Proof. These estimates follow mechanically from applying Hölder’s inequality, the Gagliardo-Nirenberg inequality, the Plancherel bound 12 , the Zhidkov energy identity 14 , and the uniform \(L^2(\mathbb{R})\) boundedness of the Hilbert transform \(H\).

As a representative example, we demonstrate the second estimate in (iii). Applying the product rule and triangle inequality yields \[\begin{align} \| \partial_x(\mathcal{N}_\delta(f_1)-\mathcal{N}_\delta(f_2))\|_{L^2} &\lesssim \| \partial_x(f_1-f_2)\|_{L^2}\| \mathcal{L}_\delta|f_1|^2\|_{L^\infty}+\| f_1-f_2\|_{L^\infty}\| \mathcal{L}_\delta\partial_x|f_1|^2\|_{L^2} \\ &\quad +\| \partial_xf_2\|_{L^2}\| \mathcal{L}_\delta(|f_1|^2-|f_2|^2)\|_{L^\infty} +\| f_2\|_{L^\infty}\| \mathcal{L}_\delta\partial_x(|f_1|^2-|f_2|^2)\|_{L^2} . \end{align}\] To bound the \(\mathcal{L}_\delta\) terms, we exploit the identity \(\mathcal{L}_{\delta}(|f_{1}|^{2}-|f_{2}|^{2}) = (T_{\delta}-H)\partial_{x}\big\{ (|f_{1}|^{2}-\rho_{f_1}^{2})-(|f_{2}|^{2}-\rho_{f_2}^{2}) \big\}\). This allows us to apply 12 and Gagliardo-Nirenberg, \[\begin{align} &\| \partial_x(\mathcal{N}_\delta(f_1)-\mathcal{N}_\delta(f_2))\|_{L^2} \\ &\lesssim \| f_1-f_2\|_{X^1}\delta^{-1}\Big( \big\| |f_1|^2-\rho_{f_1}^2\big\|_{L^2}^{\frac{1}{2}}\| \partial_x|f_1|^2\|_{L^2}^{\frac{1}{2}}+\| \partial_x|f_1|^2\|_{L^2}\Big) \\ &\quad +\| f_2\|_{X^1}\delta^{-1}\Big( \big\| (|f_1|^2-\rho_{f_1}^2)-(|f_2|^2-\rho_{f_2}^2)\big\|_{L^2}^{\frac{1}{2}}\| \partial_x(|f_1|^2-|f_2|^2)\|_{L^2}^{\frac{1}{2}}+\| \partial_x(|f_1|^2-|f_2|^2)\|_{L^2}\Big) \\ &\lesssim \delta^{-1}\Big( \big[ \widetilde{\mathcal{E}}(f_1)\big]^{\frac{1}{2}}\|f \|_{X^1} +\| f_1 \|_{X^1}^2 \Big) \| f_1-f_2\|_{X^1} \\ &\quad + \delta^{-1} \| f_2\|_{X^1} \Big( \big[ \widetilde{d}(f_1,f_2)\big]^{\frac{1}{2}} \big\{ ( \| f_1\|_{X^1} + \| f_2\|_{X^1}) \| f_1-f_2 \|_{X^1}\big\}^{\frac{1}{2}} + (\| f_1\|_{X^1}+\| f_2\|_{X^1})\| f_1-f_2\|_{X^1}\Big) . \end{align}\]

For the higher-order bounds (\(k\geq 2\)) in the second estimate of (i) and the first estimate of (ii), we use standard interpolation inequalities to control intermediate derivatives \(\| \partial_x^jf\|_{L^2}\) and \(\| \partial_x^jf\|_{L^\infty}\) (where \(1\leq j\leq k-1\)) using \(\| \partial_xf\|_{L^2}\) and \(\| \partial_x^kf\|_{L^2}\). The remaining estimates are proved analogously. ◻

Proof of Lemma 2. The first two terms are easily treated by integration by parts (one can estimate the second term without taking the real part). For the term involving \(H\), we rewrite it as \[\undefined\big( f(iH)(\overline{f}\partial_xg),g\big)_{L^2} = \undefined\big( iH\partial_x(\overline{f}g),\overline{f}g\big)_{L^2} - \undefined\big( f(iH)(g\partial_x\overline{f}),g\big)_{L^2}.\] Observe that the first term on the right-hand side vanishes by the positivity of \(H\partial_x\). The remaining term is easily bounded by applying the Cauchy-Schwarz inequality along with the uniform boundedness of \(H\) on \(L^2(\mathbb{R})\). ◻

Proof. Because the initial data satisfies \(\mathrm{supp}\,(\widehat{P_{\leq \ell}\phi}) \subset \{ |\xi| \leq 2^{\ell+1}\}\), any local solution constructed via standard iteration will preserve this frequency localization for all time, ensuring spatial smoothness, that is \(u_\ell \in \mathcal{Z}^\infty\). It therefore suffices to prove local existence in \(\mathcal{Z}^1\) for data satisfying \(\mathcal{E}^1(\phi) \leq M_0^2\). We seek a fixed point for the Duhamel map \[\Gamma[u](t) := U(t)P_{\leq \ell}\phi - i\int_0^t U(t-s)P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u(s))\,ds.\] For \(0 < T \leq 1\), we define the complete metric space \[X_T := \big\{ u \in C([0,T]; \mathcal{Z}^1) : \| \mathcal{E}^1(u)\|_{L^\infty_T} \leq 2c M_0^2 \big\}, \qquad d_T(u,v) := \| d^1(u,v)\|_{L^\infty_T},\] where \(c\) is the constant from the linear evolution bound 27 . Using the Littlewood-Paley bounds 19 and the nonlinear estimates from Lemma 1, the integral term is easily controlled in \(H^1\). For any \(u, v \in X_T\), we obtain the following bounds \[\begin{align} \Big\| \int_0^t U(t-s)P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u)\,ds \Big\|_{L^\infty_T H^1} &\lesssim T(\delta^{-1} + 2^\ell)M_0^3, \tag{32} \\ \Big\| \int_0^t U(t-s)P_{\leq \ell}\big[ \mathcal{N}(P_{\leq \ell}u) - \mathcal{N}(P_{\leq \ell}v) \big]\,ds \Big\|_{L^\infty_T H^1} &\lesssim T(\delta^{-1} + 2^\ell)(1 + M_0^3)d_T(u,v). \tag{33} \end{align}\] Applying the metric properties 17 and 18 , these estimates imply that \(\Gamma\) maps \(X_T\) into itself and satisfies the contraction property \[d_T(\Gamma[u], \Gamma[v]) \leq \frac{1}{2} d_T(u,v),\] provided we choose \(T > 0\) sufficiently small such that \(T(\delta^{-1} + 2^\ell)(1 + M_0^4) \ll 1\). By the Banach fixed-point theorem, \(\Gamma\) admits a unique local solution \(u_\ell \in X_T\).

Extending this local solution to the maximal interval \([0, T_\ell)\) and establishing the standard blow-up alternative follows from classical continuation arguments, which completes the proof. ◻

Proof. To establish the uniform bound ?? , the principal difficulty arises from a derivative loss in the highest-order energy estimate. To overcome this, we employ a modified energy method: we design a frequency-localized correction term that neutralizes the problematic nonlinear interactions in the nonlinearity of 30 .

Let \(\ell\geq 0\), and take arbitrary \(T\in (0,1]\) with \(T<T_\ell\).

We begin by controlling \(\|u_\ell\|_{X^1}\) and \(\widetilde{\mathcal{E}}(u_\ell)\). Using the integral equation, the linear bound 27 , the metric inequality 17 with \(k=1\), the Littlewood-Paley property \(\| P_{\leq \ell}u_\ell \| \leq \| u_\ell \|\), and the nonlinear bounds from Lemma 1, we can bound the Duhamel integral directly in terms of \(u_\ell\) \[\label{est:aprioriZ2-A} \begin{align} \| u_\ell \|_{L^\infty_TX^1}^2 &\lesssim \| U(t)P_{\leq \ell}\phi \|_{L^\infty_TX^1}^2+ \Big\| \int_0^t U(t-s)P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u_\ell(s))\,ds \Big\|_{L^\infty_TX^1}^2 \\ &\lesssim \| \phi \|_{X^1}^2+T^2\Big( \delta^{-1}\| u_\ell \|_{L^\infty_TX^1}^{\frac{3}{2}}\| \widetilde{\mathcal{E}}(u_\ell) \|_{L^\infty_T}^{\frac{3}{4}}+\| u_\ell \|_{L^\infty_TX^1}^2\| u_\ell \|_{L^\infty_TX^2}\Big)^2 \\ &\lesssim \| \phi \|_{X^1}^2 +T^2\| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T}^2\| u_\ell \|_{L^\infty_TX^1}^2 \,. \end{align}\tag{34}\] Similarly, for the energy functional we have \[\label{est:aprioriZ2-A39} \begin{align} \| \widetilde{\mathcal{E}}(u_\ell) \|_{L^\infty_T} &\lesssim \| \widetilde{\mathcal{E}}(U(t)P_{\leq \ell}\phi) \|_{L^\infty_T} +\Big( \| U(t)P_{\leq \ell}\phi \|_{L^\infty_TL^\infty} + \Big\| \int_0^t U(t-s)P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u_\ell)\,ds \Big\|_{L^\infty_TL^\infty}\Big) \\ & \times \Big\| \int_0^t U(t-s)P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u_\ell)\,ds \Big\|_{L^\infty_TL^2} \\ &\lesssim \widetilde{\mathcal{E}}(\phi)+\| \phi \|_{X^1}^2+\Big\{ \| \phi \|_{X^1}+T\Big( \delta^{-1}\| u_\ell \|_{L^\infty_TX^1}^{\frac{3}{2}}\| \widetilde{\mathcal{E}}(u_\ell) \|_{L^\infty_T}^{\frac{3}{4}}+\| u_\ell \|_{L^\infty_TX^1}^2\| u_\ell \|_{L^\infty_TX^2}\Big) \Big\} \\ & \times T\Big( \delta^{-1}\| u_\ell \|_{L^\infty_TX^1}\| \widetilde{\mathcal{E}}(u_\ell)\|_{L^\infty_T}+\| u_\ell \|_{L^\infty_TX^1}^2\| u_\ell \|_{L^\infty_TX^2}\Big) \\ &\lesssim \widetilde{\mathcal{E}}(\phi)+\| \phi \|_{X^1}^2+ T^2\delta^{-\frac{2}{3}}\| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T}^2\| \widetilde{\mathcal{E}}(u_\ell) \|_{L^\infty_T} +T^2\| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T}^2\| u_\ell \|_{L^\infty_TX^1}^2. \end{align}\tag{35}\]

We now estimate the highest-order component of \(\mathcal{E}^2(u_\ell)\). Since \(u_\ell\) is a smooth solution, one computes using equation 30 , \[\begin{align} \frac{d}{dt}\| \partial_x^2u_\ell(t) \|_{L^2}^2 &= 2\undefined\big( \partial_x^2\partial_t u_\ell(t),\partial_x^2u_\ell(t)\big)_{L^2} = 2\undefined\big( {-}i \partial_x^2\mathcal{N}(P_{\leq \ell}u_\ell(t)),\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &= 2\undefined\big( {-}i\partial_x^2\mathcal{N}_\delta(P_{\leq \ell}u_\ell(t)) ,\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad + 2\undefined\big( \partial_x^2\big( P_{\leq \ell}u_\ell(t)(\alpha +i\beta H)\partial_x |P_{\leq \ell}u_\ell(t)|^2\big) ,\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2}. \end{align}\] The first term is harmless and bounded directly via Lemma 1 \[\Big| \big( {-}i\partial_x^2\mathcal{N}_\delta(P_{\leq \ell}u_\ell(t)) ,\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \Big| \lesssim \delta^{-1} \Big( \big[ \widetilde{\mathcal{E}}(u_\ell(t))\big] ^{\frac{1}{2}}\|u_\ell(t)\|_{X^1}+\| u_\ell(t) \|_{X^1}^2\Big) \| \partial_xu_\ell(t) \|_{H^1}^2.\] For the second term, distributing the differentials allows us to safely estimate interactions where no function carries three derivatives, that is, \[\begin{align} &\Big| 2\undefined\big( \partial_x^2\big( P_{\leq \ell}u_\ell(t)(\alpha +i\beta H)\partial_x |P_{\leq \ell}u_\ell(t)|^2\big) ,\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad -2\undefined\big( P_{\leq \ell}u_\ell(t)(\alpha +i\beta H)\big(\partial_x^3P_{\leq \ell}u_\ell(t)\,\overline{P_{\leq \ell}u_\ell(t)}+P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\leq \ell}u_\ell(t)}\big) ,\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \Big| \\ &\lesssim \| u_\ell(t) \|_{X^2}^2\| \partial_xu_\ell(t) \|_{H^1}^2. \end{align}\]Even the terms with three derivatives can be controlled via integration by parts and Lemma 2, except for one specific interaction \[\begin{align} &\Big| 2\undefined\big( P_{\leq \ell}u_\ell(t)(\alpha +i\beta H)\big(\partial_x^3P_{\leq \ell}u_\ell(t)\,\overline{P_{\leq \ell}u_\ell(t)}+P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\leq \ell}u_\ell(t)}\big) ,\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad -2\beta\undefined\big( P_{\leq \ell}u_\ell(t)(iH)\big( P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\leq \ell}u_\ell(t)}\big) ,\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \Big| \\ &\lesssim \| u_\ell(t) \|_{X^2}^2\| \partial_xu_\ell(t) \|_{H^1}^2. \end{align}\] Summarizing these bounds, we arrive at the differential inequality \[\label{est:aprioriZ2-1} \begin{align} \frac{d}{dt}\| \partial_x^2u_\ell(t) \|_{L^2}^2 &\leq 2\beta\undefined\big( P_{\leq \ell}u_\ell(t)(iH)\big( P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\leq \ell}u_\ell(t)}\big) ,\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad +C\Big( \delta^{-1}\big[ \widetilde{\mathcal{E}}(u_\ell(t))\big] ^{\frac{1}{2}}\| u_\ell(t) \|_{X^2}+\| u_\ell(t) \|_{X^2}^2\Big) \| \partial_xu_\ell(t) \|_{H^1}^2.\end{align}\tag{36}\]

To handle the derivative loss arising from the first term on the right-hand side of 36 , we introduce a time-dependent corrector \(\mathcal{I}_2(t)\) designed precisely to cancel this interaction. Letting \(n_0\in \mathbb{Z}_{\geq 0}\) be a frequency threshold to be determined later, we define: \[\label{def:I2} \mathcal{I}_2(t):=\beta\undefined\big( P_{\leq \ell}u_\ell(t)H\partial_x\big(P_{\leq \ell}u_\ell(t)\partial_x\overline{P_{\geq n_0}P_{\leq \ell}u_\ell(t)}\big) ,\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big)_{L^2} .\tag{37}\] By Bernstein’s inequality, this corrector satisfies the bound: \[\label{bd:I2} \begin{align} |\mathcal{I}_2(t)| &\lesssim \| P_{\leq \ell}u_\ell(t) \|_{L^\infty}\| \partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t) \|_{L^2} \| P_{\leq \ell}u_\ell(t)\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t) \|_{H^1} \\ &\lesssim 2^{-n_0}\| u_\ell(t) \|_{X^2}^2\| \partial_x^2u_\ell(t) \|_{L^2}^2. \end{align}\tag{38}\] Differentiating \(\mathcal{I}_2(t)\) with respect to time, applying equation 30 , and utilizing the symmetry property 15 for \(H\partial_x\), we obtain \[\begin{align} \frac{d}{dt}\mathcal{I}_2(t)&=2\beta \undefined\big( H\partial_x\big(\partial_tP_{\leq \ell}u_\ell(t)\cdot \partial_x\overline{P_{\geq n_0}P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad +2\beta \undefined\big( H\partial_x\big(P_{\leq \ell}u_\ell(t)\cdot \partial_x\overline{P_{\geq n_0}\partial_tP_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &=2\beta \undefined\big( iH\partial_x\big(P_{\leq \ell}u_\ell(t) \partial_x^3\overline{P_{\geq n_0}P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big)_{L^2} +\mathcal{R}(t). \end{align}\] Here, the remainder \(\mathcal{R}(t)\) collects all terms with at most two derivatives on each function\[\begin{align} \mathcal{R}(t)&=-2\beta \undefined\big( {-}iH\big(\partial_x^2P_{\leq \ell}u_\ell(t)\cdot \partial_x\overline{P_{\geq n_0}P_{\leq \ell}u_\ell(t)}\big) ,\partial_x\big( \overline{P_{\leq \ell}u_\ell(t)}\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big) \big)_{L^2} \\ &\quad +2\beta \undefined\big( {-}iH\partial_x\big(P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u_\ell(t))\cdot \partial_x\overline{P_{\geq n_0}P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad -2\beta \undefined\big( iH\big(P_{\leq \ell}u_\ell(t)\cdot \partial_x\overline{P_{\geq n_0}P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u_\ell(t))}\big) ,\partial_x\big( \overline{P_{\leq \ell}u_\ell(t)}\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big) \big)_{L^2} . \end{align}\] Using Lemma 1, this remainder is controlled by \[|\mathcal{R}(t)|\lesssim \Big( \| u_\ell(t) \|_{X^2}^2+\delta^{-1}\big[ \widetilde{\mathcal{E}}(u_\ell(t))\big] ^{\frac{3}{4}}\| u_\ell(t) \|_{X^2}^{\frac{5}{2}}+\| u_\ell(t) \|_{X^2}^4\Big) \| \partial_xu_\ell(t) \|_{H^1}^2.\] Crucially, the main term in \(\frac{d}{dt}\mathcal{I}_2(t)\) compensate the first term in 36 , thus by integrating by parts and applying Bernstein’s inequality \(\| \partial_x^3P_{\leq n_0-1}P_{\leq \ell}u_\ell \|_{L^2}\lesssim 2^{n_0}\| \partial_x^2u_\ell \|_{L^2}\), the interaction between the main terms satisfies \[\begin{align} &\Big| 2\beta \undefined\big( iH\partial_x\big(P_{\leq \ell}u_\ell(t) \partial_x^3\overline{P_{\geq n_0}P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad + 2\beta \undefined\big( iH\big(P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \Big| \\ &\leq \Big| 2\beta \undefined\big( iH\big(P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\geq n_0}P_{\leq \ell}u_\ell(t)}\big) ,\partial_x\overline{P_{\leq \ell}u_\ell(t)}\cdot \partial_xP_{\geq n_0}P_{\leq \ell}u_\ell(t)\big)_{L^2} \Big| \\ &\quad +\Big| 2\beta \undefined\big( iH\big(P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\geq n_0}P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_x^2P_{\geq n_0}P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad\quad - 2\beta \undefined\big( iH\big(P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \Big| \\ &\lesssim (1+2^{n_0})\| u_\ell(t) \|_{X^2}^2\| \partial_xu_\ell(t) \|_{H^1}^2. \end{align}\] This yields the final bound for the corrector derivative \[\label{est:aprioriZ2-2} \begin{align} \frac{d}{dt}\mathcal{I}_2(t)&\leq - 2\beta \undefined\big( iH\big(P_{\leq \ell}u_\ell(t)\partial_x^3\overline{P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_x^2P_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad +C\Big( 2^{n_0}\| u_\ell(t) \|_{X^2}^2+ \delta^{-1}\big[ \widetilde{\mathcal{E}}(u_\ell(t))\big] ^{\frac{3}{4}}\| u_\ell(t) \|_{X^2}^{\frac{5}{2}}+\| u_\ell(t) \|_{X^2}^4\Big) \| \partial_xu_\ell(t) \|_{H^1}^2. \end{align}\tag{39}\]

Combining 36 and 39 , we obtain the modified energy inequality, \[\begin{align} \frac{d}{dt}\big( \| \partial_x^2u_\ell(t) \|_{L^2}^2+\mathcal{I}_2(t)\big) &\lesssim 2^{n_0}\| u_\ell(t) \|_{X^2}^4+\| u_\ell(t) \|_{X^2}^6 \\ &\quad +\delta^{-1}\big[ \widetilde{\mathcal{E}}(u_\ell(t))\big] ^{\frac{1}{2}}\| u_\ell(t) \|_{X^2}^3+\delta^{-1}\big[ \widetilde{\mathcal{E}}(u_\ell(t))\big] ^{\frac{3}{4}}\| u_\ell(t) \|_{X^2}^{\frac{9}{2}}. \end{align}\] Integrating this over \([0, t]\) and applying the corrector bound 38 , we obtain \[\label{est:aprioriZ2-C} \begin{align} \| \partial_x^2u_\ell \|_{L^\infty_TL^2}^2 &\leq \| \partial_x^2\phi \|_{L^2}^2+C \Big( \begin{aligned}[t] &(2^{-n_0}+T2^{n_0})\| u_\ell \|_{L^\infty_TX^2}^4+T\| u_\ell \|_{L^\infty_TX^2}^6 \\ &+T\delta^{-1}\| \widetilde{\mathcal{E}}(u_\ell) \|_{L^\infty_T}^{\frac{1}{2}}\| u_\ell \|_{L^\infty_TX^2}^3+T\delta^{-1}\| \widetilde{\mathcal{E}}(u_\ell) \|_{L^\infty_T}^{\frac{3}{4}}\| u_\ell \|_{L^\infty_TX^2}^{\frac{9}{2}}\Big) \end{aligned} \\ &\lesssim \| \partial_x^2\phi \|_{L^2}^2+\Big( (2^{-n_0}+T2^{n_0})\| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T}+T\| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T}^2\Big) \| u_\ell \|_{L^\infty_TX^2}^2. \end{align}\tag{40}\] Putting 34 , 35 , and 40 together, yields the full energy bound \[\label{est:aprioriZ2-D} \| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T}\lesssim \mathcal{E}^2_\delta(\phi) +\Big( (2^{-n_0}+T2^{n_0}) \| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T} +T\| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T}^2\Big) \|\mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T} .\tag{41}\] To prove the main bounds, we select the frequency threshold \(2^{n_0}=C(1+M_0^2)\) for a large constant \(C>0\) so that \(2^{-n_0}\mathcal{E}^2_\delta(\phi)\leq 2^{-n_0}M_0^2\ll 1\). Then, we choose \(T_0\) sufficiently small so that \(T_0(1+M_0^4)\ll 1\). A standard continuity argument applied to 41 then forces \[\| \mathcal{E}^2_\delta(u_\ell) \|_{L^\infty_T}\lesssim \mathcal{E}^2_\delta(\phi),\qquad 0<T<\min \{ T_0,T_\ell\}.\] By the blow-up criterion in Proposition 1, this guarantees \(T_\ell>T_0\) and establishes ?? . Plugging this back into 34 and 35 (modifying \(T_0\) if necessary) immediately yields both of the lower-order estimates listed in ?? .

Finally, the bounds for \(k \geq 3\) in ?? follow from an identical modified energy approach. For the highest component \(\| \partial_x^ku_\ell(t) \|_{L^2}\), one uses Lemmas 1, 2, and interpolation to derive \[\label{est:aprioriZk-1} \begin{align} \frac{d}{dt}\| \partial_x^ku_\ell(t) \|_{L^2}^2 &\leq 2\beta\undefined\big( P_{\leq \ell}u_\ell(t)(iH)\big( P_{\leq \ell}u_\ell(t)\partial_x^{k+1}\overline{P_{\leq \ell}u_\ell(t)}\big) ,\partial_x^kP_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad +C_k\Big( \delta^{-1}\big[ \widetilde{\mathcal{E}}(u_\ell(t))\big] ^{\frac{1}{2}}\| u_\ell(t) \|_{X^2}+\| u_\ell(t) \|_{X^2}^2\Big)\| \partial_xu_\ell(t) \|_{H^{k-1}}^2. \end{align}\tag{42}\]

Constructing the \(k\)-th order corrector \[\mathcal{I}_k(t) := \beta\undefined\big( P_{\leq \ell}u_\ell(t)H\partial_x\big(P_{\leq \ell}u_\ell(t)\partial_x^{k-1}\overline{P_{\geq n_1}P_{\leq \ell}u_\ell(t)}\big) ,\partial_x^{k-1}P_{\geq n_1}P_{\leq \ell}u_\ell(t)\big)_{L^2}\] for \(n_1 \gg 1\) yields corresponding derivative cancellations and bounds analogous to \(\mathcal{I}_2(t)\): \[\begin{align} |\mathcal{I}_k(t)| &\lesssim_k 2^{-n_1}\| u_\ell(t) \|_{X^2}^2\| \partial_x^k u_\ell(t) \|_{L^2}^2, \\ \frac{d}{dt}\mathcal{I}_k(t)&=- 2\beta \undefined\big( iH\big(P_{\leq \ell}u_\ell(t)\partial_x^{k+1}\overline{P_{\leq \ell}u_\ell(t)}\big) ,\overline{P_{\leq \ell}u_\ell(t)}\partial_x^kP_{\leq \ell}u_\ell(t)\big)_{L^2} \\ &\quad +C_k\Big\{ 2^{n_1}\| u_\ell(t)\|_{X^2}^2+ \delta^{-1}\big[ \widetilde{\mathcal{E}}(u_\ell(t))\big] ^{\frac{3}{4}}\| u_\ell(t)\|_{X^2}^{\frac{5}{2}}+\| u_\ell(t)\|_{X^2}^4\Big\} \| \partial_xu_\ell(t)\|_{H^{k-1}}^2 \end{align}\] in analogy with 38 and 39 . Setting \(2^{n_1} = C(1+M_0^2)\) with \(C\gg _k 1\) allows us to deduce \(|\mathcal{I}_k(t)| \leq \frac{1}{2} \| \partial_x^k u_\ell(t) \|_{L^2}^2\) for \(t\in[0, T_0]\) and the modified energy inequality \[\frac{d}{dt}\big( \| \partial_x^ku_\ell(t) \|_{L^2}^2+\mathcal{I}_k(t)\big) \lesssim_k (M_0^2+M_0^4)\| \partial_xu_\ell(t) \|_{H^{k-1}}^2,\qquad t\in [0,T_0].\]Since ?? ensures \(\| \partial_xu_\ell(t) \|_{H^{k-1}}^2 \lesssim \| \phi \|_{X^1}^2 + \| \partial_x^ku_\ell(t) \|_{L^2}^2\), the functional \(X_k(t) := \| \partial_x^ku_\ell(t) \|_{L^2}^2+\mathcal{I}_k(t)+\| \phi \|_{X^1}^2\) satisfies \(\frac{d}{dt}X_k(t) \leq C_k(M_0^2+M_0^4)X_k(t)\). An application of Gronwall’s inequality completes the proof of ?? . ◻

Proof. Take an arbitrary \(T\in (0,T_0]\) such that \(T(1+M_0^4)\ll 1\). Recall from Proposition 2 and the properties of the Littlewood-Paley projectors, we have the uniform a priori bounds \[\label{est:apriorid1-1} \| \mathcal{E}^2_\delta(P_{\leq \ell}u_\ell)\|_{L^\infty_T}+\| \mathcal{E}^2_\delta(P_{\leq m}v_m)\|_{L^\infty_T} \lesssim M_0^2.\tag{43}\]

We begin by controlling the \(d^1_\delta\) metric for the lower frequencies. By the definition in 11 , we must estimate \(\|u_\ell - v_m\|_{X^1}\) and \(\delta^{-4/3}\widetilde{d}(u_\ell, v_m)\). We recall from 31 and 10 , \[u_\ell(t) = U(t)P_{\leq \ell}\phi - i \int_0^t U(t-s) P_{\leq \ell} \mathcal{N}_{\delta}(P_{\leq \ell}u_\ell(s)) \, ds - i \int_0^t U(t-s) P_{\leq \ell} \mathcal{N}_{\infty}(P_{\leq \ell}u_\ell(s)) \, ds.\] Hence, we have \[\begin{align} \| u_\ell -v_m\|_{X^1}&\lesssim \| U(t)\{ P_{\leq \ell}\phi -P_{\leq m}\psi \}\|_{X^1}+\Big\| \int_0^tU(t-s)\{ P_{\leq\ell}\mathcal{N}_\delta(P_{\leq \ell}u_\ell(s))-P_{\leq m}\mathcal{N}_\delta(P_{\leq m}v_m(s))\} \,ds\Big\|_{X^1} \\ &\quad +\| \partial_x(u_\ell-v_m)\|_{L^2}+\Big\| \int_0^tU(t-s)\{ P_{\leq\ell}\mathcal{N}_\infty(P_{\leq \ell}u_\ell(s))-P_{\leq m}\mathcal{N}_\infty(P_{\leq m}v_m(s))\} \,ds\Big\|_{L^2}, \end{align}\] where we have used the embedding \(H^1(\mathbb{R})\hookrightarrow X^1\) and the integral equation to get rid of the \(\dot{H}^1\)-norm for the last term. Invoking the linear bound 27 , we obtain \[\label{est:d1-X1} \begin{align} \| u_\ell -v_m\|_{L^\infty_TX^1}&\lesssim \| P_{\leq \ell}\phi -P_{\leq m}\psi \|_{X^1}+T\| P_{\leq\ell}\mathcal{N}_\delta(P_{\leq \ell}u_\ell)-P_{\leq m}\mathcal{N}_\delta(P_{\leq m}v_m)\|_{L^\infty_TX^1}\\ &\quad +\| \partial_x(u_\ell-v_m)\|_{L^\infty_TL^2}+T\| P_{\leq\ell}\mathcal{N}_\infty(P_{\leq \ell}u_\ell)-P_{\leq m}\mathcal{N}_\infty(P_{\leq m}v_m)\|_{L^\infty_TL^2}. \end{align}\tag{44}\] A similar argument with the metric inequality 18 implies \[\label{est:d1-d} \begin{align} \|\widetilde{d}(u_\ell,v_m)\|_{L^\infty_T}&\lesssim \widetilde{d}(P_{\leq \ell}\phi,P_{\leq m}\psi)+\big( 1+T\| \mathcal{N}(P_{\leq \ell}u_\ell)\|_{L^\infty_TL^2}\big) \| P_{\leq \ell}\phi -P_{\leq m}\psi\|_{X^1}\\ &\quad +\big( \| P_{\leq m}\psi\|_{X^1}+T\| \mathcal{N}(P_{\leq \ell}u_\ell)\|_{L^\infty_TX^1}+T\| \mathcal{N}(P_{\leq m}v_m)\|_{L^\infty_TX^1}\big) \\ &\qquad\qquad \times T\| P_{\leq\ell}\mathcal{N}(P_{\leq \ell}u_\ell)-P_{\leq m}\mathcal{N}(P_{\leq m}v_m)\|_{L^\infty_TL^2}. \end{align}\tag{45}\] We now evaluate the nonlinear terms using the bounds from Lemma 1, the uniform bounds 43 and the tail estimates such as \(\| P_{\leq \ell}f-f\|_{L^2}\lesssim 2^{-\ell}\| \partial_xf\|_{L^2}\). For instance, the following bounds are available: \[\begin{align} &\| \mathcal{N}(P_{\leq \ell} u_\ell)\|_{L^\infty_TX^1}+\| \mathcal{N}(P_{\leq m}v_m)\|_{L^\infty_TX^1}\lesssim M_0^3,\\ &\| P_{\leq\ell}\mathcal{N}(P_{\leq \ell}u_\ell)-P_{\leq m}\mathcal{N}(P_{\leq m}v_m)\|_{L^\infty_TL^2}\\ &\quad \lesssim M_0^3(2^{-\ell}+2^{-m})+\delta^{\frac{1}{3}}\big( M_0^2\| P_{\leq \ell}u_\ell-P_{\leq m}v_m\|_{L^\infty_TX^1}+M_0\| \delta^{-\frac{4}{3}} \widetilde{d}(P_{\leq \ell}u_\ell ,P_{\leq m}v_m)\|_{L^\infty_T}\big) . \end{align}\] Plugging these bounds into 44 –45 and invoking 23 , we obtain \[\begin{align} \| d^1_\delta(u_\ell,v_m)\|_{L^\infty_T} &\lesssim \| \partial_x(u_\ell-v_m)\|_{L^\infty_TL^2} +(1+M_0+TM_0^3)\| \phi-\psi\|_{X^1}+\delta^{-\frac{4}{3}}\widetilde{d}(\phi,\psi) \\ &\quad +(1+M_0+TM_0^3)(M_0+TM_0^3)(2^{-\ell}+2^{-m}) \\ &\quad +(1+M_0+TM_0^3)T(M_0+M_0^2) \| d^1_\delta(u_\ell,v_m)\|_{L^\infty_T}. \end{align}\] Imposing smallness on the time step (here, \(T(1+M_0^3)\ll 1\) is sufficient), we can absorb the last term into the left-hand side, leaving us with \[\label{est:apriorid1-A} \| d^1_\delta(u_\ell,v_m)\|_{L^\infty_T} \lesssim (1+M_0)d^1_\delta(\phi,\psi)+(2^{-\ell}+2^{-m})(1+M_0^2) +\| \partial_x(u_\ell-v_m)\|_{L^\infty_TL^2}.\tag{46}\]

It remains to estimate the last term in 46 . Differentiating the \(L^2\) norm of the difference and applying the equation 30 , we have \[\frac{d}{dt}\big\| \partial_x\big( u_\ell(t)-v_m(t)\big) \big\|_{L^2}^2 = 2\undefined\big( {-}i\partial_x\big( P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u_\ell(t))-P_{\leq m}\mathcal{N}(P_{\leq m}v_m(t))\big) , \partial_x\big( u_\ell(t)-v_m(t)\big) \big)_{L^2}.\] By using Bernstein’s inequality, integration by parts, Lemma 1, and the bounds in 43 , we can shift the frequency projections to act entirely on the difference, generating a harmless commutator remainder \[\begin{align} \frac{d}{dt}\big\| \partial_x\big( u_\ell(t)-v_m(t)\big) \big\|_{L^2}^2 &= 2\undefined\big( {-}i\partial_x\big( \mathcal{N}(P_{\leq \ell}u_\ell(t))-\mathcal{N}(P_{\leq m}v_m(t))\big) , \partial_x\big( P_{\leq \ell}u_\ell(t)-P_{\leq m}v_m(t)\big) \big)_{L^2} \\ & +\mathcal{R}_1(t) \end{align}\] where the remainder satisfies \(|\mathcal{R}_1(t)|\lesssim (2^{-\ell}+2^{-m})M_0^4\). To treat the nonlinear difference, we expand it algebraically to isolate the problematic derivatives \[\begin{align} \mathcal{N}(P_{\leq \ell}u_\ell)-\mathcal{N}(P_{\leq m}v_m) &=\mathcal{N}_\delta(P_{\leq \ell}u_\ell)-\mathcal{N}_\delta(P_{\leq m}v_m) \\ &\quad +\frac{i}{2} (P_{\leq \ell}u_\ell-P_{\leq m}v_m) (\alpha +i\beta H)\partial_x (|P_{\leq \ell}u_\ell|^2+|P_{\leq m}v_m|^2) \\ &\quad +i\tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}(\alpha +i\beta H)\partial_x\left(\overline{\tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}}(P_{\leq \ell}u_\ell-P_{\leq m}v_m)\right) \\ &\quad+i\tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}(\alpha +i\beta H)\partial_x \left(\tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}\overline{(P_{\leq \ell}u_\ell-P_{\leq m}v_m)}\right) , \end{align}\] where \(\mathcal{N}_\delta(\cdot)\) is defined in 10 . Applying Lemma 1 and Lemma 2, we control \[\| \partial_x(\mathcal{N}_\delta(P_{\leq \ell}u_\ell)-\mathcal{N}_\delta(P_{\leq m}v_m))\|_{L^\infty_TL^2}\| \partial_x(P_{\leq \ell}u_\ell-P_{\leq m}v_m)\|_{L^\infty_TL^2} \lesssim (M_0+M_0^2)\| d^1_\delta(P_{\leq \ell}u_\ell,P_{\leq m}v_m)\|_{L^\infty_T}^2 .\] For the remaining terms, we use integration by parts to distribute derivatives. Most interactions are bounded by \(M_0^2\| P_{\leq \ell}u_\ell-P_{\leq m}v_m\|_{X^1}^2\) thanks to Lemma 2 and 43 . The only term that cannot be bounded directly is the one involving \(\beta H\) and two spatial derivatives on the difference. Isolating this interaction, we obtain \[\label{est:apriorid1-2} \begin{align} \frac{d}{dt}\big\| \partial_x\big( u_\ell(t)-v_m(t)\big) \big\|_{L^2}^2 &\leq 2\beta \undefined\Big( \tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}(iH)\Big(\tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}\partial_x^2\overline{(P_{\leq \ell}u_\ell-P_{\leq m}v_m)}\Big) , \\ & \partial_x\big(P_{\leq \ell}u_\ell-P_{\leq m}v_m\big) \Big)_{L^2} \\ &\quad +C(2^{-\ell}+2^{-m})M_0^4+C(1+M_0^2)\| d^1_\delta(P_{\leq \ell}u_\ell,P_{\leq m}v_m)\|_{L^\infty_T}^2. \end{align}\tag{47}\]

Since the first term on the right-hand side of 47 prevents closing the estimate, we introduce a time-dependent corrector \(\mathcal{I}(t)\) for a frequency threshold \(n_2\in \mathbb{Z}_{\geq 0}\) \[\mathcal{I}(t):=\beta \undefined\Big( \tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}H\partial_x \Big(\tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}\overline{P_{\geq n_2}\big(P_{\leq \ell}u_\ell-P_{\leq m}v_m\big)}\Big) , P_{\geq n_2}\big(P_{\leq \ell}u_\ell-P_{\leq m}v_m\big) \Big)_{L^2} .\] Differentiating \(\mathcal{I}(t)\) \[\label{est:apriorid1-3} \begin{align} \frac{d}{dt}\mathcal{I}(t) &= -2\beta \undefined\bigg( iH\Big(\tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2} \partial_x^2\overline{(P_{\leq \ell}u_\ell - P_{\leq m}v_m)}\Big) , \overline{\tfrac{P_{\leq \ell}u_\ell+P_{\leq m}v_m}{2}}\partial_x(P_{\leq \ell}u_\ell - P_{\leq m}v_m)\bigg)_{L^2} \\ &\quad + \mathcal{R}_2(t), \end{align}\tag{48}\] where \[|\mathcal{R}_2(t)| \lesssim (2^{-\ell}+2^{-m})M_0^6 + \big( (1+2^{n_2})M_0^2+M_0^4\big) \| d^1_\delta(P_{\leq \ell}u_\ell,P_{\leq m}v_m)\|_{L^\infty_T}^2.\] Combining this with 47 yields \[\label{est:apriorid1-5} \begin{align} \frac{d}{dt}\Big( \big\| \partial_x\big( u_\ell(t)-v_m(t)\big) \big\|_{L^2}^2 +\mathcal{I}(t)\Big) &\lesssim (2^{-\ell}+2^{-m})(M_0^4+M_0^6) \\ &\quad +(1+2^{n_2}M_0^2+M_0^4) \| d^1_\delta(P_{\leq \ell}u_\ell,P_{\leq m}v_m)\|_{L^\infty_T}^2. \end{align}\tag{49}\] Integrating 49 over \([0, t]\) and utilizing the bound \(\| \mathcal{I}(t)\|_{L^\infty_T}\lesssim 2^{-n_2}M_0^2\| P_{\leq \ell}u_\ell-P_{\leq m}v_m\|_{L^\infty_TX^1}^2\), we deduce \[\begin{align} \label{est:apriorid1-B} \big\| \partial_x\big( u_\ell-v_m\big) \big\|_{L^\infty_TL^2}^2 &\lesssim \Big( \|\partial_x(\phi-\psi)\|_{L^2}+(2^{-\ell}+2^{-m})M_0\Big) ^2 + (2^{-\ell}+2^{-m})T(M_0^4+M_0^6) \\ &\quad +\Big\{ T(1+2^{n_2}M_0^2+M_0^4) + 2^{-n_2}M_0^2\Big\} \Big( \| d^1_\delta(u_\ell,v_m)\|_{L^\infty_T}+(2^{-\ell}+2^{-m})(M_0+M_0^2)\Big)^2.\notag \end{align}\tag{50}\] Substituting this into 46 , we set \(2^{n_2}=C(1+M_0^2)\) and \(T \ll (1+M_0^4)^{-1}\). Absorbing the difference term into the left-hand side establishes the claimed bound ?? . ◻

Proof of Theorem 1. Let \(\phi\in \mathcal{Z}^2\) satisfy \(\mathcal{E}^2_\delta(\phi)\leq M_0^2\), and let \(T_*>0\) be the constant given in Proposition 3. We divide the proof into several steps.

Step 1. We first construct a local solution \(u\in C([0,T_*];\mathcal{Z}^1)\cap L^\infty(0,T_*;\mathcal{Z}^2)\) to 5 with initial data \(\phi\) such that \(u\in C([0,T_*];\mathcal{Z}^1_{\rho_\phi})\) and \[\begin{align} \big\| \mathcal{E}^2_\delta(u)\big\|_{L^\infty_{T_*}}&\lesssim \mathcal{E}^2_\delta(\phi), \label{bd:E2} \\ \| \widetilde{\mathcal{E}}(u)\|_{L^\infty_{T_*}}&\lesssim \widetilde{\mathcal{E}}(\phi) +\| \phi\|_{X^1}^2. \label{bd:Eti} \end{align}\] {#eq: sublabel=eq:bd:E2,eq:bd:Eti} We also show that if \(\phi\in \mathcal{Z}^k\) for some integer \(k\geq 3\), then \(u\in L^\infty(0,T_*;\mathcal{Z}^k)\) and satisfies \[\label{bd:Ek} \| u\|_{L^\infty_{T_*}X^k}\lesssim_k \| \phi\|_{X^k}.\qquad{(6)}\]

Let \(\{ u_\ell\}_{\ell\geq 0}\) be the sequence of solutions to the mollified problem 30 with initial data \(u_\ell(0)=P_{\leq \ell}\phi\), constructed in Proposition 1. Propositions 2 and 3 guarantee that \(u_\ell\in C([0,T_*];\mathcal{Z}^2)\) for all \(\ell\geq 0\) and that \(\{ u_\ell\}_{\ell\geq 0}\) is Cauchy in the complete metric space \(C([0,T_*];\mathcal{Z}^1)\). Hence, there exists \(u\in C([0,T_*];\mathcal{Z}^1)\) such that \(u_\ell\to u\) strongly in \(C([0,T_*];\mathcal{Z}^1)\). Moreover, ?? shows that the sequence \(\{ \partial_xu_\ell\}_{\ell\geq 0}\) is bounded in \(L^\infty(0,T_*;H^1(\mathbb{R}))\). By the Banach-Alaoglu theorem, we can extract a subsequence converging weakly-\(*\) to some \(\widetilde{u}\in L^\infty(0,T_*;H^1(\mathbb{R}))\). Since \(\partial_xu_\ell\to \partial_xu\) strongly in \(C([0,T_*];L^2(\mathbb{R}))\), we have \(\partial_xu=\widetilde{u}\in L^\infty(0,T_*;H^1(\mathbb{R}))\); thus \(u\in L^\infty(0,T_*;\mathcal{Z}^2)\). The bound ?? follows immediately from ?? and the weak-\(*\) lower semicontinuity of the \(L^\infty_{T_*}H^1\) norm, while ?? is a consequence of ?? and the strong convergence in \(C([0,T_*];\mathcal{Z}^1)\). The higher regularity ?? for \(k\geq 3\) is obtained analogously using ?? and ?? .

It remains to verify that the limit \(u\) satisfies the integral equation \[\label{gINLS-int} u(t)=U(t)\phi -i\int_0^tU(t-s)\mathcal{N}(u(s))\,ds,\tag{51}\] that \(u(0)=\phi\) and that \(u\in C([0,T_*];\mathcal{Z}^1_{\rho_\phi})\). The initial condition \(u(0)=\phi\) is clear from the strong convergence. To pass to the limit in the nonlinearity, we estimate \[\begin{align} &\| P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u_\ell)-\mathcal{N}(u)\|_{L^\infty_{T_*}L^2}\\ &\lesssim 2^{-\ell}\| \mathcal{N}(P_{\leq \ell}u_\ell)\|_{L^\infty_{T_*}H^1}+\| \mathcal{N}(P_{\leq \ell}u_\ell)-\mathcal{N}(u)\|_{L^\infty_{T_*}L^2} \\ &\lesssim 2^{-\ell}\delta^{\frac{1}{3}}\| \mathcal{E}^2_\delta(P_{\leq \ell}u_\ell)\|_{L^\infty_{T_*}}^{\frac{3}{2}}+\delta^{\frac{1}{3}}\Big( 1+\| \mathcal{E}^1_\delta(P_{\leq \ell}u_\ell)\|_{L^\infty_{T_*}}+\| \mathcal{E}^1_\delta(u)\|_{L^\infty_{T_*}}\Big) \| d^1_\delta(P_{\leq \ell}u_\ell,u)\|_{L^\infty_{T_*}} \\ &\lesssim_{\delta,M_0} 2^{-\ell}+\| d^1_\delta(u_\ell,u)\|_{L^\infty_{T_*}} \\ &\to \;0\qquad (\text{as } \ell\to \infty). \end{align}\] Here, we utilized Lemma 1, 20 , ?? , ?? , and the strong convergence \(u_\ell\to u\) in \(C([0,T_*];\mathcal{Z}^1)\). Consequently, \[-i\int_0^t U(t-s)P_{\leq \ell}\mathcal{N}(P_{\leq \ell}u_\ell(s))\,ds\;\to \;-i\int_0^t U(t-s)\mathcal{N}(u(s))\,ds \quad\text{strongly in } C([0,T_*];L^2(\mathbb{R})).\] Together with the convergence of the linear part, this shows that \(u\) satisfies 51 . Because \(U(t)\phi\in C(\mathbb{R};\mathcal{Z}^1_{\rho_\phi})\) by 27 and the Duhamel integral belongs to \(C([0,T_*];H^1(\mathbb{R}))\subset C([0,T_*];\mathcal{Z}^1_0)\) for \(u\in L^\infty(0,T_*;\mathcal{Z}^2)\) by Lemma 1, we conclude from 51 that \(u\in C([0,T_*];\mathcal{Z}^1_{\rho_\phi})\).

Finally, let \(\phi,\psi\in \mathcal{Z}^2\) with \(\mathcal{E}^2_\delta(\phi), \mathcal{E}^2_\delta(\psi)\leq M_0^2\) and let \(u,v\) be the corresponding solutions constructed above. Taking \(\ell,m\to \infty\) in ?? yields \[\label{bd:d1} \big\| d^1_\delta(u,v)\big\|_{L^\infty_{T_*}}\lesssim (1+M_0)\,d^1_\delta(\phi,\psi).\tag{52}\]

Step 2. We claim the following: Let \(u,v\in C([0,T];\mathcal{Z}^2)\) be two solutions to 5 on an interval \([0,T]\). Assume that \(\| \mathcal{E}^2_\delta(u)\|_{L^\infty_T}\leq M_1^2\) and \(\| \mathcal{E}^2_\delta(v)\|_{L^\infty_T}\leq M_1^2\) for some \(M_1>0\). Then, there exists \(T_1=T_1(M_1)>0\) such that for \(0\leq t\leq \min \{ T,T_1\}\) it holds that \[d^1_\delta(u(t),v(t))\lesssim (1+M_1)d^1_\delta(u(0),v(0)).\] In particular, the solution of the Cauchy problem associated with 5 is unique in \(C([0,T];\mathcal{Z}^2)\).

We remark that this claim must be discussed independently of the previous Step 1 (in particular the difference estimate 52 ), since for uniqueness we have to consider general solutions which may differ from those constructed in Step 1. However, the claim can be proven along the same lines as Proposition 3 using the original equation 5 , 51 instead of the mollified one 30 , 31 . We omit the details.

Step 3. We shall prove that the solution \(u\) constructed in Step 1 satisfies \(u \in C([0,T_*];\mathcal{Z}^2)\) and the continuity of the flow map \(u_0\mapsto u\) in \(C([0,T_*];\mathcal{Z}^2)\).

We can prove \(u\in C([0,T_*];\mathcal{Z}^k)\) for \(\phi\in \mathcal{Z}^k\) if \(k\geq 3\) in a similar manner; we omit the proof. In particular, this concludes the proof of Theorem 1. To prove the claim, we may adapt an argument due to Bona–Smith [29] or the idea of frequency envelope due to Tao [30]–[32] to the nonvanishing solutions in \(\mathcal{Z}^2\).³ Here, we outline an adaptation of the latter idea.

Fix \(\varepsilon\in (0,1)\), and for \(\delta>0\) define the frequency envelope \(\mathbf{c}_\delta [f]=\{ c_{\delta,j}[f]\}_{j\geq 0}\) of \(f\in \mathcal{Z}^2\) by⁴ \[c_{\delta,j}[f]:=\sum _{k=0}^{\infty}2^{-\varepsilon|j-k|}\| P_k\partial_xf\|_{H^1}+2^{-\varepsilon j}\big( \| f\|_{L^\infty}+\delta^{-\frac{2}{3}}\big\| |f|^2-\rho_f^2\big\|_{L^2}^{\frac{1}{2}}\big) ,\qquad j\in\mathbb{Z}_{\geq 0}.\] We collect some basic properties of \(\mathbf{c}_\delta [f]\) in the following lemma, which can be shown by a standard argument (see, e.g., [31]); we give a proof in Appendix 6 for completeness.

For any \(\delta\geq 1\), \(\phi\in \mathcal{Z}^2\) with \(\mathcal{E}^2_\delta(\phi)\leq M_0^2\) and \(j\geq 1\), let \(u\) and \(u^j\) be the unique solutions of 5 with initial condition, respectively, \(u(0)=\phi\) and \(u^j(0)=P_{\leq j}\phi\), which we have constructed in the class \(C([0,T_*];\mathcal{Z}^1)\cap L^\infty(0,T_*;\mathcal{Z}^2)\).⁵ Note that \(P_{\leq j}\phi\in \mathcal{Z}^3\) and hence \(u^j\in C([0,T_*];\mathcal{Z}^1)\cap L^\infty(0,T_*;\mathcal{Z}^3)\subset C([0,T_*];\mathcal{Z}^2)\). Using the \(X^3\) bound from ?? , the \(\mathcal{Z}^1\) difference bound 52 , and Lemma 3, we can derive the following estimate: \[\label{est:fe} \big\| P_k\partial_x^2(u^j-u^{j-1})\big\|_{L^\infty_{T_*}L^2}\lesssim_{M_0} 2^{-|k-j|}c_{\delta,j}[\phi],\qquad j\geq 1,\quad k\geq 0.\tag{53}\] This estimate is the key to prove the claim of Step 3 and shows that the initial frequency distribution is not significantly disturbed during the time of local existence. For the reader’s convenience, we also give a proof of 53 in Appendix 6.

To see \(u\in C([0,T_*];\mathcal{Z}^2)\), we first notice that the difference bound 52 shows the convergence \(u^j\to u\) in \(C([0,T_*];\mathcal{Z}^1)\), and hence \(u\in C([0,T_*];\mathcal{Z}^1)\). It then suffices to prove \(u\in C([0,T_*];\dot{H}^2(\mathbb{R}))\). We deduce from 53 that for any \(m_2>m_1\geq 0\), \[\begin{align} \| \partial_x^2(u^{m_2}-u^{m_1})\|_{L^\infty_{T_*}L^2}^2&=\bigg\| \sum_{j=m_1+1}^{m_2}\partial_x^2(u^j-u^{j-1})\bigg\|_{L^\infty_{T_*}L^2}^2 \\ &\lesssim \sum_{j_1,j_2=m_1+1}^{m_2}\sum_{\begin{smallmatrix} k_1,k_2\geq 0 \\ |k_1-k_2|\leq 1\end{smallmatrix}}\| P_{k_1}\partial_x^2(u^{j_1}-u^{j_1-1})\|_{L^\infty_{T_*}L^2}\| P_{k_2}\partial_x^2(u^{j_2}-u^{j_2-1})\|_{L^\infty_{T_*}L^2} \\ &\lesssim_{M_0} \sum_{j_1,j_2=m_1+1}^{m_2}\sum_{\begin{smallmatrix} k_1,k_2\geq 0 \\ |k_1-k_2|\leq 1\end{smallmatrix}}2^{-|k_1-j_1|}c_{\delta,j_1}[\phi] 2^{-|k_2-j_2|}c_{\delta,j_2}[\phi] \\ &\lesssim \sum_{j_1,j_2=m_1+1}^{m_2} 2^{-\frac{1}{2}|j_1-j_2|}c_{\delta,j_1}[\phi]c_{\delta,j_2}[\phi] \lesssim \sum_{j=m_1+1}^{m_2}c_{\delta,j}[\phi]^2 \leq \| \mathbf{c}_\delta[\phi]\|_{\ell^2(\mathbb{Z}_{>m_1})}^2. \end{align}\] Since \(\mathbf{c}_\delta[\phi]\in \ell^2(\mathbb{Z}_{\geq 0})\), we see that the sequence \(\{ u^j\}_{j\geq 0}\) is Cauchy in \(C([0,T_*];\dot{H}^2(\mathbb{R}))\), concluding that the limit \(u\) belongs to \(C([0,T_*];\mathcal{Z}^2)\) and that \[\label{est:cont1} \| \partial_x^2(u-u^j)\|_{L^\infty_{T_*}L^2}\lesssim_{M_0} \| \mathbf{c}_\delta[\phi]\|_{\ell^2(\mathbb{Z}_{>j})},\qquad j\geq 0.\tag{54}\]

To see the continuity of the flow map, let \(\phi, \{ \phi_{(n)} \}_{n\geq 0}\) be such that \(\mathcal{E}^2_\delta(\phi), \mathcal{E}^2_\delta(\phi_{(n)})\leq M_0^2\) and \(d^2_\delta(\phi_{(n)},\phi)\to 0\) as \(n\to \infty\). Furthermore, let \(u_{(n)}, u_{(n)}^{j} \in C([0,T_*];\mathcal{Z}^2)\) denote the solutions to 5 with \(u_{(n)}(0)=\phi_{(n)}\) and \(u_{(n)}^{j}(0)=P_{\leq j}\phi_{(n)}\). In view of the \(\mathcal{Z}^1\) difference estimate 52 , it suffices to prove \[\label{est:cont2} \lim_{n\to \infty}\| \partial_x^2(u_{(n)} - u) \|_{L_{T_*}^{\infty}L^2}=0.\tag{55}\] The triangle inequality yields \[\| \partial_x^2(u_{(n)} - u) \|_{L_{T_*}^{\infty}L^2}\leq \| \partial_x^2(u_{(n)}-u_{(n)}^j)\|_{L^\infty_{T_*}L^2}+\| \partial_x^2(u-u^j)\|_{L^\infty_{T_*}L^2}+\| \partial_x^2(u_{(n)}^j-u^j)\|_{L^\infty_{T_*}L^2} .\] Using 54 and Lemma 3 (ii), we bound the first two terms on the right-hand side by \[\| \mathbf{c}_\delta[\phi_{(n)}]\|_{\ell^2(\mathbb{Z}_{>j})}+\| \mathbf{c}_\delta[\phi]\|_{\ell^2(\mathbb{Z}_{>j})} \lesssim \| \phi_{(n)}-\phi\|_{X^2}+\big[ d^1_\delta(\phi_{(n)},\phi)\big]^{\frac{1}{2}}+\| \mathbf{c}_\delta[\phi]\|_{\ell^2(\mathbb{Z}_{>j})}.\] For the last term, we can use the \(X^3\) bound from ?? and the difference bound 52 as well as 22 : \[\label{est:cont3} \begin{align} \| \partial_x^2(u_{(n)}^j-u^j)\|_{L^\infty_{T_*}L^2}^2&\leq \big( \| \partial_x^3u_{(n)}^j\|_{L^\infty_{T_*}L^2}+\| \partial_x^3u^j\|_{L^\infty_{T_*}L^2}\big) \| d^1_\delta(u_{(n)}^j,u^j)\|_{L^\infty_{T_*}} \\ &\lesssim_{M_0} 2^j\, d^1_\delta(P_{\leq j}\phi_{(n)},P_{\leq j}\phi) \lesssim_{M_0} 2^j\,d^1_\delta(\phi_{(n)},\phi). \end{align}\tag{56}\] Putting these estimates together, we have \[\limsup_{n\to \infty}\| \partial_x^2(u_{(n)}-u)\|_{L^\infty_{T_*}L^2}\lesssim_{M_0} \| \mathbf{c}_\delta[\phi]\|_{\ell^2(\mathbb{Z}_{>j})}\] for any \(j\), and obtain 55 by letting \(j\to \infty\).

We have completed the proof of Theorem 1. ◻

Proof of Theorem 2. Let \(\{ \phi_\delta\}_{\delta\in [1,\infty]}\), \(\{ u_\delta\}_{\delta\in[1,\infty]}\) be as in the statement of the theorem, and take \(M_0>0\) such that \(\| \phi_\infty\|_{X^2}^2\leq M_0^2\). Then, by the assumption and 19 , there exists \(\delta_*\in [1,\infty)\) such that \(\mathcal{E}^2_\delta(\phi_\delta)\leq 2M_0^2\) and \(\mathcal{E}^2_\delta(P_{\leq j}\phi_\delta)\leq 6M_0^2\) for any \(\delta\in [\delta_*,\infty )\) and \(j\in \mathbb{Z}_{\geq 0}\). Let \(u_\delta^j\) denote the (unique) solution to 5 (\(\delta\in [1,\infty)\)) or 6 (\(\delta=\infty\)) with \(u_\delta^j(0)=P_{\leq j}\phi_\delta\). From the proof of Theorem 1, there exists \(T_*\gtrsim (1+M_0^4)^{-1}\) such that for any \(\delta\in [\delta_*,\infty )\) and \(j\in \mathbb{Z}_{\geq 0}\), these solutions exist on \([0,T_*]\) (in particular \(T_{\delta,\max}>T_*\)) and satisfy \[\begin{align} \| \mathcal{E}^2_\delta(u_\delta)\|_{L^\infty_{T_*}}+\| \mathcal{E}^2_\delta(u_\delta^j)\|_{L^\infty_{T_*}}&\lesssim \mathcal{E}^2_\delta(\phi_\delta)+\mathcal{E}^2_\delta(P_{\leq j}\phi_\delta)\lesssim M_0^2, \tag{57} \\ \| \widetilde{\mathcal{E}}(u_\delta)\|_{L^\infty_{T_*}}+\| \widetilde{\mathcal{E}}(u_\delta^j)\|_{L^\infty_{T_*}}&\lesssim \widetilde{\mathcal{E}}(\phi_\delta)+\widetilde{\mathcal{E}}(P_{\leq j}\phi_\delta)+\| \phi_\delta\|_{X^1}^2\lesssim \widetilde{\mathcal{E}}(\phi_\delta)+M_0^2, \tag{58} \\ \| u_\delta^j \|_{L^\infty_{T_*}X^3}&\lesssim \| P_{\leq j}\phi_\delta\|_{X^3}\lesssim 2^jM_0, \tag{59} \\ \| d^1_\delta(u_\delta^j,u_\delta)\|_{L^\infty_{T_*}}&\lesssim_{M_0} d^1_\delta(P_{\leq j}\phi_\delta,\phi_\delta)\lesssim 2^{-j}(M_0+M_0^2), \tag{60} \\ \| \partial_x^2(u_\delta-u_\delta^j)\|_{L^\infty_{T_*}L^2}&\lesssim _{M_0}\| \mathbf{c}_{\delta}[\phi_\delta]\|_{\ell^2(\mathbb{Z}_{>j})}. \tag{61} \end{align}\] Indeed, these estimates have been shown in ?? , ?? , ?? , 52 , and 54 , respectively, with the constants independent of \(\delta\in [1,\infty )\). Similarly, for \(\delta=\infty\) and any \(j\geq 0\) we have the corresponding estimates \[\begin{gather} \| u_\infty\|_{L^\infty_{T_*}X^2}+\| u_\infty^j\|_{L^\infty_{T_*}X^2}\lesssim M_0,\qquad \| u_\infty^j\|_{L^\infty_{T_*}X^3}\lesssim \| P_{\leq j}\phi_\infty\|_{X^3}\lesssim 2^jM_0, \\ \| u_\infty^j-u_\infty\|_{L^\infty_{T_*}X^1}\lesssim _{M_0}\| P_{\geq j+1}\phi_\infty\|_{X^1}\lesssim 2^{-j}M_0,\qquad \| \partial_x^2(u_\infty-u_\infty^j)\|_{L^\infty_{T_*}L^2}\lesssim _{M_0}\| \mathbf{c}_\infty[\phi_\infty]\|_{\ell^2(\mathbb{Z}_{>j})}. \end{gather}\]

We next prove the difference estimate \[\label{est:diff-delta} \| u_\delta - u_\infty \|_{L^\infty_{T_*}X^1}\lesssim \| \phi_\delta-\phi_\infty\|_{X^1} + \nu_{M_0}(\delta),\qquad \delta\in [\delta_*,\infty),\tag{62}\] where \(\nu_{M_0}(\delta)\) is a nonnegative function satisfying \(\lim\limits_{\delta\to\infty}\nu_{M_0}(\delta)=0\) for each \(M_0>0\). To see 62 , it suffices to prove \[\label{est:diff-delta-j} \| u_\delta^j-u_\infty^j\|_{L^\infty_{T_*}X^1}\lesssim \| \phi_\delta-\phi_\infty\|_{X^1} + \nu_{M_0}(\delta),\qquad \delta\in [\delta_*,\infty),\;j\geq 0\tag{63}\] and take the limit \(j\to \infty\) with the estimate 60 . Then, since \(u_\delta^j\) is smooth, we can follow the calculations in the proof of Proposition 3. In fact, the main part of the proof involving the modified energy estimate remains unchanged; we only have to replace the difference of \(\mathcal{N}_\delta\)’s for two solutions by a single \(\mathcal{N}_\delta(u_\delta^j)\). The previous argument for Proposition 3 implies the estimates \[\begin{align} \| u_\delta^j-u_\infty^j\|_{L^\infty_{T_*}X^1}&\lesssim \| P_{\leq j}(\phi_\delta-\phi_\infty)\|_{X^1}+T_*\| \mathcal{N}_\delta(u_\delta^j)\|_{L^\infty_{T_*}X^1}+\| \partial_x(u_\delta^j-u_\infty^j)\|_{L^\infty_{T_*}L^2}, \\ \| \partial_x( u_\delta^j-u_\infty^j) \|_{L^\infty_{T_*}L^2}^2 &\lesssim \| \partial_xP_{\leq j}(\phi_\delta-\phi_\infty)\|_{L^2}^2 + T_*(M_0+M_0^3)\| \mathcal{N}_\delta(u_\delta^j)\|_{L^\infty_{T_*}X^1}\\ &\quad +\big(T_*(2^{n_2}M_0^2+M_0^4) + 2^{-n_2}M_0^2\big) \| u_\delta^j-u_\infty^j\|_{L^\infty_{T_*}X^1}^2 \end{align}\] instead of 46 and 50 , respectively. Choosing \(n_2\) appropriately and modifying \(T_*\ll (1+M_0^4)^{-1}\) if necessary, we obtain \[\| u_\delta^j-u_\infty^j\|_{L^\infty_{T_*}X^1}^2\lesssim \| \phi_\delta-\phi_\infty\|_{X^1}^2+ (M_0+M_0^3)\| \mathcal{N}_\delta(u_\delta^j)\|_{L^\infty_{T_*}X^1}+\| \mathcal{N}_\delta(u_\delta^j)\|_{L^\infty_{T_*}X^1}^2.\] Now, Lemma 1, 57 , 58 , and the assumption \(\lim\limits_{\delta\to\infty}\delta^{-\frac{4}{3}}\widetilde{\mathcal{E}}(\phi_\delta)=0\) imply that \[\begin{align} \| \mathcal{N}_\delta(u_\delta^j)\|_{L^\infty_{T_*}X^1}&\lesssim \delta^{-1}\big( \| \widetilde{\mathcal{E}}(u_\delta^j)\|_{L^\infty_{T_*}}^{\frac{3}{4}}\| u_\delta^j\|_{L^\infty_{T_*}X^1}^{\frac{3}{2}}+\| u_\delta^j\|_{L^\infty_{T_*}X^1}^3\big) \\ &\lesssim \big[ \delta^{-\frac{4}{3}}\widetilde{\mathcal{E}}(\phi_\delta)\big]^{\frac{3}{4}} M_0^{\frac{3}{2}}+\delta^{-1}M_0^3 \;\to \;0\qquad (\delta \to \infty). \end{align}\] Hence, we have 63 by setting \[\nu _{M_0}(\delta):=\Big\{ (M_0+M_0^3)\Big( \big[ \delta^{-\frac{4}{3}}\widetilde{\mathcal{E}}(\phi_\delta)\big]^{\frac{3}{4}} M_0^{\frac{3}{2}}+\delta^{-1}M_0^3\Big) +\Big( \big[ \delta^{-\frac{4}{3}}\widetilde{\mathcal{E}}(\phi_\delta)\big]^{\frac{3}{4}} M_0^{\frac{3}{2}}+\delta^{-1}M_0^3\Big)^2\Big\}^{\frac{1}{2}} .\]

In view of 62 , the convergence \(\| u_\delta- u_\infty \|_{L^\infty_{T_*}X^2}\to 0\) follows once \(\| \partial_x^2(u_\delta-u_\infty)\|_{L^\infty_{T_*}L^2}\to 0\) is verified. To prove the latter convergence, we follow the argument showing continuous dependence in Step 3 of the proof of Theorem 1. Using interpolation, 59 , and 63 in place of 52 , we have the fixed-\(j\) difference estimate corresponding to 56 : \[\begin{align} \| \partial_x^2(u_\delta^j-u_\infty^j)\|_{L^\infty_{T_*}L^2}^2&\leq \big( \| \partial_x^3u_\delta^j\|_{L^\infty_{T_*}L^2}+\| \partial_x^3u_\infty^j\|_{L^\infty_{T_*}L^2}\big) \| u_\delta^j-u_\infty^j\|_{L^\infty_{T_*}X^1}\\ &\lesssim_{M_0} 2^j\big( \| \phi_\delta-\phi_\infty\|_{X^1}+\nu_{M_0}(\delta)\big) . \end{align}\] Moreover, similarly to Lemma 3 (ii) we can show that \[\| \mathbf{c}_{\delta}[\phi_\delta]- \mathbf{c}_{\infty}[\phi_\infty]\|_{\ell^2(\mathbb{Z}_{\geq 0})}\lesssim \| \phi_\delta-\phi_\infty\|_{X^2}+\big[\delta^{-\frac{4}{3}}\widetilde{\mathcal{E}}(\phi_\delta)\big]^{\frac{1}{2}}\;\to 0\qquad (\delta\to \infty).\] Combining them with 61 , we obtain \[\begin{align} \limsup_{\delta\to \infty}\| \partial_x^2(u_\delta-u_\infty)\|_{L^\infty_{T_*}L^2} &\leq \limsup_{\delta\to\infty}\Big( \| \partial_x^2(u_\delta-u_\delta^j)\|_{L^\infty_{T*}L^2}+\| \partial_x^2(u_\infty-u_\infty^j)\|_{L^\infty_{T*}L^2}+\| \partial_x^2(u_\delta^j-u_\infty^j)\|_{L^\infty_{T_*}L^2}\Big) \\ &\lesssim_{M_0} \limsup_{\delta\to \infty}\Big( \| \mathbf{c}_{\delta}[\phi_\delta]\|_{\ell^2(\mathbb{Z}_{>j})}+\| \mathbf{c}_{\infty}[\phi_\infty]\|_{\ell^2(\mathbb{Z}_{>j})}+\big[ 2^j\big( \| \phi_\delta-\phi_\infty\|_{X^1}+\nu_{M_0}(\delta) \big) \big]^{\frac{1}{2}}\Big) \\ &\lesssim \| \mathbf{c}_{\infty}[\phi_\infty]\|_{\ell^2(\mathbb{Z}_{>j})} \;\to 0\qquad (j\to \infty). \end{align}\] We have thus proven that \(T_{\delta,\max}>T_*\) for any \(\delta \in [\delta_*,\infty]\) and \(\lim\limits_{\delta\to\infty}\| u_\delta-u_\infty\|_{L^\infty_{T_*}X^2}=0\). In addition, we see from 58 that \(\delta^{-\frac{4}{3}}\widetilde{\mathcal{E}}(u_\delta(T_*))\to 0\) as \(\delta\to \infty\).

Now, we take \(T\in (0,T_{\infty,\max})\) arbitrarily. Since \(\| u_\infty (t)\|_{X^2}\) is bounded on \([0,T]\), repeating the above procedure finitely many times we see that there exists \(\delta_{**}\in [1,\infty)\) such that \(T_{\delta,\max}>T\) for any \(\delta\in [\delta_{**},\infty]\) and \(\lim\limits_{\delta\to\infty}\| u_\delta-u_\infty \|_{L^\infty_TX^2}=0\).

Finally, the case \(\phi_\infty\in \mathcal{Z}^2\), \(\lim\limits_{\delta\to \infty}d^2(\phi_\delta,\phi_\infty)=0\) can be treated by a simpler argument, and we omit the details. This completes the proof of Theorem 2. ◻

Proof. The proof follows from a direct calculation. ◻

Proof. Using the identity \(\undefined(z_1)\undefined(z_2)=\undefined(z_1\overline{z_2})-\undefined(z_1)\undefined(z_2)\) (\(z_1,z_2\in \mathbb{C}\)), we have \[\begin{align} &\int \undefined[(\partial_x\overline{u})\partial_x^2u] \partial_x(\undefined[\overline{u}\partial_xu]) =\int |\partial_x^2u|^2\undefined[u\partial_x\overline{u}] -\int \undefined[(\partial_x\overline{u})\partial_x^2u] \undefined[\overline{u}\partial_x^2u] \\ &=\int |\partial_x^2u|^2\cdot \frac{1}{2}\partial_x(|u|^2) -\int \frac{1}{2} \partial_x(|\partial_xu|^2)\cdot \Big( \frac{1}{2} \partial_x^2(|u|^2)-|\partial_xu|^2\Big) \\ &=\frac{1}{2}\int |\partial_x^2u|^2\partial_x(|u|^2) -\frac{1}{4}\int \partial_x(|\partial_xu|^2)\cdot \partial_x^2(|u|^2) , \end{align}\] as desired. ◻

Proof of Theorem 4. First, it is easy to see that \(\mathcal{H}_1(u)\) and \(\mathcal{H}_2(u)\) are well-defined for \(u\in \mathcal{Z}^2_\rho\), noticing \(u\in L^\infty(\mathbb{R})\), \(\partial_xu\in H^1(\mathbb{R})\), \(|u|^2-\rho^2\in H^1(\mathbb{R})\), and that \(T_{\delta}\partial_x\) maps \(H^1(\mathbb{R})\) into \(L^2(\mathbb{R})\) by 13 . Moreover, we may assume that \(u\in \mathcal{Z}^\infty_\rho\) by a standard approximation argument, which relies on continuity of the flow map and persistence of regularity in the local well-posedness Theorem 1.

For (i), we begin by observing, from Lemma 4, that \[\partial_t \int |\partial_xu|^2 =2\alpha \int |\partial_xu|^2\partial_x(|u|^2) +2\beta \int \undefined[\overline{u}\partial_xu] T_{\delta}\partial_x^2(|u|^2) .\] To cancel out two terms on the right-hand side, we introduce two correctors \[I_1^{(1)}(u):=\int (|u|^2-\rho^2)\undefined[ \overline{u}\partial_{x}u],\qquad I_1^{(2)}(u):= \int (|u|^2-\rho^2)T_{\delta}\partial_{x}(|u|^2-\rho^2) .\] Using Lemma 4 and integration by parts, we see that \[\begin{align} \partial_tI_1^{(1)}(u)&=-2\int |\partial_xu|^2\partial_x(|u|^2) -2\alpha \int |u|^2(|u|^2-\rho^2) \partial_x(\undefined[\overline{u}\partial_xu]) \\ &\quad -\beta \int \partial_x(|u|^4)\cdot T_{\delta}\partial_x(|u|^2-\rho^2) ,\\ \partial_tI_1^{(2)}(u)&=-4\int \undefined[\overline{u}\partial_xu] T_{\delta}\partial_x^2(|u|^2) +2\alpha \int\partial_x(|u|^4)\cdot T_{\delta}\partial_x(|u|^2-\rho^2) . \end{align}\] Here, we have used the symmetry of \(T_{\delta}\partial_x\) 15 to verify \[\int \partial_x(|u|^2)\cdot T_{\delta}\partial_x(|u|^2-\rho^2)=\int \partial_x(|u|^2-\rho^2)\cdot T_{\delta}\partial_x(|u|^2-\rho^2)=0.\] From these calculations, we see that \[\partial_t\Big( \int |\partial_xu|^2 +\alpha I_1^{(1)}(u)+\frac{\beta}{2}I_1^{(2)}(u)\Big) =-2\alpha ^2\int |u|^2(|u|^2-\rho^2)\partial_x(\undefined[\overline{u}\partial_xu]).\] To cancel out the right-hand side, we further add two correctors \[I_1^{(3)}(u):= \int (|u|^2-\rho^2)^3,\qquad I_1^{(4)}(u):=\int (|u|^2-\rho^2)^2 .\] Using Lemma 4 and integration by parts, we see that \[\begin{align} \partial_tI_1^{(3)}(u)&=6\int (|u|^2-\rho^2)^2\partial_x (\undefined[\overline{u}\partial_xu]),\\ \partial_tI_1^{(4)}(u)&=4\int (|u|^2-\rho^2)\partial_x (\undefined[\overline{u}\partial_xu]) . \end{align}\] Here, we have calculated as follows: for \(k\geq 1\), \[\begin{align} \int (|u|^2-\rho^2)^k\partial_x(|u|^4) &=\int (|u|^2-\rho^2)^k\partial_x\Big( (|u|^2-\rho^2)^2 +2\rho^2(|u|^2-\rho^2)+\rho^4 \Big) \\ &=\int \partial_x\Big( \frac{2}{k+2}(|u|^2-\rho^2)^{k+2}+\frac{2\rho^2}{k+1}(|u|^2-\rho^2)^{k+1}\Big) =0. \end{align}\] From the above calculations, we have \[2\alpha ^2\partial_t\Big( \frac{1}{6}I_1^{(3)}(u)+\frac{\rho^2}{4}I_1^{(4)}(u)\Big) =2\alpha ^2\int |u|^2(|u|^2-\rho^2)\partial_x(\undefined[\overline{u}\partial_xu]).\] Consequently, the quantity \[\mathcal{H}_1(u):=\int |\partial_xu|^2 +\alpha I_1^{(1)}(u)+\frac{\beta}{2}I_1^{(2)}(u)+\frac{\alpha^2}{3}I_1^{(3)}(u)+\frac{\alpha^2\rho^2}{2}I_1^{(4)}(u)\] is conserved in time. This finishes the proof of (i).

In a similar manner, to compute \(\mathcal{H}_2\), we begin with \[\begin{align} \partial_t\int |\partial_x^2u|^2 &=2\alpha \int |\partial_x^2u|^2\partial_x(|u|^2) +4\alpha \int \partial_x(|\partial_xu|^2)\cdot \partial_x^2(|u|^2) \\ &\quad +4\beta \int \undefined[(\partial_x\overline{u})\partial_x^2u] T_{\delta}\partial_x^2(|u|^2) +2\beta \int \partial_x(\undefined[\overline{u}\partial_xu])\cdot T_{\delta}\partial_x^3(|u|^2) , \end{align}\] and to cancel out the terms on the right-hand side we introduce five correctors⁶\[\begin{gather} I_2^{(1)}(u):=\int |\partial_xu|^2\undefined[\overline{u}\partial_xu] ,\qquad I_2^{(2)}(u):=\int \partial_x^2(|u|^2)\cdot \undefined[\overline{u}\partial_xu] ,\\ I_2^{(3)}(u):=\int |\partial_xu|^2T_{\delta}\partial_x(|u|^2-\rho^2) ,\qquad I_2^{(4)}(u):=\int \partial_x^2(|u|^2)\cdot T_{\delta}\partial_x(|u|^2-\rho^2) ,\\ I_2^{(5)}(u):=\int \undefined[\overline{u}\partial_xu]T_{\delta}\partial_x(\undefined[\overline{u}\partial_xu]) . \end{gather}\] By a calculation involving Lemmas 4–5, we see that \[\begin{align} \partial_tI_2^{(1)}(u)&= -\int |\partial_x^2u|^2\partial_x(|u|^2) +\int \partial_x(|\partial_xu|^2)\cdot \partial_x^2(|u|^2) \;+R_1(u),\\ \partial_tI_2^{(2)}(u)&=2\int \partial_x(|\partial_xu|^2)\cdot \partial_x^2(|u|^2) \;+R_2(u) ,\\ \partial_tI_2^{(3)}(u)&=-2\int \undefined[\partial_x\overline{u}\cdot \partial_x^2u]T_{\delta}\partial_x^2(|u|^2) -2\int \undefined[\overline{u}\partial_xu]T_{\delta}\partial_x^2(|\partial_xu|^2) \;+R_3(u),\\ \partial_tI_2^{(4)}(u)&=4\int \partial_x(\undefined[\overline{u}\partial_xu])\cdot T_{\delta}\partial_x^3(|u|^2) \;+R_4(u),\\ \partial_tI_2^{(5)}(u)&=\int \partial_x(\undefined[\overline{u}\partial_xu])\cdot T_{\delta}\partial_x^3(|u|^2) +4\int \undefined[\overline{u}\partial_xu]T_{\delta}\partial_x^2(|\partial_xu|^2) \;+R_5(u). \end{align}\] Here, each of the remainder terms \(R_j(u)\) (\(j=1,\dots,5\)) is a linear combination of integrals with six \(u\)’s and four spatial differentials, and also of the form \[\begin{gather} \text{either}\qquad \int fh_1h_2\qquad \text{or}\qquad \int g_1g_2h \intertext{with} f\in \big\{ 1,|u|^2\big\},\qquad g_1,g_2\in \big\{ \partial_x(|u|^2),\undefined[\overline{u}\partial_xu],T_{\delta}\partial_x(|u|^2) \big\},\\ h,h_1,h_2\in \big\{ |\partial_xu|^2,\partial_x^2(|u|^2), \partial_x^2(|u|^4), \partial_x(\undefined[\overline{u}\partial_xu]) ,T_{\delta}\partial_x^2(|u|^2), T_{\delta}\partial_x^2(|u|^4), T_{\delta}\partial_x(\undefined[\overline{u}\partial_xu]) \big\} . \end{gather}\] By using Hölder, Gagliardo-Nirenberg \(\| \partial_xu\|_{L^\infty}\leq \| \partial_xu\|_{L^2}^{1/2}\| \partial_x^2u\|_{L^2}^{1/2}\), and the estimate 13 , we estimate these remainder terms as \[\label{remainder-conservation32laws} \begin{align} |R_j(u)|&\lesssim_{\alpha,\beta ,\delta} \Big( \| u\|_{L^\infty}+\| \partial_xu\|_{L^2}\Big) ^4 \Big( \| u\|_{L^\infty}^2+\| \partial_xu\|_{L^2}^2+\| \partial_x^2u\|_{L^2}^2 \Big) \\ &\lesssim \Big(\mathcal{E}^1(u)+\big[\mathcal{E}^1(u)\big]^{\frac{3}{2}}\Big) \Big( 1+\| \partial_x^2u\|_{L^2}^2\Big) \notag. \end{align}\tag{64}\] We also see that \[\mathcal{H}_2(u):= \int |\partial_x^2u|^2 +2\alpha I_2^{(1)}(u)-3\alpha I_2^{(2)}(u)+2\beta I_2^{(3)}(u) -\frac{3}{4}\beta I_2^{(4)}(u) +\beta I_2^{(5)}(u)\] is the only choice of the coefficients to cancel out the right-hand side of \(\partial_t\int |\partial_x^2u|^2\) and yield \[\partial_t |\mathcal{H}_2(u)| \lesssim _{\alpha,\beta}\sum_{j=1}^5|R_j(u)| \lesssim _{\alpha,\beta,\delta} \Big( \mathcal{E}^1(u)+\big[\mathcal{E}^1(u)\big]^{\frac{3}{2}}\Big) \Big(1+\| \partial_x^2u\|_{L^2}^2\Big) .\] Integrating in \(t\), we obtain ?? and thus (ii).

It remains to prove (iii). By Young’s inequality, we see that \[\Big| \alpha \int (|u|^2-\rho^2)\undefined[\overline{u}\partial_xu]\Big| \leq \frac{5}{6}\int |\partial_xu|^2+\frac{3}{10}\alpha^2\int |u|^2(|u|^2-\rho^2)^2 .\] Thus, noticing the positivity of \(T_{\delta}\partial_x\) in 15 and the assumption \(\beta \geq 0\), we see that \[\mathcal{H}_1(u)\geq \frac{1}{6}\| \partial_xu\|_{L^2}^2+\frac{\alpha^2\rho^2}{6}\| |u|^2-\rho^2\|_{L^2}^2.\] Using the inequality 16 and that \(\alpha^2\rho^2>0\), we conclude \[\label{est:I1below} \| \partial_xu\|_{L^2}^2\lesssim \mathcal{H}_1(u),\quad \| |u|^2-\rho^2\|_{L^2}\lesssim_{\alpha,\rho}\mathcal{H}_1(u)^{\frac{1}{2}},\quad \| u\|_{L^\infty}^2\lesssim_{\alpha,\rho}\mathcal{H}_1(u)^{\frac{2}{3}}+1.\tag{65}\] On the other hand, using 13 –14 we easily estimate \(\mathcal{H}_1(\phi)\) as \[\label{est:I1above} \begin{align} \mathcal{H}_1(\phi)&\lesssim_{\alpha,\beta,\rho,\delta} \| \partial_x\phi\|_{L^2}^2+\| |\phi|^2-\rho^2\|_{L^2}\| \phi\|_{L^\infty}\| \partial_x\phi\|_{L^2}+\Big( \| \phi\|_{L^\infty}^2+1\Big) \| |\phi|^2-\rho^2\|_{L^2}^2 \\ &\lesssim \big[\mathcal{E}^1(\phi)\big]^{\frac{1}{2}}+\big[\mathcal{E}^1(\phi)\big]^{\frac{3}{2}}. \end{align}\tag{66}\] From 65 , 66 and the conservation of \(\mathcal{H}_1(u(t))\) in (i), we obtain ?? .

Finally, we prove ?? . Using 13 –14 , Hölder’s inequality and the a priori bound ?? , we see that \[\begin{gather} \| \partial_x^2(|u|^2)\|_{L^2}+\| |\partial_xu|^2\|_{L^2}+\| T_{\delta}\partial_x(\undefined[\overline{u}\partial_xu])\|_{L^2}\lesssim 1+\| \partial_x^2u\|_{L^2},\\ \| \undefined[\overline{u}\partial_xu]\|_{L^2}+\| T_{\delta}\partial_x(|u|^2)\|_{L^2}\lesssim 1, \intertext{and then, by Young's inequality, that} \Big| \mathcal{H}_2(u)-\int |\partial_x^2u|^2\Big| \leq \frac{1}{2}\int |\partial_x^2u|^2+C_0, \end{gather}\] where \(C_0>0\) and other implicit constants may depend on \(\alpha,\beta,\delta,\rho, \mathcal{E}^1(\phi)\). In particular, we have \[\label{est:equivI2} \int |\partial_x^2u|^2\leq 2\mathcal{H}_2(u)+2C_0,\qquad \mathcal{H}_2(u)\leq \frac{3}{2}\int|\partial_x^2u|^2+C_0.\tag{67}\] The desired a priori bound ?? follows from ?? , ?? , 67 and Gronwall’s inequality.

This is the end of the proof of Theorem 4. ◻

Outline of proof of Corollary 2. In the integrable case of 5 where \(\beta =\pm |\alpha|\), we can further repeat the energy correction process (as in the proof of Theorem 4 (ii)) to obtain an exactly conserved quantity \(\mathcal{H}^{\mathrm{INLS}}_2(u)\), by introducing the following new correctors of order six with two spatial differentials \[\begin{gather} I_2^{(6)}(u):=\int |u|^4|u_x|^2 ,\qquad I_2^{(7)}(u):=\int |u|^2\big[(|u|^2)_x\big]^2 ,\\ I_2^{(8)}(u):=\int |u|^2\undefined[\overline{u}u_x]T_{\delta}\partial_x(|u|^2-\rho^2) ,\qquad I_2^{(9)}(u):=\int \undefined[\overline{u}u_x]T_{\delta}\partial_x(|u|^4-\rho^4) ,\\ I_2^{(10)}(u):=\int |u|^2\big[ T_{\delta}\partial_x(|u|^2-\rho^2)\big]^2, \end{gather}\] the correctors of order eight with one differential \[\begin{gather} I_2^{(11)}(u):=\int (|u|^6-\rho^6)\undefined[\overline{u}u_x],\\ I_2^{(12)}(u):=\int (|u|^6-\rho^6)T_{\delta}\partial_x(|u|^2-\rho^2),\qquad I_2^{(13)}(u):=\int (|u|^4-\rho^4)T_{\delta}\partial_x(|u|^4-\rho^4), \end{gather}\] and finally, the corrector of order ten with no differential \[\begin{align} I_2^{(14)}(u):=D_1\int (|u|^2-\rho^2)^5 +D_2\rho^2\int (|u|^2-\rho^2)^4+D_3\rho^4\int (|u|^2-\rho^2)^3+D_4\rho^6\int (|u|^2-\rho^2)^2\\ (D_1,\dots ,D_4\in \mathbb{R}), \end{align}\] and successively determining the coefficients to cancel the remainder in 64 .

Second, to prove ?? , we recall that in the proof of ?? in Theorem 4, we have already seen that \[\sum_{j=1}^5 |I_2^{(j)}(u)|\lesssim 1+\| \partial_x^2u\|_{L^2}.\] Similarly, using Hölder’s inequality and 13 , we see that \[\sum_{j=6}^{14}|I_2^{(j)}(u)|\lesssim 1.\] Here, the implicit constants depend on \(\alpha,\delta,\rho\) and \(\mathcal{E}^1(u)\). As we did for 67 , by Young’s inequality we deduce \[\Big| \mathcal{H}_2^{\mathrm{INLS}}(u(t))-\int |\partial_x^2u(t)|^2\Big| \leq \frac{1}{2}\int |\partial_x^2u(t)|^2 +C_4\big(\alpha,\delta,\rho,\mathcal{E}^1(u(t))\big)\qquad (t\in [0,T])\] for some constant \(C_4>0\). Then, from the a priori bound ?? for \(\mathcal{E}^1(u(t))\) proven in Theorem 4 (here we use the defocusing assumption \(\beta =|\alpha|>0\)) and conservation of \(\mathcal{H}_2^{\mathrm{INLS}}(u(t))\), we infer that \[\begin{align} \| \partial_x^2u(t)\|_{L^2}^2&\leq 2\mathcal{H}_2^{\mathrm{INLS}}(u(t))+2C_4\big(\alpha,\delta,\rho,\mathcal{E}^1(u(t))\big) \\ &\leq 2\mathcal{H}_2^{\mathrm{INLS}}(\phi)+2C_4\big(\alpha,\delta,\rho,C_1(\alpha,\delta,\rho,\mathcal{E}^1(\phi)\big) \\ &\leq 3\|\partial_x^2\phi\|_{L^2}^2+2C_4\big(\alpha,\delta,\rho,\mathcal{E}^1(\phi)\big)+2C_4\big(\alpha,\delta,\rho,C_1(\alpha,\delta,\rho,\mathcal{E}^1(\phi))\big), \end{align}\] where \(C_1>0\) is the constant in ?? . This completes the proof. ◻

Proof of Lemma 6. Let \(0<\varepsilon <1/2\) be a constant to be taken \(\varepsilon \to 0\). We consider the perimeter of the rectangle \(\{ z \in \mathbb{C} \colon -R \leq \undefined[z] \leq R,~0\leq \undefined[z]\leq 2\pi \}\). Then, an elementary complex analysis involving the residue theorem shows that \[\begin{align} \mathcal{F}\left[\frac{e^{\varepsilon x}}{e^{x}+1}\right](\xi) &= \int_{-\infty}^{\infty} \frac{e^{(\varepsilon -i\xi) x}}{e^{x}+1} \,dx \\ &=2 i \pi\,\mathrm{Res} \left(\frac{e^{(\varepsilon -i\xi)z}}{e^{z}+1}; i\pi \right)+\lim_{R\to \infty}\int_{-R}^{R} \frac{e^{(\varepsilon -i\xi) (x+2 i \pi )}}{e^{x+2i \pi}+1} \,dx \\ &\quad -\lim_{R\to \infty}\int_{0}^{2\pi} \frac{e^{(\varepsilon -i\xi) (R+iy)}}{e^{R+iy}+1} i \,dy+\lim_{R\to \infty}\int_{0}^{2\pi} \frac{e^{(\varepsilon -i\xi) (-R+iy)}}{e^{-R+iy}+1} i\,dy\\ &=-2i \pi e^{\pi(\xi + i\varepsilon)}+e^{2\pi(\xi + i\varepsilon)}\int_{-\infty}^{\infty} \frac{e^{(\varepsilon -i\xi) x}}{e^{x}+1} \,dx \\ &=-2i \pi e^{\pi(\xi + i\varepsilon)}+ e^{2\pi(\xi + i\varepsilon)}\mathcal{F}\left[ \frac{e^{\varepsilon x}}{e^{x}+1}\right](\xi), \end{align}\] which implies that \[\label{2547747194710:53} \mathcal{F}\left[\frac{e^{\varepsilon x}}{e^{x}+1}\right](\xi)=\frac{-2i \pi e^{\pi(\xi + i\varepsilon)}}{1-e^{2\pi(\xi + i\varepsilon)}}=\frac{2i \pi }{e^{\pi(\xi + i\varepsilon)}-e^{-\pi(\xi + i\varepsilon)}}.\tag{68}\] Moreover, by \(\mathcal{F}[f(-x)](\xi) = \mathcal{F}[f](-\xi)\), we see that \[\label{2547747194710:7} \mathcal{F}\left[\frac{e^{-\varepsilon x}}{e^{-x}+1}\right](\xi)= \mathcal{F}\left[\frac{e^{\varepsilon x}}{e^{x}+1}\right](-\xi)= \frac{2i \pi }{e^{-\pi(\xi - i\varepsilon)}-e^{\pi(\xi - i\varepsilon)}}.\tag{69}\] By 68 , 69 , and \[\lim_{\varepsilon \downarrow 0}\left( \frac{e^{\varepsilon x}}{e^{x}+1}-\frac{e^{-\varepsilon x}}{e^{-x}+1}\right) =\frac{1}{e^{x}+1} -\frac{1}{e^{-x}+1}\qquad for all x\in \mathbb{R},\] it suffices for ?? to show that \[\label{2547747194711:25} \begin{align} F_{\varepsilon}(\xi) &:= \frac{1}{2i \pi}\mathcal{F}\left[\frac{e^{\varepsilon x}}{e^{x}+1}- \frac{e^{-\varepsilon x}}{e^{-x}+1}\right](\xi) \\ &\;=\frac{1}{e^{i \pi \varepsilon} e^{\pi \xi }-e^{-i \pi \varepsilon}e^{-\pi \xi}}-\frac{1}{e^{i\pi \varepsilon}e^{-\pi \xi}-e^{-i\pi\varepsilon }e^{\pi \xi}} \\ &\;\to \textrm{p.v.}\operatorname{cosech}{(\pi \xi)}\quad in the tempered distribution sense, as \varepsilon \downarrow 0. \end{align}\tag{70}\] Let \(\phi\) be a Schwartz function on \(\mathbb{R}\), and \(0< r \leq 1\) a constant to be taken \(r\to 0\). Since \(F_{\varepsilon}\) is an odd function, we see that \[\label{2547747194711:39} \int_{\mathbb{R}} F_{\varepsilon}(\xi) \phi(\xi) \,d\xi =\int_{|\xi|>r} F_{\varepsilon}(\xi) \phi(\xi) \,d\xi +\int_{-r}^{r} F_{\varepsilon}(\xi) \left\{ \phi(\xi)-\phi(0)\right\} \,d\xi .\tag{71}\] Note that \(F_{\varepsilon}(\xi)\) converges to \(\operatorname{cosech}{(\pi \xi)}\) for all \(x\in \mathbb{R}\setminus \{ 0\}\) as \(\varepsilon \downarrow 0\). Moreover, we see that \[\label{2547747194711:48} \begin{align} \left| F_{\varepsilon}(\xi) \right| &=\left| \frac{1}{e^{i \pi \varepsilon} e^{\pi \xi }-e^{-i \pi \varepsilon}e^{-\pi \xi}}-\frac{1}{e^{i\pi \varepsilon}e^{-\pi \xi}-e^{- i\pi \varepsilon}e^{\pi \xi}}\right| \\ &=\left| \frac{(e^{i\pi \varepsilon} + e^{-i \pi \varepsilon}) (e^{-\pi \xi} - e^{\pi \xi})}{e^{2i \pi \varepsilon} +e^{-2i \pi \varepsilon} -( e^{2\pi \xi } + e^{-2\pi \xi})}\right| =\left| \frac{2 \cos{(\pi \varepsilon)}(e^{-\pi \xi} - e^{\pi \xi})}{2 \cos{(2 \pi \varepsilon)} - (e^{2\pi \xi } + e^{-2\pi \xi})}\right| \\ &\leq \frac{2 |e^{-\pi \xi} - e^{\pi \xi}|}{e^{2\pi \xi } + e^{-2\pi \xi} - 2}=\frac{2 |e^{-\pi \xi} - e^{\pi \xi}|}{|e^{\pi \xi } - e^{-\pi \xi}|^{2} }=\frac{2 }{|e^{\pi \xi } - e^{-\pi \xi}|}. \end{align}\tag{72}\] By 72 , we see that \(F_{\varepsilon}\) is uniformly bounded with respect to \(\varepsilon\) for \(|\xi|>r\); for instance, we have \[\left| F_{\varepsilon}(\xi) \right| \leq \frac{2}{e^{\pi r } - e^{-\pi r}}.\] Thus, Lebesgue’s dominated convergence theorem shows that \[\label{2547747194712:37} \lim_{\varepsilon \downarrow 0}\int_{|\xi|>r} F_{\varepsilon}(\xi) \phi(\xi) \,d\xi =\int_{|\xi|>r}\operatorname{cosech}{(\pi \xi)}\phi(\xi)\,d\xi.\tag{73}\] Furthermore, by 72 , we see that \[\label{2547747204710:29} \lim_{\varepsilon \downarrow 0}\left| \int_{-r}^{r} F_{\varepsilon}(\xi) \left\{ \phi(\xi) -\phi(0) \right\} d\xi \right| \leq \int_{-r}^{r} \frac{2e^{\pi \xi} |\xi|}{|e^{2\pi \xi} - 1|}\left| \frac{\phi(\xi) -\phi(0)}{\xi} \right| d\xi \lesssim |r| \| \partial_\xi \phi \|_{L^{\infty}},\tag{74}\] where the implicit constant is independent of \(\varepsilon\) and \(r\). Putting 71 , 73 and 74 together, we see that \[\label{2547747194712:55} \begin{align} \lim_{\varepsilon \downarrow 0}\int_{\mathbb{R}} F_{\varepsilon}(\xi) \phi(\xi) \,d\xi &=\lim_{r\downarrow 0} \lim_{\varepsilon \downarrow 0}\bigg\{ \int_{|\xi|>r} F_{\varepsilon}(\xi) \phi(\xi) \,d\xi + \int_{-r}^{r} F_{\varepsilon}(\xi) \left\{ \phi(\xi) -\phi(0) \right\} d\xi \bigg\} \\ &=\lim_{r\downarrow 0}\int_{|\xi|>r}\operatorname{cosech}{(\pi \xi)}\phi(\xi)\,d\xi=\langle \textrm{p.v.}\operatorname{cosech}{(\pi \xi)}, \phi \rangle. \end{align}\tag{75}\] Thus, we have proven 70 and the lemma. ◻

Proof of 8 . It is sufficient to prove that \[\label{254774712477:1} \mathcal{F}[\textrm{p.v.}\coth{(\cdot)}](\xi) = -i \pi \, \textrm{p.v.}\coth{\Big(\frac{\pi \xi}{2}\Big)} .\tag{76}\] Indeed, 76 together with basic properties of Fourier transformations show that for any Schwartz function \(f\) on \(\mathbb{R}\), \[\begin{align} \mathcal{F}[T_{\delta}f](\xi) &= \frac{1}{2\delta} \mathcal{F}\left[ \textrm{p.v.}\coth{\left( \frac{\pi \cdot}{2\delta}\right)}* f\right](\xi)=\frac{1}{2\delta} \mathcal{F}\left[\textrm{p.v.}\coth{\left( \frac{\pi \cdot}{2\delta}\right)}\right](\xi) \mathcal{F}[f](\xi) \\ &=\frac{1}{\pi} \mathcal{F}\left[\textrm{p.v.}\coth{(\cdot)}\right]\Big(\frac{2\delta \xi}{\pi}\Big)\mathcal{F}[f](\xi)=-i\, \textrm{p.v.}\coth{\left(\delta \xi \right)}\mathcal{F}[f](\xi). \end{align}\]

Note that \[\frac{1}{e^{x}+1} - \frac{1}{e^{-x}+1}=\frac{e^{-\frac{x}{2}}-e^{\frac{x}{2}}}{e^{\frac{x}{2}}+e^{-\frac{x}{2}}}=-\tanh{\left( \frac{x}{2}\right)}.\] Hence, by Lemma 6, we see that \[\label{2547747204710:52} \mathcal{F}\left[ \tanh{\left(\frac{x}{2}\right)} \right]\left(\xi \right) =-\mathcal{F}\left[ \frac{1}{e^{x}+1} - \frac{1}{e^{-x}+1}\right]\left( \xi\right)=-2i \pi\, \textrm{p.v.}\operatorname{cosech}{\left( \pi \xi \right)}.\tag{77}\] Furthermore, by 77 , and \(\mathcal{F}^{2}[f](x) = 2\pi f(-x)\), we see that \[\label{2547747204712:1} \begin{align} \mathcal{F}\left[ \textrm{p.v.}\operatorname{cosech}{\left( x \right)}\right](\xi) &=\frac{i}{2}\mathcal{F}\left[ -2 i \pi\, \textrm{p.v.}\operatorname{cosech}{\left(\pi x \right)}\right]\left(\pi \xi \right)\\ &= \frac{i}{2}\mathcal{F}^{2}\left[ \tanh{\left(\frac{x}{2}\right)} \right]\left(\pi \xi\right)=i\pi \tanh{\Big( \frac{-\pi \xi}{2}\Big)}=-i\pi \tanh{\Big( \frac{\pi \xi}{2}\Big)}. \end{align}\tag{78}\] Note that \[\label{2547747204712:19} \begin{align} \coth{(x)} &=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\frac{(e^{\frac{x}{2}} - e^{-\frac{x}{2}})^{2} + 2}{e^{x}-e^{-x}}=\frac{e^{\frac{x}{2}} - e^{-\frac{x}{2}}}{e^{\frac{x}{2}}+ e^{-\frac{x}{2}}}+\frac{2}{ e^{x}- e^{-x}}\\ &=\tanh{\left(\frac{x}{2} \right)} + \operatorname{cosech}{(x)}. \end{align}\tag{79}\] Moreover, we see that \[\label{2547747204713:3} \operatorname{cosech}{\left(x \right)} + \coth{(x)} =\frac{2+ e^{x} + e^{-x}}{e^{x}- e^{-x}}= \frac{(e^{\frac{x}{2}} + e^{-\frac{x}{2}})^{2}}{e^{x}-e^{-x}}=\coth{\left( \frac{x}{2}\right)}.\tag{80}\] Using 79 twice, 77 , 78 , and 80 , we see that \[\label{2547747204712:55} \begin{align} \mathcal{F}\left[\textrm{p.v.}\coth{(x)}\right](\xi) &=\mathcal{F}\left[ \tanh{\left(\frac{x}{2} \right)}\right](\xi) + \mathcal{F}\left[\textrm{p.v.}\operatorname{cosech}{(x)}\right](\xi)\\ &=-2i \pi\, \textrm{p.v.}\operatorname{cosech}{\left( \pi \xi \right)}-i\pi \tanh{\Big( \frac{\pi \xi}{2}\Big)}\\ &=-i \pi\, \textrm{p.v.}\operatorname{cosech}{\left( \pi \xi \right)}-i\pi\,\textrm{p.v.}\coth{(\pi \xi)}\\ &=-i\pi\,\textrm{p.v.}\coth{\Big( \frac{\pi \xi}{2} \Big)}. \end{align}\tag{81}\] Thus, we have proven 76 ; thus 8 is true. ◻

Proof of Lemma 3. (i) The first inequality follows immediately from the definition of \(c_{\delta,j}[f]\). We also see that \(c_{\delta,j\pm 1}[f]\leq 2^{\varepsilon}c_{\delta,j}[f]\), which implies the second inequality. For the last one, we easily see \(\mathcal{E}^2_\delta(f)\lesssim \| \mathbf{c}_\delta[f]\|_{\ell^2}^2\), while the converse is shown by the (discrete) Young inequality.

(ii) Using the inequality \(\big| \| \phi \|_{L^2}^{1/2} -\| \psi \|_{L^2}^{1/2}\big|\leq \| \phi -\psi \|_{L^2}^{1/2}\), we have \[|c_{\delta,j}[f] - c_{\delta,j}[g]| \leq \sum_{k=0}^{\infty} 2^{-\varepsilon|j-k|} \big\| P_{k}\partial_x(f-g)\big\|_{H^1}+2^{-\varepsilon j}\Big( \| f-g\|_{L^\infty} +\delta^{-\frac{2}{3}}\big\| (|f|^2-\rho_f^2)-(|g|^2-\rho_g^2)\big\|_{L^2}^{\frac{1}{2}}\Big) .\] The claim follows by Young’s inequality as before.

(iii) By Bernstein’s inequality, (i), and that \(\varepsilon\in (0,1)\), we see that \[\| P_{\leq j} \partial_x f \|_{H^2} \lesssim \sum_{k=0}^{j} 2^{k} \| P_{k} \partial_{x} f \|_{H^1} \leq \sum_{k=0}^{j} 2^{k} 2^{\varepsilon(j-k)} c_{\delta,j}[f] \lesssim 2^{j} c_{\delta,j}[f].\] On the other hand, we see that \(\| P_{\leq j}f\|_{L^\infty}\leq \| f\|_{L^\infty}\leq 2^{\varepsilon j}c_{\delta,j}[f]\leq 2^jc_{\delta,j}[f]\), and the claim follows from these estimates. Thus, we have completed the proof of Lemma 3. ◻

Proof of 53 . By Bernstein’s inequality, the \(X^3\) bound from ?? , and Lemma 3 (iii), we see that \[\begin{align} \| P_k\partial_x^2(u^j-u^{j-1})\|_{L^\infty_{T_*}L^2}&\lesssim 2^{-k}\big( \| \partial_xu^j\|_{L^\infty_{T_*}H^2}+ \| \partial_xu^{j-1}\|_{L^\infty_{T_*}H^2}\big) \lesssim 2^{-k}\big( \| P_{\leq j}\phi\|_{X^3}+\| P_{\leq j-1}\phi\|_{X^3}\big) \\ &\lesssim 2^{-k+j}c_{\delta,j}[\phi] . \end{align}\] This gives 53 if \(j\leq k\). When \(k<j\), we use the difference bound 52 instead, to obtain \[\begin{align} \| P_k\partial_x^2(u^j-u^{j-1})\|_{L^\infty_TL^2}&\lesssim 2^k\| \partial_x(u^j-u^{j-1})\|_{L^\infty_TL^2}\lesssim_{M_0} 2^k d^1_\delta (P_{\leq j}\phi,P_{\leq j-1}\phi) \\ &\lesssim 2^k\big( \| P_j\partial_x\phi\|_{L^2}+\| P_j\phi\|_{L^\infty}+\delta^{-\frac{4}{3}}\| \phi\|_{L^\infty}\| P_j\phi\|_{L^2}\big) \\ &\lesssim 2^k\big( 2^{-j}+2^{-\frac{3}{2}j}+M_02^{-2j}\big) \| P_j\partial_x\phi\|_{H^1} \lesssim_{M_0} 2^{k-j}c_{\delta,j}[\phi], \end{align}\] as desired. ◻