{4\kappa^2 - \partial^2}\|_{H^{-\frac12}_\varkappa}^2\lesssim \|q\|_{H^{-\frac12}_\varkappa}^2\\ &\left|\kappa^2\int \mathbf R[\tfrac{q'}{4\kappa^2 - \partial^2},\tfrac{r'}{4\kappa^2 - \partial^2}]\,(\psi_h^{12})"\,dx\right|\lesssim \kappa^2\|\tfrac{q'}{4\kappa^2 - \partial^2}\|_{H^{-\frac12}_\varkappa}^2 \lesssim \|q\|_{H^{-\frac12}_\varkappa}^2,\\ &\left|\kappa^2\int \mathbf R[\tfrac{q}{4\kappa^2 - \partial^2},\tfrac{r}{4\kappa^2 - \partial^2}]\,(\psi_h^{12})""\,dx\right|\lesssim \kappa^2\|\tfrac q{4\kappa^2 - \partial^2}\|_{H^{-\frac12}_\varkappa}^2\lesssim \kappa^{-2}\|q\|_{H^{-\frac12}_\varkappa}^2. \end{align*} $$

For the first commutator term, we estimate

$$ \begin{align*} &\left|\int \Big(\mathbf R[\tfrac{q"'}{4\kappa^2 - \partial^2},\tfrac{r"'}{4\kappa^2 - \partial^2}]\,\psi_h^{12} - \mathbf R[\tfrac{(\psi_h^{6} q)"'}{4\kappa^2 - \partial^2},\tfrac{(\psi_h^{6}r)"'}{4\kappa^2 - \partial^2}]\Big)\,dx\right|\\ &\qquad\lesssim \|[\psi_h^{6},\tfrac{\partial^3}{(2\varkappa - \partial)(4\kappa^2 - \partial^2)}]q\|_{H^{\frac12}_\varkappa}\|\tfrac{\partial^3}{4\kappa^2 - \partial^2}(\psi_h^{6} q)\|_{H^{-\frac12}_\varkappa}\\ &\qquad \quad + \Big\{\|[\psi_h^{6},\tfrac{\partial^3}{(2\varkappa - \partial)(4\kappa^2 - \partial^2)}]q\|_{H^{\frac12}_\varkappa}+ \|\tfrac{\partial^3}{4\kappa^2 - \partial^2}(\psi_h^{6} q)\|_{H^{-\frac12}_\varkappa}\Big\}\|[\psi_h^{6},\tfrac{\partial^3}{4\kappa^2 - \partial^2}]q\|_{H^{-\frac12}_\varkappa}\\ &\qquad\lesssim \|q\|_{H^{-\frac12}_\varkappa}\|\tfrac{(\psi_h^{6}q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{H^{-\frac12}_\varkappa} + \|q\|_{H^{-\frac12}_\varkappa}^2. \end{align*} $$

The second commutator term is bounded similarly:

$$ \begin{align*} &\left|\kappa^2\int \Big(\mathbf R[\tfrac{q"}{4\kappa^2 - \partial^2},\tfrac{r"}{4\kappa^2 - \partial^2}]\,\psi_h^{12} - \mathbf R[\tfrac{(\psi_h^{6}q)"}{4\kappa^2 - \partial^2},\tfrac{(\psi_h^{6}r)"}{4\kappa^2 - \partial^2}]\Big)\,dx\right|\lesssim \|q\|_{H^{-\frac12}_\varkappa}\|\tfrac{(\psi_h^{6}q)"}{\sqrt{4\kappa^2 - \partial^2}}\|_{H^{-\frac12}_\varkappa} + \|q\|_{H^{-\frac12}_\varkappa}^2. \end{align*} $$

This completes the proof of the lemma.

Proof. By space-translation invariance, we may assume $h=0$ .

By Bernstein’s inequality, for $0\leq j\leq 2$ , we may bound

$$ \begin{align*} \tfrac{N^j}{(\kappa + N)^2}\|(\psi^3q)_N\|_{L^2_{t,x}}& \lesssim \kappa^{-2} N^{j-s-\frac52} \min\{N^{\frac52},\kappa N,\kappa^2 \} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}\\ \tfrac{N^j}{(\kappa + N)^2}\|\big(\psi^3\tfrac f{2\varkappa + \partial}\big)_N\|_{L^2_{t,x}}& \lesssim |\varkappa|^{-(s+\frac12)}(|\varkappa| + N)^{s-\frac12} \kappa^{-2} N^{j-s-\frac52} \min\{N^{\frac52},\kappa N,\kappa^2 \} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}, \end{align*} $$

which we will use to estimate high-frequency terms. Using Bernstein’s inequality again, we also find that for $0\leq j\leq 2$ ,

$$ \begin{align*} &\sum\limits_{M\leq N}\tfrac{M^j}{(\kappa + M)^2}\|(\psi^3q)_M\|_{L^\infty_{t,x}}\lesssim \kappa^{-2}N^{\frac12-s} \min\{ N^j,\kappa^j\}\delta,\\ &\sum\limits_{M\leq N}\tfrac{M^j}{(\kappa + M)^2}\|\big(\psi^3\tfrac f{2\varkappa + \partial}\big)_M\|_{L^\infty_{t,x}}\!\lesssim |\varkappa|^{-(s+\frac12)}(|\varkappa| + N)^{s-\frac12}\kappa^{-2}N^{\frac12-s} \min\{ N^j,\kappa^j\}\delta, \end{align*} $$

which we will use to estimate low frequency terms.

For dyadic $N_j\geq 1$ , we write

$$ \begin{align*} m_{N_1,N_2,N_3,N_4} := m\big[\psi^{-3}(\psi^3q)_{N_1},\psi^{-3}(\psi^3r)_{N_2},\psi^{-3}(\psi^3q)_{N_3},\psi^9(\psi^3\tfrac f{2\varkappa + \partial})_{N_4}\big], \end{align*} $$

so that

$$\begin{align*}\big\| m[q,r,q,\psi^{12}\tfrac f{2\varkappa + \partial}] \big\|_{L^1_{t,x}} \leq \sum\limits_{N_j\geq 1} \big\| m_{N_1,N_2,N_3,N_4} \big\|_{L^1_{t,x}}. \end{align*}$$

As the estimates will be symmetric in the first three terms, we may assume that $N_1\geq N_2\geq N_3$ . Our strategy will be to bound the two highest frequency terms in $L^2_{t,x}$ to take advantage of the local smoothing norms, and the two lowest frequencies in $L^\infty _{t,x}$ . Concretely, when $N_4\leq N_2$ , we apply Lemma 2.8, to obtain

$$ \begin{align*} \big\| m_{N_1,N_2,N_3,N_4} \big\|_{L^1_{t,x}}&\lesssim\|\psi^3\tfrac{\partial^{\beta_1}}{4\kappa^2 - \partial^2}\psi^{-3}(\psi^3q)_{N_1}\|_{L^2_{t,x}}\|\psi^3\tfrac{\partial^{\beta_2}}{4\kappa^2 - \partial^2}\psi^{-3}(\psi^3q)_{N_2}\|_{L^2_{t,x}}\\ &\quad\times \|\psi^3\tfrac{\partial^{\beta_3}}{4\kappa^2 - \partial^2}\psi^{-3}(\psi^3q)_{N_3}\|_{L^\infty_{t,x}}\|\psi^{-9}\tfrac{\partial^{\beta_4}}{4\kappa^2 - \partial^2}\psi^9(\psi^3\tfrac f{2\varkappa + \partial})_{N_4}\|_{L^\infty_{t,x}}\\ &\lesssim \tfrac{N_1^{\beta_1}N_2^{\beta_2}N_3^{\beta_3}N_4^{\beta_4}}{(\kappa + N_1)^2(\kappa + N_2)^2(\kappa + N_3)^2(\kappa + N_4)^2}\|(\psi^3q)_{N_1}\|_{L^2_{t,x}}\|(\psi^3q)_{N_2}\|_{L^2_{t,x}}\\ &\quad\times\|(\psi^3q)_{N_3}\|_{L^\infty_{t,x}}\|(\psi^3\tfrac f{2\varkappa + \partial})_{N_4}\|_{L^\infty_{t,x}}, \end{align*} $$

whereas, when $N_4>N_2$ , we obtain instead

$$ \begin{align*} \big\| m_{N_1,N_2,N_3,N_4} \big\|_{L^1_{t,x}}&\lesssim\tfrac{N_1^{\beta_1}N_2^{\beta_2}N_3^{\beta_3}N_4^{\beta_4}}{(\kappa + N_1)^2(\kappa + N_2)^2(\kappa + N_3)^2(\kappa + N_4)^2}\|(\psi^3q)_{N_1}\|_{L^2_{t,x}}\|(\psi^3q)_{N_2}\|_{L^\infty_{t,x}}\\ &\qquad\qquad\times\|(\psi^3q)_{N_3}\|_{L^\infty_{t,x}}\|\big(\psi^3 \tfrac f{2\varkappa + \partial}\big)_{N_4}\|_{L^2_{t,x}}. \end{align*} $$

We then sum over the lowest two frequencies and invoke the estimates laid out above. When $N_4\leq N_2$ , this leads to a bound of the form

$$ \begin{align*}\sum_{N_1\geq N_2} \Gamma_{N_1,N_2} \delta^2 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2, \end{align*} $$

where $\Gamma $ is the matrix

$$ \begin{align*} \Gamma_{N_1,N_2} := |\varkappa|^{-(s+\frac12)}&(|\varkappa| + N_2)^{s-\frac12}\kappa^{-8}N_1^{\beta_1 - \frac52 - s}N_2^{\beta_2 - \frac32 - 3s}\\ &\times \min\{N_1^{\frac52},\kappa N_1,\kappa^2\}\min\{N_2^{\beta_3 + \beta_4 + \frac52},\kappa N_2^{\beta_3 + \beta_4 + 1},\kappa^{\beta_3 + \beta_4 + 2}\}. \end{align*} $$

When on the other hand $N_4> N_2$ , we are led to a bound of the form

$$ \begin{align*}\sum_{N_1\geq N_4} \Gamma_{N_1,N_4} \delta^2 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2 + \sum_{N_4\geq N_1} \Big[\tfrac{|\varkappa| + N_4}{|\varkappa| + N_1}\Big]^{s-\frac12}\Gamma_{N_4,N_1} \delta^2 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2 , \end{align*} $$

with corresponding permutations of the indices $\beta $ .

In this way, we see that the proof can be completed by proving

$$ \begin{align*} \sum_{N \geq M } \Gamma_{N,M} \lesssim \kappa^{|\beta| - 7 -\frac83(s+\frac12)}. \end{align*} $$

As the matrix entries are monotone in $|\varkappa |$ , it suffices to prove the bound when $|\varkappa |=\kappa ^{2/3}$ . Summing first in N, we are left to estimate

(5.26) $$ \begin{align} \sum_{M\geq 1} \kappa^{-8-\frac13(2s+1)} (\kappa^{\frac23} + M&)^{s-\frac12} M^{\beta_2-\frac32(2s+1)}\min\Big\{M^{\beta_3+\beta_4+\frac52}[\kappa^{\beta_1-\frac12-s}+ \kappa^{\frac23(\beta_1-s)}], \notag\\ &\kappa^2 M^{\beta_3+\beta_4+1} [\kappa^{\beta_1-\frac32-s}+ M^{\beta_1-\frac32-s}], \ \kappa^{\beta_3+\beta_4 +4} M^{\beta_1-\frac52-s}\Big\}. \end{align} $$

From here, one need only consider the cases $M\leq \kappa ^{\frac 23}$ , $\kappa ^{\frac 23}\leq M\leq \kappa $ , and $M\geq \kappa $ .

Proof. The proof is essentially identical to that of Lemma 5.10. By space-translation symmetry, we may assume $h=0$ .

We will reuse the $L^\infty _{t,x}$ bounds appearing in the proof of Lemma 5.10; however, the $L^2_{t,x}$ local smoothing bounds used to treat the high-frequency terms must be adapted to the mKdV setting. Specifically, we will use

$$ \begin{align*} \tfrac{N^j}{(\kappa + N)^2}\|(\psi^3q)_N\|_{L^2_{t,x}} &\lesssim \kappa^{-2} N^{j-3-s} \min\{N^3,\kappa N, \kappa^2 \} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa},\\ \tfrac{N^j}{(\kappa + N)^2}\|\big(\psi^3\tfrac f{2\varkappa + \partial}\big)_N\|_{L^2_{t,x}} &\lesssim |\varkappa|^{-(s+\frac12)}(|\varkappa| + N)^{s - \frac12} \kappa^{-2} N^{j-3-s} \min\{N^3,\kappa N, \kappa^2 \} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}. \end{align*} $$

Proceeding as in the proof of (5.25), we take

$$ \begin{align*} m_{N_1,N_2,N_3,N_4} := m\big[\psi^{-3}(\psi^3q)_{N_1},\psi^{-3}(\psi^3r)_{N_2},\psi^{-3}(\psi^3q)_{N_3},\psi^{9}(\psi^3\tfrac f{2\varkappa + \partial})_{N_4}\big], \end{align*} $$

so that

$$\begin{align*}\|m[q,r,q,\psi^{12}\tfrac f{2\varkappa + \partial}]\|_{L^1_{t,x}} \leq \sum\limits_{N_j\geq 1}\|m_{N_1,N_2,N_3,N_4}\|_{L^1_{t,x}}, \end{align*}$$

or

$$ \begin{align*} \tilde m_{N_1,N_2,N_3,N_4} := m\big[\psi^{-3}(\psi^3q)_{N_1},\psi^{-3}(\psi^3r)_{N_2},\psi^{-3}(\psi^3q)_{N_3},\psi^{-3}(\psi^3\tfrac f{2\varkappa + \partial})_{N_4}\big], \end{align*} $$

$$\begin{align*}\|m[q,r,q,\tfrac f{2\varkappa + \partial}]\,\psi^{12}\|_{L^1_{t,x}} \leq \sum\limits_{N_j\geq 1}\|\tilde m_{N_1,N_2,N_3,N_4}\,\psi^{12}\|_{L^1_{t,x}}. \end{align*}$$

As in the proof of (5.25), it suffices to restrict our attention to the case $N_1\geq N_2\geq N_3\geq N_4$ . With $\ell =3,9$ , we may bound

$$ \begin{align*} \text{LHS}(5.27)&\lesssim \!\!\sum\limits_{N_1\geq N_2\geq N_3\geq N_4}\!\!\|\psi^3\tfrac{\partial^{\beta_1}}{4\kappa^2 - \partial^2}\psi^{-3}(\psi^3q)_{N_1}\|_{L^2_{t,x}}\|\psi^3\tfrac{\partial^{\beta_2}}{4\kappa^2 - \partial^2}\psi^{-3}(\psi^3q)_{N_2}\|_{L^2_{t,x}}\\ &\qquad\quad\times \|\psi^3\tfrac{\partial^{\beta_3}}{4\kappa^2 - \partial^2}\psi^{-3}(\psi^3q)_{N_3}\|_{L^\infty_{t,x}}\|\psi^{-\ell}\tfrac{\partial^{\beta_4}}{4\kappa^2 - \partial^2}\psi^\ell(\psi^3\tfrac f{2\varkappa + \partial})_{N_4}\|_{L^\infty_{t,x}}\\ &\lesssim \!\!\!\sum\limits_{N_1\geq N_2\geq N_3\geq N_4}\!\!\!\!\tfrac{N_1^{\beta_1}N_2^{\beta_2}N_3^{\beta_3}N_4^{\beta_4}}{(\kappa + N_1)^2(\kappa + N_2)^2(\kappa + N_3)^2(\kappa + N_4)^2}\|(\psi^3q)_{N_1}\|_{L^2_{t,x}}\|(\psi^3q)_{N_2}\|_{L^2_{t,x}}\\ &\qquad\quad\times\|(\psi^3q)_{N_3}\|_{L^\infty_{t,x}}\|(\psi^3\tfrac f{2\varkappa + \partial})_{N_4}\|_{L^\infty_{t,x}}. \end{align*} $$

Summing in $N_3\geq N_4\geq 1$ we obtain a bound of a constant multiple of

$$ \begin{align*} &\sum\limits_{N_1\geq N_2}|\varkappa|^{-(s+\frac12)}(|\varkappa| + N_2)^{s-\frac12}\kappa^{-8}N_1^{\beta_1-3-s}N_2^{\beta_2 - 2 - 3s}\min\{N_1^3,\kappa N_1,\kappa^2\}\\ &\qquad \qquad\qquad\qquad\times\min\{N_2^{\beta_3 + \beta_4 + 3},\kappa N_2^{\beta_3 + \beta_4 + 1}, \kappa^{\beta_3 + \beta_4 + 2}\}\delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2. \end{align*} $$

Proceeding as in Lemma 5.10 and summing in $N_1$ , we are led to control the following analogue of (5.26):

$$ \begin{align*} \sum_{M\geq 1} \kappa^{-8} |\varkappa|^{-(s+\frac12)} (|\varkappa|& + M)^{s-\frac12} M^{\beta_2-2-3s}\min\Big\{M^{\beta_3+\beta_4+3}[\kappa^{\beta_1-1-s}+ \kappa^{\frac12(\beta_1-s)}], \notag\\ &\kappa^2 M^{\beta_3+\beta_4+1} [\kappa^{\beta_1-2-s}+ M^{\beta_1-2-s}], \ \kappa^{\beta_3+\beta_4 +4} M^{\beta_1-3-s}\Big\}. \end{align*} $$

Once again, this requires consideration of individual cases. Unlike in Lemma 5.10, the final bound depends upon $|\varkappa |$ and so we cannot exploit monotonicity; thus, we need to treat separately $\kappa ^{\frac 12}\leq |\varkappa |\leq \kappa $ and $|\varkappa |\geq \kappa $ . Evaluating these sums carefully reveals that (5.27) can be improved to

(5.29) $$ \begin{align} \text{LHS(5.27)} \lesssim \Big[\tfrac{\kappa^{-2s}}{\kappa+|\varkappa|}+|\varkappa|^{-2(2s+1)}\log\big|\tfrac{4\varkappa^2}{\kappa}\big|\Big]\kappa^{|\beta|-8}\delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2 \end{align} $$

in two cases: (i) if $|\beta |\geq 3$ or (ii) if $|\beta |=2$ and no individual $\beta _j=2$ . These bounds suffice to prove (5.28) because if $|\beta |=2$ and some factor has two derivatives (i.e., some $\beta _j=2$ ), then we may integrate by parts to redistribute one of the derivatives and recover case (ii).

Proof. By translation invariance, we may take $h=0$ . Decomposing dyadically and using (2.22) yields

$$\begin{align*}\mathrm{LHS}(5.30) \lesssim \sum_{N\geq 1} (|\varkappa| + N)^{-1}\log\big(4 + \tfrac{N^2}{|\varkappa|^2}\big) \big\|P_N( \psi f_1\cdot \psi\tfrac {f_2}{2\varkappa - \partial}\big)\big\|_{L^2_{t,x}}^2, \end{align*}$$

in which we then substitute the bound

$$\begin{align*}\big\|P_N( \psi f_1\cdot \psi\tfrac {f_2}{2\varkappa - \partial}\big)\big\|_{L^2_{t,x}}\leq \bigg\| \sum_{N_1,N_2\geq 1}\big\|P_N\big( (\psi f_1)_{N_1}\cdot \big(\psi\tfrac {f_2}{2\varkappa - \partial}\big)_{N_2}\big)\big\|_{L^2}\bigg\|_{L^2_t}. \end{align*}$$

We then proceed using the Littlewood–Paley trichotomy:

$\underline{\mathrm{Case\ 1{:}}\ N_2 \ll N_1 \approx N}$ . Applying Bernstein’s inequality, we bound

$$ \begin{align*} \big\|P_N\big( (\psi f_1)_{N_1}\cdot \big(\psi\tfrac {f_2}{2\varkappa - \partial}\big)_{N_2}\big)\big\|_{L^2_{t,x}} &\lesssim \|(\psi f_1)_{N_1}\|_{L^2_{t,x}}\|\big(\psi\tfrac {f_2}{2\varkappa - \partial}\big)_{N_2}\|_{L^\infty_{t,x}}\\ &\lesssim N_1^{-s-\frac32}\min\{N_1^{\frac32},\kappa + N_1\} N_2^{\frac12-s}(|\varkappa| + N_2)^{-1}\delta {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}. \end{align*} $$

Observing that for fixed $N\geq 1$ , we have

$$\begin{align*}\sum_{1\leq N_2\lesssim N}N_2^{\frac12-s}(|\varkappa| + N_2)^{-1}\lesssim |\varkappa|^{-1}(N\wedge |\varkappa|)^{\frac12-s}, \end{align*}$$

we are led to estimate

$$ \begin{align*} &\quad \sum_{N\geq 1}|\varkappa|^{-2}(N\wedge |\varkappa|)^{1-2s}(|\varkappa| + N)^{-1}\log(4 + \tfrac{N^2}{|\varkappa|^2}) N^{-2s-3}\min\{N^3,(\kappa + N)^2\} \\ &\qquad\qquad\lesssim |\varkappa|^{-3} \big[ \kappa^{\frac23 - \frac{8s}3}+|\varkappa|^{-4s} \big]. \end{align*} $$

$\underline{\mathrm{Case\ 2{:}}\ N_1 \ll N_2\approx N}$ . A similar argument yields the estimate

$$ \begin{align*} &\big\|P_N\big( (\psi f_1)_{N_1}\cdot \big(\psi\tfrac {f_2}{2\varkappa - \partial}\big)_{N_2}\big)\big\|_{L^2_{t,x}}\lesssim N_1^{\frac12-s}(|\varkappa| + N_2)^{-1}N_2^{-s-\frac32}\min\{N_2^{\frac32},\kappa + N_2\} \delta {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}. \end{align*} $$

This then leads us to evaluate

$$\begin{align*}\sum_{N\geq 1}(|\varkappa| + N)^{-3}\log(4 + \tfrac{N^2}{\varkappa^2})N^{-2(1+2s)}\min\{N^3,(\kappa + N)^2\}, \end{align*}$$

which yields the same bound as in Case 1.

$\underline{\mathrm{Case\ 3{:}}\ N_1\approx N_2 \gtrsim N}$ . Bernstein’s inequality implies

$$ \begin{align*} \big\|P_N\big( (\psi f_1)_{N_1}\cdot &\big(\psi\tfrac {f_2}{2\varkappa - \partial}\big)_{N_2}\big\|_{L^2} \\ &\lesssim N^{\frac12} \big\|(\psi f_1)_{N_1} \big\|_{L^2} \cdot N_1^{-s} (|\varkappa|+N_1)^{-1} \big\|(2\varkappa-\partial)\big(\psi\tfrac {f_2}{2\varkappa - \partial}\big)_{N_2}\big\|_{H^{s}}. \end{align*} $$

Thus, applying Cauchy–Schwarz to the sum, we obtain

$$ \begin{align*} \bigg\| \sum_{N_1\approx N_2\gtrsim N}\big\|P_N\big( (\psi & f_1)_{N_1}\cdot \big(\psi\tfrac {f_2}{2\varkappa - \partial}\big)_{N_2}\big)\big\|_{L^2}\bigg\|_{L^2_t}^2\\ &\lesssim N \sum\limits_{N_1\gtrsim N} N_1^{-4s-3}(|\varkappa| + N_1)^{-2}\min\{N_1^3,(\kappa + N_1)^2\} \delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2. \end{align*} $$

We are then left to evaluate the sum

$$ \begin{align*} &\quad \sum_{N\geq 1} \sum_{N_1\gtrsim N} \tfrac N{|\varkappa| + N}\log\big(4 + \tfrac{N^2}{|\varkappa|^2}\big) N_1^{-4s-3}(|\varkappa| + N_1)^{-2}\min\{N_1^3,(\kappa + N_1)^2\} \\ &\lesssim \sum\limits_{N_1\geq 1}|\varkappa|^{-1}(|\varkappa| + N_1)^{-2} N_1^{-4s-2}\min\{N_1^3,(\kappa + N_1)^2\}, \end{align*} $$

which ultimately yields a contribution identical to that of Cases 1 and 2.

Proof. The estimate (5.31) follows from the same argument used to prove (5.30); all that changes are the specific powers inside the sums.

Thus, it remains to consider the estimate (5.32). Proceeding as in the proof of (5.30), we may assume that $h=0$ and bound

$$\begin{align*}\mathrm{LHS}(5.32) \lesssim \sum\limits_{N\geq 1}(|\varkappa| + N)^{-1}\log\big(4 + \tfrac{N^2}{\varkappa^2}\big)\big\|P_N\big(\psi f_1\cdot \psi\tfrac{f_2}{2\varkappa - \partial}\cdot\psi\tfrac{f_3}{2\varkappa + \partial}\big)\big\|_{L^2_{t,x}}^2. \end{align*}$$

We then decompose further by frequency, using

$$ \begin{align*} &\big\|P_N\big(\psi f_1\cdot \psi\tfrac{f_2}{2\varkappa - \partial}\cdot\psi\tfrac{f_3}{2\varkappa + p}\big)\big\|_{L^2_{t,x}}\\ &\qquad \leq \bigg\| \sum\limits_{N_1,N_2,N_3\geq 1} \big\|P_N\big((\psi f_1)_{N_1}\cdot \big(\psi\tfrac{f_2}{2\varkappa - \partial}\big)_{N_2}\cdot\big(\psi\tfrac{f_3}{2\varkappa + \partial}\big)_{N_3}\big)\big\|_{L^2} \bigg\|_{L^2_t}. \end{align*} $$

As everything is symmetric under the $N_2\leftrightarrow N_3$ interchange, we may reduce matters to four possible cases:

$\underline{\mathrm{Case\ 1{:}}\ \min \{N_1,N\}\geq \max \{N_2,N_3\}}$ . Here, we apply Bernstein’s inequality to bound

$$ \begin{align*} &\big\|P_N\big((\psi f_1)_{N_1}\cdot \big(\psi\tfrac{f_2}{2\varkappa - \partial}\big)_{N_2}\cdot\big(\psi\tfrac{f_3}{2\varkappa + \partial}\big)_{N_3}\big)\big\|_{L^2_{t,x}}\\ &\qquad \leq \|(\psi f_1)_{N_1}\|_{L^2_{t,x}}\|\big(\psi\tfrac{f_2}{2\varkappa - \partial}\big)_{N_2}\|_{L^\infty_{t,x}}\|\big(\psi\tfrac{f_3}{2\varkappa + \partial}\big)_{N_3}\|_{L^\infty_{t,x}}\\ &\qquad \lesssim N_1^{-s-2}\min\{N_1^2,\kappa + N_1\}N_2^{\frac12-s}(|\varkappa| + N_2)^{-1}N_3^{\frac12-s}(|\varkappa| + N_3)^{-1}\delta^2 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}. \end{align*} $$

Summing in $N_2,N_3$ and then in $N_1\approx N$ using the Cauchy-Schwarz inequality, we obtain a contribution to $\mathrm {RHS}$ (5.32) that is

$$ \begin{align*} &\lesssim \sum\limits_{N\geq 1}|\varkappa|^{-4}(N\wedge |\varkappa|)^{2-4s}(|\varkappa| + N)^{-1}\log\big(4 + \tfrac{N^2}{\varkappa^2}\big)N^{-2s-4}\min\{N^4,(\kappa + N)^2\} \delta^4 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2 \\ &\lesssim |\varkappa|^{-5} \big[\kappa^{1-3s}+ \big(1+\tfrac{\kappa^2}{\varkappa^2}\big)|\varkappa|^{-6s} \log|2\varkappa| \big]\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2. \end{align*} $$

$\underline{\mathrm{Case\ 1{:}}\ \min \{N_2,N\}\geq \max \{N_1,N_3\}}$ . A similar argument, this time placing $\psi \frac {f_2}{2\varkappa - \partial }$ in $L^2_{t,x}$ yields the estimate

$$ \begin{align*} &\big\|P_N\big((\psi f_1)_{N_1}\cdot \big(\psi\tfrac{f_2}{2\varkappa - \partial}\big)_{N_2}\cdot\big(\psi\tfrac{f_3}{2\varkappa + \partial}\big)_{N_3}\big)\big\|_{L^2_{t,x}}\\ &\qquad \lesssim N_1^{\frac12-s}(|\varkappa| + N_2)^{-1}N_2^{-s-2}\min\{N_2^2,\kappa + N_2\}N_3^{\frac12-s}(|\varkappa| + N_3)^{-1}\delta^2 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}, \end{align*} $$

which yields a contribution to $\mathrm {RHS}$ (5.32) of

$$\begin{align*}\lesssim \sum\limits_{N\geq 1}|\varkappa|^{-2}(N\wedge |\varkappa|)^{1-2s}(|\varkappa| + N)^{-3}\log\big(4 + \tfrac{N^2}{\varkappa^2}\big)N^{-4s-3}\min\{N^4,(\kappa + N)^2\} \delta^4 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2, \end{align*}$$

which yields an identical contribution to Case 1.

$\underline{\mathrm{Case\ 3{:}}\ \min \{N_1,N_2\}\geq \max \{N_3,N\}}$ . Here, we apply Bernstein’s inequality at the output frequency and sum using the Cauchy-Schwarz inequality in $N_1\approx N_2$ so that for fixed $N\geq 1$ , we obtain

$$ \begin{align*} &\left\|\sum\limits_{N_1\approx N_2\gtrsim N_3,N}\big\|P_N\big((\psi f_1)_{N_1}\cdot \big(\psi\tfrac{f_2}{2\varkappa - \partial}\big)_{N_2}\cdot\big(\psi\tfrac{f_3}{2\varkappa + \partial}\big)_{N_3}\big)\big\|_{L^2}\right\|_{L^2_t}^2\\ &\qquad\lesssim N \left\| \sum\limits_{N_1\approx N_2\gtrsim N_3,N} \|(\psi f_1)_{N_1}\|_{L^2}\|\big(\psi\tfrac{f_2}{2\varkappa - \partial}\big)_{N_2}\|_{L^2}\|\big(\psi\tfrac{f_3}{2\varkappa + \partial}\big)_{N_3}\|_{L^\infty}\right\|_{L^2_t}^2\\ &\qquad \lesssim N \sum\limits_{N_1\gtrsim N}N_1^{-4s-4}\min\{N_1^4,(\kappa + N_1)^2\}(|\varkappa| + N_1)^{-2}|\varkappa|^{-2}(N_1\wedge |\varkappa|)^{1-2s}\delta^4 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2. \end{align*} $$

We then obtain a contribution to $\mathrm {RHS}$ (5.32) of

$$ \begin{align*} &\lesssim \sum\limits_{N\geq 1} \tfrac N{|\varkappa| + N}\log\big(4 + \tfrac{N^2}{\varkappa^2}\big) \sum\limits_{N_1\gtrsim N}N_1^{-4s-4}\min\{N_1^4,(\kappa + N_1)^2\}\\ &\qquad\qquad\qquad\times(|\varkappa| + N_1)^{-2}|\varkappa|^{-2}(N_1\wedge |\varkappa|)^{1-2s}\delta^4 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2\\ &\lesssim \sum\limits_{N_1\geq 1} |\varkappa|^{-3}N_1^{-4s-3}\min\{N_1^4,(\kappa + N_1)^2\}(|\varkappa| + N_1)^{-2}(N_1\wedge |\varkappa|)^{1-2s}\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2, \end{align*} $$

which gives an identical contribution to Cases 1 and 2.

$\underline{\mathrm{Case\ 4{:}}\ \min \{N_2,N_3\}\geq \max \{N_1,N\}}$ . Arguing as in Case 3, for fixed $N\geq 1$ , we may bound

$$ \begin{align*} &\left\|\sum\limits_{N_2\approx N_3\gtrsim N_1,N}\big\|P_N\big((\psi f_1)_{N_1}\cdot \big(\psi\tfrac{f_2}{2\varkappa - \partial}\big)_{N_2}\cdot\big(\psi\tfrac{f_3}{2\varkappa + \partial}\big)_{N_3}\big)\big\|_{L^2}\right\|_{L^2_t}^2\\ &\qquad \lesssim N \sum\limits_{N_2\gtrsim N}N_2^{-6s-3}\min\{N_2^4,(\kappa + N_2)^2\}(|\varkappa| + N_2)^{-4}\delta^4 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2, \end{align*} $$

which yields a contribution to $\mathrm {RHS}$ (5.32) of

$$\begin{align*}\sum\limits_{N_2\geq 1} |\varkappa|^{-1}N_2^{-6s-2}\min\{N_2^4,(\kappa + N_2)^2\}(|\varkappa| + N_2)^{-4}\delta^4 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2. \end{align*}$$

This gives an identical contribution to the previous cases.

Proof. By translation invariance, it suffices to consider the case $h=0$ . Our basic technique, here, is to expand using the series (3.19), commute copies of $\psi $ , and then use Hölder’s inequality in trace ideals. We first exhibit this technique to prove the auxiliary result (5.37) before turning our attention to the principal claims.

Given a test function $F\in \mathcal C([-1,1];\mathcal {S})$ , using (3.19), we may write

$$\begin{align*}\operatorname{\mathrm{sgn}}(\varkappa) \int F\,g_{12}(\varkappa)^{[\geq 3]} \psi^4 \,dx = \sum_{\ell=1}^\infty (-1)^{\ell-1} \operatorname{\mathrm{tr}}\left\{\Lambda(\Gamma\Lambda)^{\ell}(\varkappa + \partial)^{-\frac12}\psi^4 F(\varkappa - \partial)^{-\frac12}\right\}. \end{align*}$$

Applying Lemma 2.8 followed by the operator estimates (3.7) and (2.25), we obtain

$$ \begin{align*} &\left\|\operatorname{\mathrm{tr}}\left\{\Lambda(\Gamma\Lambda)^{\ell}(\varkappa + \partial)^{-\frac12}\psi^4 F(\varkappa - \partial)^{-\frac12}\right\}\right\|_{L^1_t}\\ &\qquad\lesssim \|\Lambda\|_{L^\infty_t\mathfrak I_2}^{2\ell-2}\|(\varkappa - \partial)^{-\frac12}(\psi q)(\varkappa + \partial)^{-\frac12}\|_{L^4_t\mathfrak I_4}^3\|(\varkappa + \partial)^{-\frac12}(\psi F)(\varkappa - \partial)^{-\frac12}\|_{L^4_t\mathfrak I_4}\\ &\qquad\lesssim C^\ell |\varkappa|^{-(2s+1)(\ell-1)-3}\big[\kappa^{\frac23 - \frac{8s}3} + |\varkappa|^{-4s}\big]\delta^{2(\ell-1)}\\ &\qquad\qquad\qquad\qquad\times\|q\|_{L^\infty_tH^s}^{\frac32}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^{\frac32}\|F\|_{L^\infty_t H^s}^{\frac12}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert F \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^{\frac12}, \end{align*} $$

where the implicit constant is independent of $\ell $ . Summing in $\ell $ and applying Young’s inequality, we obtain

(5.37) $$ \begin{align} \Big\|\int F\,& g_{12}^{[\geq 3]}(\varkappa)\,\psi^4 \,dx\Big\|_{L^1_t}\lesssim |\varkappa|^{-3}\big[\kappa^{\frac23 - \frac{8s}3} + |\varkappa|^{-4s}\big]\delta{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}\Big(\|F\|_{L^\infty_t H^s}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa} + \delta{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert F \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}\Big). \end{align} $$

The estimate (5.33) follows immediately from (5.37) by setting $\varkappa =\pm \kappa $ , $F=\frac {\chi }{\psi ^4}\frac f{2\varkappa + \partial }$ and using (2.8) and (2.14) to bound ${\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert F \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }_{\mathrm {NLS}_\kappa }\lesssim |\varkappa |^{-1} {\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert q \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }_{\mathrm {NLS}_\kappa }$ .

We turn now to (5.34) and recall that

(5.38) $$ \begin{align} \big(\tfrac{g_{12}}{2 + \gamma}\big)^{[\geq 3]}= \tfrac12 g_{12}^{[\geq 3]} -\tfrac{g_{12}\gamma}{2(2 + \gamma)}. \end{align} $$

By (5.37), the contribution of the first term to the left-hand side of (5.34) is easily seen to be acceptable. To estimate the contribution of the second term on the right-hand side of (5.38), we take $f_1 = \chi f/\psi ^4$ and $f_2 = (2\varkappa - \partial ) \tfrac {g_{12}(\varkappa )}{2 + \gamma (\varkappa )}$ and apply the estimate (5.30) to bound

$$\begin{align*}\|(\varkappa \pm \partial)^{-\frac12}(\psi^2 f_1\tfrac{f_2}{2\varkappa-\partial})(\varkappa \pm \partial)^{-\frac12}\|_{L^2_t \mathfrak I_2} \lesssim |\varkappa|^{-\frac32}\big[\kappa^{\frac13 - \frac{4s}3} + |\varkappa|^{-2s}\big]\delta{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}, \end{align*}$$

where we have used (2.14) with the estimates (3.41), (3.43) to bound

$$ \begin{align*}{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert f_2 \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa} = {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (2\varkappa - \partial) \tfrac{g_{12}(\varkappa)}{2 + \gamma(\varkappa)} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}\lesssim {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}. \end{align*} $$

We then use (3.19) to write

$$ \begin{align*} \int \!\tfrac{\chi f g_{12}(\varkappa)}{2(2 + \gamma(\varkappa))}\, \gamma\,dx = \operatorname{\mathrm{sgn}}(\varkappa)\sum\limits_{\ell=1}^\infty(-1)^{\ell}\Big[&\operatorname{\mathrm{tr}}\left\{(\Lambda\Gamma)^\ell(\varkappa -\partial)^{-\frac12}(\psi^4 f_1\,\tfrac{f_2}{2\varkappa-\partial})(\varkappa - \partial)^{-\frac12}\right\}\\ +{}& \operatorname{\mathrm{tr}}\left\{(\Gamma\Lambda)^\ell(\varkappa +\partial)^{-\frac12}(\psi^4 f_1\,\tfrac{f_2}{2\varkappa-\partial})(\varkappa + \partial)^{-\frac12}\right\}\Big]. \end{align*} $$

Repeating our basic technique using (2.25), we obtain

(5.39) $$ \begin{align} \left\|\int f_1\,\tfrac{f_2}{2\varkappa-\partial}\, \gamma(\varkappa)\,\psi^{4 }\,dx\right\|_{L^1_t}&\lesssim \|(\varkappa - \partial)^{-\frac12}(\psi q)(\varkappa + \partial)^{-\frac12}\|_{L^4_t\mathfrak I_4}^2\|(\varkappa - \partial)^{-\frac12}(\psi^2 f_1\,\tfrac{f_2}{2\varkappa-\partial}) (\varkappa - \partial)^{-\frac12}\|_{L^2_t \mathfrak I_2}\notag\\ &\lesssim |\varkappa|^{-3}\big[\kappa^{\frac23 - \frac{8s}3} + |\varkappa|^{-4s}\big]\delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2,\end{align} $$

which completes the proof of (5.34). The estimate (5.35) follows analogously using (2.25) with $\varkappa = \kappa $ :

(5.40) $$ \begin{align} \bigg\|\int F\, \gamma^{[\geq 4]}(\kappa)\,\chi\,dx\bigg\|_{L^1_t}&\lesssim \|(\kappa - \partial)^{-\frac12}(\psi q)(\kappa + \partial)^{-\frac12}\|_{L^4_t\mathfrak I_4}^4 \|(\kappa \pm \partial)^{-\frac12}\|_{\mathrm{op}}^2 \|\chi F \psi^{-4}\|_{L^\infty_{t,x}} \notag\\ &\lesssim \kappa^{- 2 - \frac43(2s+1)}\delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2\|F\|_{L^\infty_{t,x}}. \end{align} $$

Finally, we consider (5.36). Arguing in the same style, we bound

$$ \begin{align*} & \left\|\int g_{12}^{[\geq 5]}(\kappa)\, \tfrac f{2\varkappa + \partial}\psi^{12}\,dx\right\|_{L^1_t} \\ &\quad \lesssim \|\Lambda\|_{L^\infty_t\mathfrak I_2} \|(\kappa - \partial)^{-\frac12}(\psi q)(\kappa + \partial)^{-\frac12}\|_{L^4_t\mathfrak I_4}^4 \|(\kappa - \partial)^{-\frac12}( \psi^8 \tfrac f{2\varkappa + \partial})(\kappa + \partial)^{-\frac12}\|_{L^\infty_t\mathrm{op}}\\ &\quad \lesssim \kappa^{- 2 - \frac{11}6(2s+1)}\delta^3{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2\|\tfrac f{2\varkappa + \partial}\|_{L^\infty_{t,x}}\\ &\quad \lesssim \kappa^{- 2 - \frac{13}6(2s+1)}\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2, \end{align*} $$

where we have used that $|\varkappa |\geq \kappa ^{\frac 23}$ to estimate $ \|\tfrac f{2\varkappa + \partial }\|_{L^\infty _{t,x}}\lesssim \kappa ^{-\frac 13(2s+1)}\delta $ . For the remaining term, we observe that integrating by parts we may write

$$\begin{align*}\int g_{12}^{[3]}(\kappa)\,\tfrac f{2\varkappa + \partial}\,\psi^{12}\,dx = \int m[q,r,q,\psi^{12} \tfrac f{2\varkappa + \partial}]\,dx, \end{align*}$$

where the symbol

$$\begin{align*}m(\xi_1,\xi_2,\xi_3,\xi_4) = \tfrac2{(2\kappa + i\xi_2) (2\kappa - i\xi_3)(2\kappa + i\xi_4)} \end{align*}$$

is a sum of terms in $\kappa ^{5 - |\beta |}S(\beta ;2\kappa )$ for $0\leq |\beta |\leq 5$ , where at most one $\beta _j = 2$ . In particular, when considering the sum $g_{12}^{[3]}(\kappa ) + g_{12}^{[3]}(-\kappa )$ , we see that the terms with even $|\beta |$ cancel, and hence

$$\begin{align*}\int \Big(g_{12}^{[3]}(\kappa) + g_{12}^{[3]}(-\kappa)\Big)\,\tfrac f{2\varkappa + \partial}\,\psi^{12}\,dx = \int \widetilde m[q,r,q,\psi^{12} \tfrac f{2\varkappa + \partial}]\,dx, \end{align*}$$

where the symbol of $\widetilde m$ is given by a sum of terms in $\kappa ^{5 - |\beta |}S(\beta ;2\kappa )$ for $|\beta | = 1,3,5$ and at most one $\beta _j = 2$ . Applying the estimate (5.25), we then obtain

$$\begin{align*}\left\|\int \Big(g_{12}^{[3]}(\kappa) + g_{12}^{[3]}(-\kappa)\Big)\,\tfrac f{2\varkappa + \partial}\,\psi^{12}\,dx\right\|_{L^1_t}\lesssim \kappa^{-2 - \frac43(2s+1)}\delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{NLS}_\kappa}^2, \end{align*}$$

which completes the proof of (5.36).

Proof. The basic technique is that used to prove Lemma 5.15; however, new cancellations need to be exhibited. We begin with the estimates on $\gamma $ .

Mimicking (5.40) but using Lemma 2.7 with $p=4$ yields (5.46). When taking $p=6$ , we obtain instead

(5.49) $$ \begin{align} \big\| \gamma(\pm\kappa)^{[\geq 6]}\chi \big\|_{L^1_{t,x}}&\lesssim [\kappa^{-5-3s}+\kappa^{-6-6s}\log(2\kappa)]\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2 \\ &\lesssim \kappa^{-4-2s}\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2.\notag \end{align} $$

This estimate reduces (5.47) and (5.48) to consideration of the quartic terms, for which we turn to (3.23). Evidently, every term in (5.47) and (5.48) can be written as a sum of paraproducts with symbols conforming to (5.23); however, by forming these particular linear combinations, we eliminate all terms with $|\beta |=0$ . Thus, we may apply Lemma 5.11 (with $\varkappa =\kappa $ ) and so deduce (5.47) and (5.48).

Applying our basic technique to $g_{12}$ using Lemma 2.7 with $p=4$ yields

$$ \begin{align*} \bigg\|\int f\, g_{12}^{[3]}(\varkappa)\,\chi_h\,dx\bigg\|_{L^1_t} + \bigg\|\int f g_{12}^{[\geq 3]}(\varkappa)\,\chi\,dx\bigg\|_{L^1_t} \lesssim \text{RHS(5.41)}. \end{align*} $$

Taking $p=5$ and using also (3.7) yields

$$ \begin{align*} \bigg\|\int f g_{12}^{[\geq 5]}(\varkappa)\,\chi\,dx\bigg\|_{L^1_t} \lesssim \text{RHS(5.42)} , \end{align*} $$

These constitute a significant step toward proving (5.41) and (5.42). In view of (5.38), the proof of (5.41) is completed by the following:

(5.50) $$ \begin{align} \bigg\|\int f \tfrac{g_{12}}{2+\gamma} \gamma \,\chi\,dx\bigg\|_{L^1_t} &\lesssim |\varkappa|^{-3} \big[\kappa^{\frac12 - 2s} + |\varkappa|^{-1-4s}\log^4|2\varkappa|\big] \delta^2 {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2, \end{align} $$

which is a consequence of the argument used in (5.39) but using Lemma 2.7 and (5.31) in place of their NLS analogues.

To prove (5.42), we use (3.32) and $\gamma ^{[2]} = 2g_{12}^{[1]}g_{21}^{[1]}$ to rewrite (3.40) as

(5.51) $$ \begin{align} \big(\tfrac{g_{12}}{2 + \gamma}\big)^{[\geq 5]} = \tfrac12g_{12}^{[\geq 5]} - \tfrac{g_{12}(4 + \gamma)}{4(2+\gamma)}\gamma^{[\geq 4]} + \tfrac{g_{12}g_{21}}{2 + \gamma}g_{12}^{[\geq 3]} - \tfrac12g_{12}^{[1]}g_{21}^{[1]} g_{12}^{[\geq 3]} + \tfrac{g_{12}}{2 + \gamma}g_{12}^{[1]}g_{21}^{[\geq 3]}. \end{align} $$

The contribution of the first term was handled already.

Consider, now, the second term in (5.51). Applying Lemma 2.4 together with the estimates (3.27), (3.35), (3.41), and (3.43), we find that

$$\begin{align*}F=\chi f\tfrac{g_{12}(4 + \gamma)}{2(2+\gamma)\psi^5} \quad\text{satisfies}\quad {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert F \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}\lesssim |\varkappa|^{-(s+\frac12)}\delta{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}. \end{align*}$$

Thus, applying the basic technique and using Lemma 2.7 with $p=5$ shows

$$ \begin{align*} \left\|\int \tfrac{g_{12}(4 + \gamma)}{2(2+\gamma)}\gamma^{[\geq 4]} \,\chi\,dx\right\|_{L^1_t} &\lesssim \text{RHS(5.42)}. \end{align*} $$

The remaining three terms in (5.51) are handled in a parallel fashion, which we demonstrate using the first term. Set $F= f_1\,\frac {f_2}{2\varkappa -\partial } \, \frac {f_3}{2\varkappa +\partial }$ with $f_1 = \chi f/\psi ^6$ , $f_2 = (2\varkappa - \partial )\frac {g_{12}}{2 + \gamma }$ , and $f_3 = (2\varkappa + \partial )g_{21}$ . Then (5.32) implies

$$ \begin{align*} \|(\varkappa - \partial)^{-\frac12}\psi^3 & F(\varkappa + \partial)^{-\frac12}\|_{L^2_t\mathfrak I_2}^2 \\ &\lesssim |\varkappa|^{-5}\big[ \kappa^{1-3s} + \big(1+\tfrac{\kappa^2}{\varkappa^2}\big) |\varkappa|^{-6s} \log |2\varkappa|\big]\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2. \end{align*} $$

Thus, applying Lemma 2.7 with $p=6$ , we find

$$ \begin{align*} \left\|\int F\,g_{12}^{[\geq 3]}(\varkappa)\,\psi^6\,dx\right\|_{L^1_t} &\lesssim |\varkappa|^{-5}\big[ \kappa^{1-3s} + \big(1+\tfrac{\kappa^2}{\varkappa^2}\big) |\varkappa|^{-6s} \log^6\! |2\varkappa|\big]\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2, \end{align*} $$

which is no larger than RHS(5.42). This completes the proof of (5.42).

We turn now to (5.43). Combining (5.42) with (3.21) and Lemma 2.4 yields

$$ \begin{align*} \bigg\|\int g_{21}^{[3]}(\pm\varkappa)\, \big(\tfrac{g_{12}}{2 + \gamma}\big)^{[\geq 5]}(\varkappa)\,\chi\,dx\bigg\|_{L^1_t} &\lesssim |\varkappa|^{-1-(1+2s)} \cdot\text{RHS(5.42)} \lesssim \text{RHS(5.43)}. \end{align*} $$

To continue, we employ (3.38). From Lemma 2.7 and (2.20), we find that

$$ \begin{align*} \bigg|\iint g_{21}^{[3]}\,F\psi^3\,dx\,dt\bigg| &\lesssim\|(\varkappa-\partial)^{-\frac12}\psi q(\varkappa + \partial)^{-\frac12}\|_{L^6_t\mathfrak I_6}^3\|(\varkappa - \partial)^{-\frac12}F(\varkappa + \partial)^{-\frac12}\|_{L^2_t\mathfrak I_2}\\ &\lesssim |\varkappa|^{-3}\big[\kappa^{\frac12 - \frac{3s}2}+ (1+\tfrac{\kappa}{|\varkappa|}\big)|\varkappa|^{-3s}\log^3\!|2\varkappa| \big] \delta^2{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}\|F\|_{L^2_{t,x}} \end{align*} $$

and consequently, that

$$ \begin{align*} \bigg\|\int g_{21}^{[3]}(\pm\varkappa)\, g_{12}^{[3]}(\varkappa)\,\chi\,dx\bigg\|_{L^1_t} &\lesssim\big\|g_{21}^{[3]}(\pm\varkappa)\psi^3\big\|_{L^2_{t,x}}^2 \lesssim \text{RHS(5.43)}. \end{align*} $$

On the other hand, using Lemma 2.7, (3.22), and (5.32), we get

$$ \begin{align*} &\bigg\|\int g_{21}^{[3]}(\pm\varkappa)\, g_{12}^{[1]}(\varkappa) \gamma^{[2]}(\varkappa)\,\chi\,dx\bigg\|_{L^1_t}\\ &\lesssim\|(\varkappa-\partial)^{-\frac12}(\psi q)(\varkappa + \partial)^{-\frac12}\|_{L^6_t\mathfrak I_6}^3\|(\varkappa - \partial)^{-\frac12}\psi^3 g_{12}^{[1]}(\varkappa) \gamma^{[2]}(\varkappa)(\varkappa + \partial)^{-\frac12}\|_{L^2_t\mathfrak I_2}\\ &\lesssim |\varkappa|^{-6}\big[ \kappa^{1-3s} + \big(1+\tfrac{\kappa^2}{\varkappa^2}\big) |\varkappa|^{-6s} \log^6\! |2\varkappa|\big]\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2 \lesssim \text{RHS(5.43)}. \end{align*} $$

This completes the proof of (5.43).

It remains to prove (5.44) and (5.45). We begin by reducing matters to the quartic terms. As $|\varkappa |\geq \sqrt \kappa $ , so $\|\tfrac f{2\varkappa + \partial }\|_{L^\infty _{t,x}}\lesssim \kappa ^{-\frac 12(s+\frac 12)}\delta $ . Thus, we find

$$\begin{align*}\left\|\int \tfrac f{2\varkappa + \partial}\cdot g_{12}^{[\geq 5]}(\pm\kappa)\,\chi\,dx\right\|_{L^1_t}\lesssim \kappa^{-\frac92-3s}\delta^4{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert q \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{\mathrm{mKdV}_\kappa}^2, \end{align*}$$

by applying Lemma 2.7 with $p=5$ .

Regarding the quartic terms, we observe that

$$\begin{align*}\int \tfrac f{2\varkappa + \partial}\cdot \Big(g_{12}^{[3]}(\kappa) - \tfrac1{4\kappa^3}q^2r\Big)\,\chi\,dx = \int m[q,r,q,\chi \tfrac f{2\varkappa + \partial}]\,dx, \end{align*}$$

where the lowest order terms cancel to give a symbol m that is a sum of terms in $\kappa ^{5 - |\beta |}S(\beta ;2\kappa )$ for $1\leq |\beta |\leq 8$ . Thus, (5.27) may be applied, which then yields (5.44). To obtain (5.45), we use (5.28) instead. This is possible due to the absence of any $|\beta |=1$ terms in the multiplier.

Proof of Lemma 5.3.

We consider each of the currents in turn.

Proof of (5.8). From Corollary 4.14 and (4.14),

$$\begin{align*}j_{\mathrm{NLS}}^{[\geq 4]} = - i (2\varkappa + \partial)q\cdot \big(\tfrac{g_{21}}{2 + \gamma}\big)^{[\geq 3]} + i (2\varkappa - \partial) r\cdot \big(\tfrac{g_{12}}{2 + \gamma}\big)^{[\geq 3]}. \end{align*}$$

Writing

$$ \begin{align*}\psi_h^6 (2\varkappa + \partial)q= 2\varkappa (\psi_h^6 q) - (\psi_h^6)' q + (\psi_h^6 q)^{\prime}_{\leq \varkappa} +(\psi_h^6 q)^{\prime}_{>\varkappa} \end{align*} $$

and invoking (5.34) and (3.44), we estimate