Cauchy-Schwarz inequality and estimates (7-12), (7-6) and (6-21) we obtain that the term in (7-25) is bounded from above by $\|u\|_{H^s}^6$. Consider the terms in (7-22) and (7-24). Recalling (7-5) and (7-1) we write

$$ \begin{aligned} \mathrm{Re}(T_{(\xi)2n} X_{\mathrm{H}^{(4)}_{\mathrm{NLS}}}^{+\mathrm{res}}(Z), z_n)_{L^2} &= \mathrm{Re}(T_{(\xi)2n} X_{\mathrm{H}^{(4)}_{\mathrm{NLS}}}^{+\mathrm{res}}(Z), T_{(\xi)2n} z)_{L^2} + \mathrm{Re}(T_{(\xi)2n} X_{\mathrm{H}^{(4)}_{\mathrm{NLS}}}^{+\mathrm{res}}(Z), T_{(1+\mathcal{L})^{n-(\xi)2n}} z)_{L^2} \\ &= \mathrm{Re}(T_{(\xi)2n} X_{\mathrm{H}^{(4)}_{\mathrm{NLS}}}^{+\mathrm{res}}(Z), T_{(1+\mathcal{L})^{n-(\xi)2n}} z)_{L^2} \quad \text{(by (7-16))}. \end{aligned} $$

Moreover we write

$$ \mathrm{Re}(B_n^{(2)}(Z), z_n)_{L^2} = \mathrm{Re}(B_n^{(2)}(Z), T_{(\xi)2n} z)_{L^2} + \mathrm{Re}(B_n^{(2)}(Z), T_{(1+\mathcal{L})^{n-(\xi)2n}} z)_{L^2}. $$

Using the bound (7-3) in Lemma 7.1 to estimate the operator $T_{(1+\mathcal{L})^{n-(\xi)2n}}$ and Lemma 3.7 and (7-11) to estimate the operator $B_n^{(2)}(Z)$, we get

$$ |\mathrm{Re}(T_{(\xi)2n} X_{\mathrm{H}^{(4)}_{\mathrm{NLS}}}^{+\mathrm{res}}(Z), T_{(1+\mathcal{L})^{n-(\xi)2n}} z)_{L^2}| + |\mathrm{Re}(B_n^{(2)}(Z), T_{(1+\mathcal{L})^{n-(\xi)2n}} z)_{L^2}| \lesssim \|u\|_{H^s}^{10}, $$

which means that these remainders can be absorbed in the term $\mathcal{B}_{>5}(t)$. Then we set

$$ \mathcal{B}(t) := 2 \mathrm{Re}(B_n^{(2)}(Z), T_{(\xi)2n} z)_{L^2}. $$

Formulas (7-20) follow by an explicit computation using (7-9), (7-11). $\square$

**7A2. Estimates of nonresonant terms.** In this subsection we provide estimates on the term $\mathcal{B}(t)$ appearing in (7-19). We state the main result of this section.

**Proposition 7.6.** Let $N > 0$. Then there is $s_0 = s_0(N_0)$, where $N_0 > 0$ is given by Proposition 2.1, such that, if Hypothesis 6.1 holds with $s \ge s_0$, one has

$$ \left| \int_0^t \mathcal{B}(\sigma) \, d\sigma \right| \lesssim \|u\|_{L^\infty H^s}^{10} TN + \|u\|_{L^\infty H^s}^6 T + \|u\|_{L^\infty H^s}^4 TN^{-1} + \|u\|_{L^\infty H^s}^4, \quad (7-26) $$

where $\mathcal{B}(t)$ is in (7-20).

We need some preliminary results. We consider the trilinear maps

$$ \mathcal{B}_i = \mathcal{B}_i[z_1, z_2, z_3], \quad \hat{\mathcal{B}}_i(\xi) = \frac{1}{(2\pi)^d} \sum_{\eta, \zeta \in \mathbb{Z}^d} b_i(\xi, \eta, \zeta) \hat{z}_1(\xi - \eta - \zeta) \hat{z}_2(\eta) \hat{z}_3(\zeta), \quad i = 1, 2, \quad (7-27) $$

$$ \mathcal{T}_< = \mathcal{T}_<[z_1, z_2, z_3], \quad \hat{\mathcal{T}}_< (\xi) = \frac{1}{(2\pi)^d} \sum_{\eta, \zeta \in \mathbb{Z}^d} t_<(\xi, \eta, \zeta) \hat{z}_1(\xi - \eta - \zeta) \hat{z}_2(\eta) \hat{z}_3(\zeta), \quad (7-28) $$

where

$$ b_1(\xi, \eta, \zeta) = b(\xi, \eta, \zeta) 1_{\{\max(|\xi - \eta - \zeta|, |\eta|, |\zeta|) \le N\}}, \quad (7-29) $$

$$ b_2(\xi, \eta, \zeta) = b(\xi, \eta, \zeta) 1_{\{\max(|\xi - \eta - \zeta|, |\eta|, |\zeta|) > N\}}, \quad (7-30) $$

$$ t_<(\xi, \eta, \zeta) = \frac{-1}{i\omega_{\mathrm{NLS}}(\xi, \eta, \zeta)} b_1(\xi, \eta, \zeta), \quad (7-31) $$

where $b(\xi, \eta, \zeta)$ are the coefficients in (7-20), and $\omega_{\mathrm{NLS}}$ is the phase in (2-1). We remark that if $(\xi, \eta, \zeta) \in \mathcal{R}$ (see Definition 3.9) then the coefficients $b(\xi, \eta, \zeta)$ are equal to zero (see (7-20), (7-9), (7-11)). Therefore, since $\omega_{\mathrm{NLS}}$ is nonresonant (see Proposition 2.1), the coefficients in (7-31) are well-defined. We now prove an abstract results on the trilinear maps introduced in (7-27)–(7-28).
**Lemma 7.7.** One has that, for $s = 2n > \frac{1}{2}d + 4$,

$$ \|\mathcal{B}_2[z_1, z_2, z_3]\|_{L^2} \lesssim N^{-1} \sum_{i=1}^{3} \|z_i\|_{H^s} \prod_{i \neq k} \|z_k\|_{H^{d/2+4+\epsilon}} \quad \text{for all } \epsilon > 0. \quad (7-32) $$

There is $s_0(N_0) > 0$ ($N_0 > 0$ given by Proposition 2.1) such that for $s \ge s_0(N_0)$ one has

$$ \|\mathcal{T}_{<}[z_1, z_2, z_3]\|_{H^p} \lesssim N \sum_{i=1}^{3} \|z_i\|_{H^{s+p-2}} \prod_{i \neq k} \|z_k\|_{H^{s_0}}, \quad p \in \mathbb{N}, \quad (7-33) $$

$$ \|\mathcal{T}_{<}[z_1, z_2, z_3]\|_{L^2} \lesssim \sum_{i=1}^{3} \|z_i\|_{H^s} \prod_{i \neq k} \|z_k\|_{H^{s_0}}. \quad (7-34) $$

*Proof.* Using (7-30), (7-20), (7-11) we get

$$ \begin{aligned} \|\mathcal{B}_2[z_1, z_2, z_3]\|_{L^2}^2 & \lesssim \sum_{\xi \in \mathbb{Z}^d} \left( \sum_{\eta, \zeta \in \mathbb{Z}^d} |\mathfrak{b}_2(\xi, \eta, \zeta)| |\hat{z}_1(\xi - \eta - \zeta)| |\hat{z}_2(\eta)| |\hat{z}_3(\zeta)| \right)^2 \\ & \lesssim N^{-2} \sum_{\xi \in \mathbb{Z}^d} \left( \sum_{\eta, \zeta \in \mathbb{Z}^d} \langle \xi \rangle^{2n} \max_2 \{|\xi - \eta - \zeta|, |\eta|, |\zeta|\}^4 |\hat{z}_1(\xi - \eta - \zeta)| |\hat{z}_2(\eta)| |\hat{z}_3(\zeta)| \right)^2. \end{aligned} $$

Then, by reasoning as in the proof of Lemma 3.7, one obtains (7-32). Let us prove the bound (7-33) for $p=0$; the others are similar. Using (7-31), (2-2), (7-20), (7-11) we have

$$ \begin{aligned} \|\mathcal{T}_{<}[z_1, z_2, z_3]\|_{L^2}^2 & \lesssim \sum_{\xi \in \mathbb{Z}^d} \left( \sum_{\eta, \zeta \in \mathbb{Z}^d} |t_{<}(\xi, \eta, \zeta)| |\hat{z}_1(\xi - \eta - \zeta)| |\hat{z}_2(\eta)| |\hat{z}_3(\zeta)| \right)^2 \\ & \lesssim_N N^2 \sum_{\xi \in \mathbb{Z}^d} \left( \sum_{\eta, \zeta \in \mathbb{Z}^d} \frac{\langle \xi \rangle^{2n} \max_2 \{|\xi - \eta - \zeta|, |\eta|, |\zeta|\}^{N_0+4}}{\max_1 \{|\xi - \eta - \zeta|, |\eta|, |\zeta|\}^2} |\hat{z}_1(\xi - \eta - \zeta)| |\hat{z}_2(\eta)| |\hat{z}_3(\zeta)| \right)^2. \end{aligned} $$

Again, reasoning as in the proof of Lemma 3.7, one obtains (7-33). Inequality (7-34) follows similarly. □

*Proof of Proposition 7.6.* By (7-27), (7-29), (7-30) and recalling the definition of $\mathcal{B}$ in (7-20), we can write

$$ \int_0^t \mathcal{B}(\sigma) \, d\sigma = \int_0^t (\mathcal{B}_1[z, \bar{z}, z], T_{\langle\xi\rangle^{2n}z})_{L^2} \, d\sigma + \int_0^t (\mathcal{B}_2[z, \bar{z}, z], T_{\langle\xi\rangle^{2n}z})_{L^2} \, d\sigma. \quad (7-35) $$

By Lemma 7.7 we have

$$ \left| \int_0^t (\mathcal{B}_2[z, \bar{z}, z], T_{\langle\xi\rangle^{2n}z})_{L^2} d\sigma \right| \stackrel{(7-32)}{\lesssim} N^{-1} \int_0^t \|z\|_{H^s}^4 d\sigma \stackrel{(6-21)}{\lesssim} N^{-1} \int_0^t \|u\|_{H^s}^4 d\sigma. \quad (7-36) $$

Consider now the first summand in the right-hand side of (7-35). We claim that we have the identity

$$ \begin{aligned} \int_0^t (\mathcal{B}_1[z, \bar{z}, z], T_{\langle\xi\rangle^{2n}z})_{L^2} d\sigma = & \int_0^t (\mathcal{T}_{<}[z, \bar{z}, z], T_{\langle\xi\rangle^{2n}(\partial_t + i\Lambda_{\text{NLS}})z})_{L^2} d\sigma \\ & + \int_0^t (\mathcal{T}_{<}[(\partial_t + i\Lambda_{\text{NLS}})z, \bar{z}, z], T_{\langle\xi\rangle^{2n}z})_{L^2} d\sigma \\ & + \int_0^t (\mathcal{T}_{<}[z, \bar{z}, (\partial_t + i\Lambda_{\text{NLS}})z], T_{\langle\xi\rangle^{2n}z})_{L^2} d\sigma \\ & + \int_0^t (\mathcal{T}_{<}[z, \overline{(\partial_t + i\Lambda_{\text{NLS}})z}, z], T_{\langle\xi\rangle^{2n}z})_{L^2} d\sigma + O(\|u\|_{H^s}^4). \end{aligned} \quad (7-37) $$
We use the claim, postponing its proof. Consider the first summand in the right-hand side of (7-37). Using the self-adjointness of $T_{\langle \xi \rangle}^2$ and (7-7) we write

$$ (\mathcal{T}_{<}[z, \bar{z}, z], T_{\langle \xi \rangle}^{2n}(\partial_t + i\Lambda_{\text{NLS}})z)_{L^2} = (T_{\langle \xi \rangle}^2 \mathcal{T}_{<}[z, \bar{z}, z], -T_{\langle \xi \rangle}^{2n-2}iT_{\Sigma}z)_{L^2} + (\mathcal{T}_{<}[z, \bar{z}, z], T_{\langle \xi \rangle}^{2n}(X_{H_{\text{NLS}}^{(4)}}^{+}(Z) + R_{5}^{(2,+)}(U)))_{L^2}. $$

We estimate the first summand in the right-hand side by means of the Cauchy–Schwarz inequality, (7-33) with $p = 2$ and (7-3); analogously we estimate the second summand by means of the Cauchy–Schwarz inequality, (7-34), (6-23) and (6-24), obtaining

$$ \left| \int_0^t (\mathcal{T}_{<}[z, \bar{z}, z], T_{\langle \xi \rangle}^{2n}(\partial_t + i\Lambda_{\text{NLS}})z)_{L^2} d\sigma \right| \le \int_0^t \|u(\sigma)\|_{H^s}^{10} N + \|u(\sigma)\|_{H^s}^6 d\sigma. $$

The other terms in (7-37) are estimated in a similar way. We eventually obtain (7-26).

We now prove the claim (7-37). Recalling (7-7) we have

$$ \partial_t \hat{z}(\xi) = -i\Lambda_{\text{NLS}}(\xi)\hat{z}(\xi) + \hat{\mathcal{D}}(\xi), \quad \xi \in \mathbb{Z}^d, \quad \mathcal{D} := -iT_{\Sigma}z + X_{H_{\text{NLS}}^{(4)}}^{+}(Z) + R_{5}^{(2,+)}(U). $$

We define $\hat{g}(\xi) := e^{it\Lambda_{\text{NLS}}(\xi)}\hat{z}(\xi)$ for all $\xi \in \mathbb{Z}^d$. One can note that $\hat{g}(\xi)$ satisfies

$$ \partial_t \hat{g}(\xi) = e^{it\Lambda_{\text{NLS}}(\xi)} \hat{\mathcal{D}}(\xi) = e^{it\Lambda_{\text{NLS}}(\xi)} (\partial_t + i\Lambda_{\text{NLS}}) \hat{z}(\xi) \quad \text{for all } \xi \in \mathbb{Z}^d. \quad (7-38) $$

According to this notation and using (7-27) and (2-1) we have

$$ \int_0^t (\mathcal{B}_1[z, \bar{z}, z], T_{\langle \xi \rangle}^{2n} z)_{L^2} d\sigma = \int_0^t \sum_{\xi, \eta, \zeta \in \mathbb{Z}^d} \frac{1}{(2\pi)^d} b_1(\xi, \eta, \zeta) e^{-i\sigma\omega_{\text{NLS}}(\xi, \eta, \zeta)} \hat{g}(\xi - \eta - \zeta) \hat{\bar{g}}(\eta) \hat{g}(\zeta) \hat{\bar{g}}(-\xi) \langle \xi \rangle^{2n} d\sigma. $$

By integrating by parts in $\sigma$ and using (7-38) one gets (7-37) with

$$ O(\|u\|_{H^s}^4) = (\mathcal{T}_{<}[z(t), \bar{z}(t), z(t)], T_{\langle \xi \rangle}^{2n} z(t))_{L^2} - (\mathcal{T}_{<}[z(0), \bar{z}(0), z(0)], T_{\langle \xi \rangle}^{2n} z(0))_{L^2}. $$

The remainder above is bounded from above by $\|u\|_{L^\infty H^s}$ using Cauchy–Schwarz and (7-34). $\square$

**7B. Estimates for the KG.** In this section we provide a priori energy estimates on the variable Z solving (6-48). This implies similar estimates on the solution U of the system (4-44) thanks to the equivalence (6-47). In Section 7B1 we introduce an equivalent energy norm and we provide a first energy inequality. This is the content of Proposition 7.10. Then in Section 7B2 we give improved bounds on the nonresonant terms.

**7B1. First energy inequality.** We recall that the system (6-48) is diagonal up to smoothing terms plus some higher degree of homogeneity remainder. Hence, for simplicity, we pass to the scalar equation

$$ \partial_t z + i\Lambda_{\text{KG}}z = -i \text{Op}^{\text{BW}}(\tilde{a}_2^+(x, \xi) \Lambda_{\text{KG}}(\xi))z + X_{H_{\text{KG}}^{(4)}}^+(Z) + R_4^{(2,+)}(u), \quad (7-39) $$
where (recall (6-49)) $X_{H_{\text{KG}}^{(4)}}^{+}(Z) = -i \text{Op}^{\text{BW}}(a_0(x, \xi))z + Q_{H_{\text{KG}}^{(4)}}^{+}(Z)$. For $n \in \mathbb{R}$ we define

$$z_n := \langle D \rangle^n z, \quad Z_n = \begin{bmatrix} z_n \\ \bar{z}_n \end{bmatrix} := \mathbb{1} \langle D \rangle^n Z, \quad Z = \begin{bmatrix} z \\ \bar{z} \end{bmatrix}. \qquad (7-40)$$

**Lemma 7.8.** Fix $n := n(d) \gg 1$ large enough and recall (7-39). One has that the function $z_n$ defined in (7-40) solves the problem

$$\partial_t z_n = -i \text{Op}^{\text{BW}}((1 + \tilde{a}_2^+(x, \xi))\Lambda_{\text{KG}}(\xi))z_n + \langle D \rangle^n X_{H_{\text{KG}}^{(4)}}^{+\text{res}}(Z) + B_n^{(1)}(Z) + B_n^{(2)}(Z) + R_{4,n}(U), \quad (7-41)$$

where the resonant vector field $X_{H_{\text{KG}}^{(4)}}^{+\text{res}}$ is defined as in Definition 3.9 (see also Remark 3.11), the cubic terms $B_n^{(i)}$, $i = 1, 2$, have the form

$$\widehat{B_n^{(1)}}(Z)(\xi) = \frac{1}{(2\pi)^d} \sum_{\substack{\sigma_1, \sigma_2 \in \{\pm\} \\ \eta, \zeta \in \mathbb{Z}^d}} b_1^{\sigma_1, \sigma_2}(\xi, \eta, \zeta) \widehat{z}^{\sigma_1}(\xi - \eta - \zeta) \widehat{z}^{\sigma_2}(\eta) \hat{z}_n(\zeta), \quad (7-42)$$

$$\widehat{B_n^{(2)}}(Z)(\xi) = \frac{1}{(2\pi)^d} \sum_{\substack{\sigma_1, \sigma_2, \sigma_3 \in \{\pm\} \\ \eta, \zeta \in \mathbb{Z}^d}} b_{2,n}^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta) \widehat{z}^{\sigma_1}(\xi - \eta - \zeta) \widehat{z}^{\sigma_2}(\eta) \widehat{z}^{\sigma_3}(\zeta), \quad (7-43)$$

with (recall Remark 4.5)

$$b_1^{\sigma_1, \sigma_2}(\xi, \eta, \zeta) := -i a_0^{\sigma_1, \sigma_2} \left( \xi - \zeta, \eta, \frac{\xi + \zeta}{2} \right) \chi_{\epsilon} \left( \frac{|\xi - \zeta|}{|\xi + \zeta|} \right) 1_{\mathcal{R}^c}(\xi, \eta, \zeta), \quad (7-44)$$

$$|b_{2,n}^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta)| \lesssim \frac{\langle \xi \rangle^n \max_2 \{|\xi - \eta - \zeta|, |\eta|, |\zeta|\}^{\mu}}{\max_1 \{\langle \xi - \eta - \zeta \rangle, \langle \eta \rangle, \langle \zeta \rangle\}} 1_{\mathcal{R}^c}(\xi, \eta, \zeta) \quad (7-45)$$

for some $\mu > 1$. The remainder satisfies

$$\| R_{4,n}(U) \|_{L^2} \lesssim \| u \|_{H^n}^4. \qquad (7-46)$$

*Proof.* Recalling the definition of resonant vector fields in Definition 3.9 we set

$$X_{H_{\text{KG}}^{(4)}}^{+,-}(Z) := X_{H_{\text{KG}}^{(4)}}^{+}(Z) - X_{H_{\text{KG}}^{(4)}}^{+\text{res}}(Z), \quad (7-47)$$

which represents the nonresonant terms in the cubic vector field of (7-39). By differentiating in $t$ (7-40) and using (7-39) we get

$$\partial_t z_n = -i \text{Op}^{\text{BW}}((1 + \tilde{a}_2^+(x, \xi))\Lambda_{\text{KG}}(\xi))z_n + \langle D \rangle^n X_{H_{\text{KG}}^{(4)}}^{+\text{res}}(Z) \\ - i[\langle D \rangle^n, \text{Op}^{\text{BW}}((1 + \tilde{a}_2^+(x, \xi))\Lambda_{\text{KG}}(\xi))]z \quad (7-48)$$

$$+ \langle D \rangle^n X_{H_{\text{KG}}^{(4)}}^{+,-}(Z) \qquad (7-49)$$

$$+ \langle D \rangle^n R_4^{(2,+)}(u). \qquad (7-50)$$

We analyze each summand above separately. By estimate (6-50) we deduce $\|(7-50)\|_{L^2} \lesssim \|u\|_{H^n}^4$. Let us now consider the commutator term in (7-48). By Lemma 3.1, Proposition 3.2 and the estimate on the seminorm of the symbol $\tilde{a}_2^+(x, \xi)$ in (6-38), we obtain that $\|(7-48)\|_{L^2} \lesssim \|u\|_{H^n}^3 \|z\|_{H^n} \lesssim \|u\|_{H^n}^4$; we have
used also (6-47). The term in (7-49) is the most delicate. By (6-49) and (7-47) (recall also Remark 4.5 and (3-6))

$$ \langle D \rangle^n X_{H_{\text{KG}}^{(4)}}^{+,-}(Z) = B_n^{(1)}(Z) + \mathcal{C}_1 + \mathcal{C}_2, \quad (7-51) $$

with $B_n^{(1)}(Z)$ as in (7-42) and coefficients as in (7-44), the term $\mathcal{C}_1$ has the form

$$ \hat{\mathcal{C}}_1(\xi) = \frac{1}{(2\pi)^d} \sum_{\substack{\sigma_1, \sigma_2 \in \{\pm\} \\ \eta, \zeta \in \mathbb{Z}^d}} c_1^{\sigma_1, \sigma_2}(\xi, \eta, \zeta) \hat{z}^{\sigma_1}(\xi - \eta - \zeta) \hat{z}^{\sigma_2}(\eta) \hat{z}(\zeta), \quad (7-52) $$

$$ c_1^{\sigma_1, \sigma_2}(\xi, \eta, \zeta) = -i a_0^{\sigma_1, \sigma_2} \left( \xi - \zeta, \eta, \frac{\xi + \zeta}{2} \right) \chi_{\epsilon} \left( \frac{|\xi - \zeta|}{|\xi + \zeta|} \right) [\langle \xi \rangle^n - \langle \zeta \rangle^n] 1_{\mathcal{R}^c}(\xi, \eta, \zeta), $$

and the term $\mathcal{C}_2$ has the form (7-43) with coefficients (see (5-30))

$$ c_2^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta) := q_{H_{\text{KG}}^{(4)}}^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta) (\xi)^n 1_{\mathcal{R}^c}(\xi, \eta, \zeta). \quad (7-53) $$

In order to conclude the proof we need to show that the coefficients in (7-52), (7-53) satisfy the bound (7-45). This is true for the coefficients in (7-53) thanks to the bound (5-31). Moreover notice that

$$ |\langle \xi \rangle^n - \langle \zeta \rangle^n| \lesssim |\xi - \zeta| \max\{\langle \xi \rangle, \langle \zeta \rangle\}^{n-1}. $$

Then the coefficients in (7-52) satisfy (7-45) by using Remark 4.5 and Lemma 3.8. $\square$

**Remark 7.9.** In view of Remarks 4.6, 4.8, 6.14 if (KG) is semilinear then the symbol $\tilde{a}_2^+$ in (7-41) is equal to zero and the coefficients $b_{2,n}^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta)$ in (7-43) satisfy the bound (7-45) with the better denominator $\max_1\{\langle \xi - \eta - \zeta \rangle, \langle \eta \rangle, \langle \zeta \rangle\}^2$.

In view of Lemma 7.8 we deduce the following.

**Proposition 7.10.** Let $\psi(t, x)$ be a solution of (KG) satisfying Hypothesis 6.8 and consider the function $z_n$ in (7-40) (see also (6-46), (6-42)). Then, setting $s=n=n(d) \gg 1$ we have $\|z_n\|_{L^2} \sim \|\psi\|_{H^{s+1/2}} + \|\dot{\psi}\|_{H^{s-1/2}}$ and

$$ \partial_t \|z_n(t)\|_{L^2}^2 = \mathcal{B}(t) + \mathcal{B}_{>4}(t), \quad t \in [0, T), \quad (7-54) $$

where:

* The term $\mathcal{B}(t)$ has the form

$$ \mathcal{B}(t) = \sum_{\substack{\sigma_1, \sigma_2, \sigma_3 \in \{\pm\} \\ \xi, \eta, \zeta \in \mathbb{Z}^d}} \langle \xi \rangle^{2n} b^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta) \hat{z}^{\sigma_1}(\xi - \eta - \zeta) \hat{z}^{\sigma_2}(\eta) \hat{z}^{\sigma_3}(\zeta) \hat{z}(-\xi), \quad (7-55) $$

where $b^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta) \in \mathbb{C}$ satisfy, for $\xi, \eta, \zeta \in \mathbb{Z}^d$,

$$ |b^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta)| \lesssim \frac{\max_2\{|\xi - \eta - \zeta|, |\eta|, |\zeta|\}^{\mu}}{\max_1\{\langle \xi - \eta - \zeta \rangle, \langle \eta \rangle, \langle \zeta \rangle\}} 1_{\mathcal{R}^c}(\xi, \eta, \zeta) \quad (7-56) $$

for some $\mu > 1$.

* The term $\mathcal{B}_{>5}(t)$ satisfies

$$ |\mathcal{B}_{>4}(t)| \lesssim \|u\|_{H^s}^5, \quad t \in [0, T). \quad (7-57) $$
*Proof.* The equivalence between $\|z_n\|_{L^2}$ and $\|\psi\|_{H^{s+1/2}} + \|\dot{\psi}\|_{H^{s-1/2}}$ follows by Remarks 6.12 and 6.9. By using (7-41) we get

$$ \frac{1}{2} \partial_t \|z_n(t)\|_{L^2}^2 = \mathrm{Re}(-i \mathrm{Op}^{\mathrm{BW}}((1 + \tilde{a}_2^+(x, \xi))\Lambda_{\mathrm{KG}}(\xi))z_n, z_n)_{L^2} \quad (7-58) $$

$$ + \mathrm{Re}(\langle D \rangle^n X_{H_{\mathrm{KG}}^{(4)}}^{+,{\mathrm{res}}}(Z), z_n)_{L^2} \quad (7-59) $$

$$ + \mathrm{Re}(B_n^{(1)}(Z), z_n)_{L^2} \quad (7-60) $$

$$ + \mathrm{Re}(B_n^{(2)}(Z), z_n)_{L^2} \quad (7-61) $$

$$ + \mathrm{Re}(R_{4,n}(Z), z_n)_{L^2}. \quad (7-62) $$

By (6-34), (6-33) and (4-24) we have that the symbol $(1 + \tilde{a}_2^+(x, \xi))\Lambda_{\mathrm{KG}}(\xi)$ is real-valued. Hence the operator $i \mathrm{Op}^{\mathrm{BW}}((1 + \tilde{a}_2^+(x, \xi))\Lambda_{\mathrm{KG}}(\xi))$ is skew-self-adjoint. We deduce (7-58) $\equiv 0$. By Lemma 3.10 (see also Remark 3.11) we have $(7-59) \equiv 0$. We also have $(7-60) \equiv 0$; to see this one can reason as done in the proof of Lemma 7.4, by using Remark 4.5, in particular (4-43). By formula (7-43) and estimates (7-45) we have that the term in (7-61) has the form (7-55) with coefficients satisfying (7-56). By the Cauchy-Schwarz inequality and estimate (7-46) we get that the term in (7-62) satisfies the bound (7-57). $\square$

**Remark 7.11.** In view of Remark 7.9, if (KG) is semilinear, then the coefficients $b^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta)$ of the energy in (7-55) satisfy the bound (7-56) with the better denominator $\max_1\{\langle\xi - \eta - \zeta\rangle, \langle\eta\rangle, \langle\zeta\rangle\}^2$.

**7B2. Estimates of nonresonant terms.** In Proposition 7.10 we provided a precise structure of the term $\mathcal{B}(t)$ of degree 4 in (7-54). In this section we show that, actually, $\mathcal{B}(t)$ satisfies better bounds with respect to a general quartic multilinear map by using that it is *nonresonant*. We state the main result of this section.

**Proposition 7.12.** Let $N > 0$ and let $\beta$ be as in Proposition 2.2. Then there is $s_0 = s_0(N_0)$, where $N_0 > 0$ is given by Proposition 2.2, such that, if Hypothesis 6.8 holds with $s \ge s_0$, one has

$$ \left| \int_0^t \mathcal{B}(\sigma) d\sigma \right| \lesssim \|u\|_{L^\infty H^s}^6 T N^{\beta-1} + \|u\|_{L^\infty H^s}^7 N^\beta T + \|u\|_{L^\infty H^s}^4 T N^{-1} + N^{\beta-1} \|u\|_{L^\infty H^s}^4, \quad (7-63) $$

where $\mathcal{B}(t)$ is in (7-55).

We first introduce some notation. Let $\vec{\sigma} := (\sigma_1, \sigma_2, \sigma_3) \in \{\pm\}^3$ and consider the trilinear maps

$$ \mathcal{B}_i^{\vec{\sigma}} = \mathcal{B}_i^{\vec{\sigma}}[z_1, z_2, z_3], \quad \hat{\mathcal{B}}_i^{\vec{\sigma}}(\xi) = \frac{1}{(2\pi)^d} \sum_{\eta, \zeta \in \mathbb{Z}^d} \langle \xi \rangle^s \mathcal{B}_i^{\vec{\sigma}}(\xi, \eta, \zeta) \hat{z}_{1}^{\sigma_1}(\xi - \eta - \zeta) \hat{z}_{2}^{\sigma_2}(\eta) \hat{z}_{3}^{\sigma_3}(\zeta), \quad (7-64) $$

$$ \mathcal{T}_<^{\vec{\sigma}} = \mathcal{T}_<^{\vec{\sigma}}[z_1, z_2, z_3], \quad \hat{\mathcal{T}}_<^{\vec{\sigma}}(\xi) = \frac{1}{(2\pi)^d} \sum_{\eta, \zeta \in \mathbb{Z}^d} \langle \xi \rangle^s t_{<}^{\vec{\sigma}}(\xi, \eta, \zeta) \hat{z}_1^{\sigma_1}(\xi - \eta - \zeta) \hat{z}_2^{\sigma_2}(\eta) \hat{z}_3^{\sigma_3}(\zeta), \quad (7-65) $$

where

$$ \mathbf{b}_1^{\vec{\sigma}}(\xi, \eta, \zeta) = \mathbf{b}^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta) 1_{\{\max(|\xi - \eta - \zeta|, |\eta|, |\zeta|) \le N\}}, \quad (7-66) $$

$$ \mathbf{b}_2^{\vec{\sigma}}(\xi, \eta, \zeta) = \mathbf{b}^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta) 1_{\{\max(|\xi - \eta - \zeta|, |\eta|, |\zeta|) > N\}}, \quad (7-67) $$

$$ t_{<}^{\vec{\sigma}}(\xi, \eta, \zeta) = \frac{-1}{i \omega_{\mathrm{KG}}^{\vec{\sigma}}(\xi, \eta, \zeta)} \mathbf{b}_1^{\vec{\sigma}}(\xi, \eta, \zeta), \quad (7-68) $$
where $b^{\sigma_1, \sigma_2, \sigma_3}(\xi, \eta, \zeta)$ are the coefficients in (7-55), and $\omega_{\text{KG}}^{\vec{\sigma}}$ is the phase in (2-4). We remark that if $(\xi, \eta, \zeta) \in \mathbb{R}$ (see Definition 3.9) then the coefficients $b(\xi, \eta, \zeta)$ are equal to zero (see (7-55), (7-43), (7-45)). Therefore, since $\omega_{\text{KG}}^{\vec{\sigma}}$ is nonresonant (see Proposition 2.2), the coefficients in (7-68) are well-defined. We now state an abstract result on the trilinear maps introduced in (7-64)–(7-65).

**Lemma 7.13.** Let $\mu > 1$ as in (7-56). One has that, for $s > \frac{1}{2}d + \mu$,

$$ \|\mathcal{B}_2^{\vec{\sigma}}[z_1, z_2, z_3]\|_{L^2} \lesssim N^{-1} \sum_{i=1}^{3} \|z_i\|_{H^s} \prod_{i \neq k} \|z_k\|_{H^{d/2+\mu+\epsilon}} \quad (7-69) $$

for any $\vec{\sigma} \in \{\pm\}^3$ and any $\epsilon > 0$. There is $s_0(N_0) > 0$ ($N_0 > 0$ given by Proposition 2.2) such that for $s \ge s_0(N_0)$ one has

$$ \|\mathcal{T}_{<}[z_1, z_2, z_3]\|_{H^p} \lesssim N^{\beta} \sum_{i=1}^{3} \|z_i\|_{H^{s+p-1}} \prod_{i \neq k} \|z_k\|_{H^{s_0}}, \quad p \in \mathbb{N}, \quad (7-70) $$

$$ \|\mathcal{T}_{<}[z_1, z_2, z_3]\|_{L^2} \lesssim N^{\beta-1} \sum_{i=1}^{3} \|z_i\|_{H^s} \prod_{i \neq k} \|z_k\|_{H^{s_0}}, \quad (7-71) $$

where $\beta$ is defined in Proposition 2.2.

*Proof.* The proof is similar to that of Lemma 7.7. One has to use Proposition 2.2 instead of Proposition 2.1 to estimate the small divisors. $\square$

**Remark 7.14.** In view of Remark 7.11, if (KG) is semilinear we may improve (7-69) and (7-71) with

$$ \|\mathcal{B}_2^{\vec{\sigma}}[z_1, z_2, z_3]\|_{L^2} \lesssim N^{-2} \sum_{i=1}^{3} \|z_i\|_{H^s} \prod_{i \neq k} \|z_k\|_{H^{d/2+\mu+\epsilon}}, $$

$$ \|\mathcal{T}_{<}[z_1, z_2, z_3]\|_{L^2} \lesssim N^{\beta-2} \sum_{i=1}^{3} \|z_i\|_{H^s} \prod_{i \neq k} \|z_k\|_{H^{s_0}}. $$

We are now in a position to prove Proposition 7.12.

*Proof of Proposition 7.12.* By (7-64), (7-66), (7-67), and recalling the definition of $\mathcal{B}$ in (7-55), we can write

$$ \int_0^t \mathcal{B}(\tau) d\tau = \sum_{\vec{\sigma} \in \{\pm\}^3} \int_0^t (\mathcal{B}_1^{\vec{\sigma}}[z, z, z], \langle D \rangle^s z)_{L^2} d\tau + \sum_{\vec{\sigma} \in \{\pm\}^3} \int_0^t (\mathcal{B}_2^{\vec{\sigma}}[z, z, z], \langle D \rangle^s z)_{L^2} d\tau. \quad (7-72) $$

By Lemma 7.13 we have

$$ \left| \int_0^t (\mathcal{B}_2^{\vec{\sigma}}[z, z, z], \langle D \rangle^s z)_{L^2} d\sigma \right| \stackrel{(7-69)}{\lesssim} N^{-1} \int_0^t \|z\|_{H^s}^4 d\tau \stackrel{(6-47)}{\lesssim} N^{-1} \int_0^t \|u\|_{H^s}^4 d\tau. \quad (7-73) $$
Consider now the first summand in the right-hand side of (7-72). Integrating by parts as done in the proof of Proposition 7.6 we have

$$
\begin{align}
\int_0^t (\mathcal{B}_1^{\bar{\sigma}}[z, \bar{z}, z], \langle D \rangle^s z)_{L^2} d\tau &= \int_0^t (\mathcal{T}_<^{\bar{\sigma}}[z, z, z], \langle D \rangle^s (\partial_t + i\Lambda_{\mathrm{KG}})z)_{L^2} d\tau \nonumber \\
&\quad + \int_0^t (\mathcal{T}_<^{\bar{\sigma}}[(\partial_t + i\Lambda_{\mathrm{KG}})z, z, z], \langle D \rangle^s z)_{L^2} d\tau \nonumber \\
&\quad + \int_0^t (\mathcal{T}_<^{\bar{\sigma}}[z, \bar{z}, (\partial_t + i\Lambda_{\mathrm{KG}})z], \langle D \rangle^s z)_{L^2} d\tau \nonumber \\
&\quad + \int_0^t (\mathcal{T}_<^{\bar{\sigma}}[z, (\partial_t + i\Lambda_{\mathrm{KG}})z, z], \langle D \rangle^s z)_{L^2} d\tau + R, \tag{7-74}
\end{align}
$$

where

$$
R = (\mathcal{T}_{<}^{\bar{\sigma}}[z(t), z(t), z(t)], \langle D \rangle^s z(t))_{L^2} - (\mathcal{T}_{<}^{\bar{\sigma}}[z(0), z(0), z(0)], \langle D \rangle^s z(0))_{L^2}.
$$

The remainder $R$ above is bounded from above by $N^\beta \|u\|_{L^\infty H^s}$ using Cauchy–Schwarz and (7-70). Let us now consider the first summand in the right-hand side of (7-74). Using that the operator $\langle D \rangle$ is self-adjoint and recalling (7-39) we have

$$
\begin{align}
(\mathcal{T}_{<}^{\bar{\sigma}}[z, z, z], \langle D \rangle^s (\partial_t + i\Lambda_{\mathrm{KG}})z)_{L^2} &= ((D) \mathcal{T}_{<}^{\bar{\sigma}}[z, z, z], \langle D \rangle^{s-1} (\partial_t + i\Lambda_{\mathrm{KG}})z)_{L^2} \nonumber \\
&= ((D) \mathcal{T}_{<}^{\bar{\sigma}}[z, z, z], \langle D \rangle^{s-1} \mathrm{Op}^{\mathrm{BW}}(-i\tilde{a}_2^+(x, \xi) \Lambda_{\mathrm{KG}}(\xi))z)_{L^2} \tag{7-75} \\
&\quad + (\mathcal{T}_{<}^{\bar{\sigma}}[z, z, z], \langle D \rangle^s (X_{H_{\mathrm{KG}}^{(4)}}^+(Z) + R_4^{(2,+)}(u)))_{L^2}. \tag{7-76}
\end{align}
$$

By the Cauchy–Schwarz inequality, estimate (7-70) with $p=1$, estimate (6-38) on the seminorm of the symbol $\tilde{a}_2^+(x, \xi)$, Lemma 3.1 and the equivalence (6-47), we get $|(7-75)| \lesssim \|u\|_{H^s}^7 N^\beta$. Consider the term in (7-76). First of all notice that, by (4-31) and Lemma 3.1, and by (5-31) and Lemma 3.7, the field $X_{H_{\mathrm{KG}}^{(4)}}(Z)$ in (6-49) satisfies the same estimates (4-32) as the field $X_{H_{\mathrm{KG}}^{(4)}}$. Therefore, using (7-71) and (6-50), we obtain $|(7-76)| \lesssim \|u\|_{H^s}^6 N^{\beta-1}$. Using that (see Hypothesis 6.8) $\|u\|_{H^s} \ll 1$, we conclude that the first summand in the right-hand side of (7-74) is bounded from above by $N^\beta \int_0^t \|u(\tau)\|^7 d\tau + N^{\beta-1} \int_0^t \|u(\tau)\|^6 d\tau$. The other terms in (7-74) are estimated in a similar way. We eventually obtain (7-63). $\square$

**Remark 7.15.** In view of Remarks 4.6, 4.8, 6.14, 7.9, 7.11 and 7.14, if (KG) is semilinear we have the better (with respect to (7-63)) estimate

$$
\left| \int_0^t \mathcal{B}(\sigma) \, d\sigma \right| \lesssim \|u\|_{L^\infty H^s}^6 T N^{\beta-2} + \|u\|_{L^\infty H^s}^4 T N^{-2} + N^{\beta-2} \|u\|_{L^\infty H^s}^4. \quad (7-77)
$$

8. Proof of the main results

In this section we conclude the proof of our main theorems.

**Proof of Theorem 1.** Consider (NLS) and let $u_0$ be as in the statement of Theorem 1. By the result in [Feola and Iandoli 2022] we have that there is $T > 0$ and a unique solution $u(t, x)$ of (NLS) with $V \equiv 0$
such that Hypothesis 6.1 is satisfied. To recover the result when $V \neq 0$ one can argue as done in [Feola and Iandoli 2019]. Consider a potential $V$ as in (1-5), with $\vec{x} \in \partial \mathcal{N}$, where $\mathcal{N}$ is the zero measure set given in Proposition 2.1. We claim that we have the following a priori estimate: Fix any $0 < N$. Then for any $t \in [0, T)$, with $T$ as in Hypothesis 6.1, one has

$$ \|u(t)\|_{H^s}^2 \le 2\|u_0\|_{H^s}^2 + C(\|u\|_{L^\infty H^s}^{10} TN + \|u\|_{L^\infty H^s}^6 T + \|u\|_{L^\infty H^s}^4 TN^{-1} + \|u\|_{L^\infty H^s}^4) \quad (8-1) $$

for some $C > 0$ depending on $s$. To prove the claim we reason as follows. By Proposition 4.2 we have that (NLS) is equivalent to the system (4-12). By Propositions 6.4, 6.6 and Lemma 7.3 we can construct a function $z_n$ with $2n = s$ such that if $u(t, x)$ solves the (NLS) then $z_n$ solves (7-8). Moreover by Proposition 7.5 we have the equivalence (7-18), and we deduce

$$ \|u(t)\|_{H^s}^2 \le 2^{1/2} \|z_n(t)\|_{L^2}^2 \le 2 \|u_0\|_{H^s}^2 + 2 \left| \int_0^t \mathcal{B}(\sigma) \, d\sigma \right| + 2 \left| \int_0^t \mathcal{B}_{>5}(\sigma) \, d\sigma \right|. \quad (8-2) $$

Propositions 7.5 and 7.6 apply; therefore, by (7-26) and (7-21), we obtain (8-1). The thesis of Theorem 1 follows from the following lemma.

**Lemma 8.1 (main bootstrap).** Let $u(t, x)$ be a solution of (NLS) with $t \in [0, T)$ and initial condition $u_0 \in H^s(\mathbb{T}^d; \mathcal{C})$. Then, for $s \gg 1$ large enough, there exist $\epsilon_0, c_0 > 0$ such that, for any $0 < \epsilon \le \epsilon_0$, if

$$ \|u_0\|_{H^s} \le \frac{1}{4}\epsilon, \quad \sup_{t \in [0,T)} \|u(t)\|_{H^s} \le \epsilon, \quad T \le c_0\epsilon^{-4}, \quad (8-3) $$

then we have the improved bound $\sup_{t \in [0,T)} \|u(t)\|_{H^s} \le \frac{1}{2}\epsilon$.

*Proof.* For $\epsilon$ small enough the bound (8-1) holds true, and we fix $N := \epsilon^{-3}$. Therefore, there is $C = C(s) > 0$ such that, for any $t \in [0, T)$,

$$ \begin{aligned} \|u(t)\|_{H^s}^2 &\le 2\|u_0\|_{H^s}^2 + C(\|u\|_{L^\infty H^s}^4 + \|u\|_{L^\infty H^s}^{10} T \epsilon^{-3} + \|u\|_{L^\infty H^s}^6 T + \|u\|_{L^\infty H^s}^4 T \epsilon^3) \\ &\le \frac{1}{8}\epsilon^2 + C(\epsilon^4 + 2\epsilon^7 T + \epsilon^6 T) \quad (\text{by (8-3)}) \\ &\le \frac{1}{4}\epsilon^2(\frac{1}{2} + 4C(\epsilon^4 + 2\epsilon c_0 + c_0)) \le \frac{1}{4}\epsilon^2, \end{aligned} \quad (8-4) $$

where in the last inequality we have chosen $c_0$ and $\epsilon$ sufficiently small. This implies the thesis. $\square$

**Proof of Theorem 2.** One has to follow almost word by word the proof of Theorem 1. The only difference relies on the estimates on the small divisors, which in this case are given by Proposition 2.1(ii).

**Proof of Theorem 3.** Consider (KG) and let $(\psi_0, \psi_1)$ be as in the statement of Theorem 3. Let $\psi(t, x)$ be a solution of (KG) satisfying the condition in Hypothesis 6.8. By Proposition 4.7, recall (3-77), the function

$$ U := \begin{bmatrix} u \\ \bar{u} \end{bmatrix} $$

solves (4-12) with initial condition

$$ u_0 = \frac{1}{\sqrt{2}} (\Lambda_{\mathrm{KG}}^{1/2} \psi_0 + i \Lambda_{\mathrm{KG}}^{-1/2} \psi_1). $$
Moreover, by Hypothesis 6.8 and Remark 6.9 one has $\|u_0\|_{H^s} \le \frac{1}{16}\epsilon$. By Remark 6.9, in order to get (1-8), we have to show that the bound $\sup_{t \in [0,T)} \|u\|_{H^s} \le \frac{1}{4}\epsilon$ holds for time $T \gtrsim \epsilon^{-3-}$ if $d=2$ and $T \gtrsim \epsilon^{-8/3-}$ if $d \ge 3$. Fix $\beta$ as in Proposition 2.2 and let $m \in \mathcal{C}_\beta$. By Propositions 6.11, 6.13 and Lemma 7.8 we can construct a function $z_n$ with $n=s$ such that if $\psi(t, x)$ solves (KG) then $z_n$ solves (7-41). By Proposition 7.10 and Remark 6.12 we get

$$ \|u(t)\|_{H^s}^2 \le 2^{1/2} \|z_n(t)\|_{L^2}^2 \le 2 \|u_0\|_{H^s}^2 + 2 \left| \int_0^t \mathcal{B}(\sigma) d\sigma \right| + 2 \left| \int_0^t \mathcal{B}_{>5}(\sigma) d\sigma \right| \quad (8-5) $$

Propositions 7.10 and 7.12 apply, therefore, by (7-63) and (7-57), we obtain the following a priori estimate: Fix any $0 < N$. Then for any $t \in [0, T)$, with $T$ as in Hypothesis 6.8, one has

$$ \|u(t)\|_{H^s}^2 \le 2\|u_0\|_{H^s}^2 + C\left(\|u\|_{L^\infty H^s}^6 TN^{\beta-1} + \|u\|_{L^\infty H^s}^7 TN^\beta + \|u\|_{L^\infty H^s}^6 T + \|u\|_{L^\infty H^s}^4 TN^{-1} + N^{\beta-1} \|u\|_{L^\infty H^s}^4\right) \quad (8-6) $$

for some $C > 0$ depending on $s$. The thesis of Theorem 3 follows from the lemma below.

**Lemma 8.2 (main bootstrap).** Let $u(t, x)$ be a solution of (4-44) with $t \in [0, T)$ and initial condition $u_0 \in H^s(\mathbb{T}^d; \mathbb{C})$. Define $a = 3$ if $d=2$ and $a = \frac{8}{3}$ if $d \ge 3$. Then, for $s \gg 1$ large enough and any $\delta > 0$, there exists $\epsilon_0 = \epsilon_0(d, s, m, \delta) > 0$ such that, for any $0 < \epsilon \le \epsilon_0$, if

$$ \|u_0\|_{H^s} \le \frac{1}{16}\epsilon, \quad \sup_{t \in [0,T)} \|u(t)\|_{H^s} \le \frac{1}{4}\epsilon, \quad T \le \epsilon^{-a+\delta}, \quad (8-7) $$

then we have the improved bound $\sup_{t \in [0,T)} \|u(t)\|_{H^s} \le \frac{1}{8}\epsilon$.

*Proof.* We start with $d \ge 3$. For $\epsilon$ small enough the bound (8-6) holds true. Let $\delta > 0$ and $0 < \sigma \ll \delta$. Define

$$ \beta := 3 + \sigma, \quad N := \epsilon^{-2/(3+\sigma)}. \quad (8-8) $$

By (8-6), (8-7), (8-8), there is $C = C(s) > 0$ such that, for any $t \in [0, T)$,

$$ \|u(t)\|_{H^s}^2 \le 2 \frac{1}{16^2} \epsilon^2 + C \epsilon^2 \epsilon^{2/(3+\sigma)} + 2CT \epsilon^2 (\epsilon^3 + \epsilon^{2+2/(3+\sigma)}) \le \frac{1}{64} \epsilon^2, \quad (8-9) $$

where in the last inequality we have chosen $\epsilon$ sufficiently small and we used the choice of $T$ in (8-7) and that $\sigma \ll \delta$. This implies the thesis for $d \ge 3$. In the case $d=2$ the proof is similar setting $\beta = 2+\sigma$ and $N = \epsilon^{-2/(2+\sigma)}$. $\square$

**Proof of Theorem 4.** Using Remarks 4.6, 4.8, 6.14, 7.9, 7.11, 7.14, 7.15 one deduces the result by reasoning as in the proof of Theorem 3 and using in particular the estimate (7-77).

Acknowledgements

We would like to warmly thank Prof. Fabio Pusateri, Joackim Bernier and the anonymous referees for their useful comments.
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Received 5 Oct 2020. Revised 14 Sep 2021. Accepted 7 Jan 2022.

ROBERTO FEOLA: roberto.feola@uniroma3.it

Dipartimento di Matematica e Fisica, Università degli studi Roma Tre, Rome, Italy

BENOÎT GRÉBERT: benoit.grebert@univ-nantes.fr

Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, Nantes, France

FELICE IANDOLI: felice.iandoli@unical.it

Dipartimento di Matematica e Informatica, Università della Calabria, Rende, Italy
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# Analysis & PDE

msp.org/apde

## EDITORS-IN-CHIEF

**Patrick Gérard** Université Paris Sud XI, France
patrick.gerard@universite-paris-saclay.fr

**Clément Mouhot** Cambridge University, UK
c.mouhot@dpmms.cam.ac.uk

## BOARD OF EDITORS

<table><tr><td>Massimiliano Berti</td><td>Scuola Intern. Sup. di Studi Avanzati, Italy<br/>berti@sissa.it</td><td>William Minicozzi II</td><td>Johns Hopkins University, USA<br/>minicozz@math.jhu.edu</td></tr><tr><td>Zbigniew Błocki</td><td>Uniwersytet Jagielloński, Poland<br/>zbigniew.blocki@uj.edu.pl</td><td>Werner Müller</td><td>Universität Bonn, Germany<br/>mueller@math.uni-bonn.de</td></tr><tr><td>Charles Fefferman</td><td>Princeton University, USA<br/>cf@math.princeton.edu</td><td>Igor Rodnianski</td><td>Princeton University, USA<br/>irod@math.princeton.edu</td></tr><tr><td>David Gérard-Varet</td><td>Université de Paris, France<br/>david.gerard-varet@imj-prg.fr</td><td>Yum-Tong Siu</td><td>Harvard University, USA<br/>siu@math.harvard.edu</td></tr><tr><td>Colin Guillarmou</td><td>Université Paris-Saclay, France<br/>colin.guillarmou@universite-paris-saclay.fr</td><td>Terence Tao</td><td>University of California, Los Angeles, USA<br/>tao@math.ucla.edu</td></tr><tr><td>Ursula Hamenstaedt</td><td>Universität Bonn, Germany<br/>ursula@math.uni-bonn.de</td><td>Michael E. Taylor</td><td>Univ. of North Carolina, Chapel Hill, USA<br/>met@math.unc.edu</td></tr><tr><td>Vadim Kaloshin</td><td>University of Maryland, USA<br/>vadim.kaloshin@gmail.com</td><td>Gunther Uhlmann</td><td>University of Washington, USA<br/>gunther@math.washington.edu</td></tr><tr><td>Izabella Laba</td><td>University of British Columbia, Canada<br/>ilaba@math.ubc.ca</td><td>András Vasy</td><td>Stanford University, USA<br/>andras@math.stanford.edu</td></tr><tr><td>Anna L. Mazzucato</td><td>Penn State University, USA<br/>alm24@psu.edu</td><td>Dan Virgil Voiculescu</td><td>University of California, Berkeley, USA<br/>dvv@math.berkeley.edu</td></tr><tr><td>Richard B. Melrose</td><td>Massachussets Inst. of Tech., USA<br/>rbm@math.mit.edu</td><td>Jim Wright</td><td>University of Edinburgh, UK<br/>j.r.wright@ed.ac.uk</td></tr><tr><td>Frank Merle</td><td>Université de Cergy-Pontoise, France<br/>merle@ihes.fr</td><td>Maciej Zworski</td><td>University of California, Berkeley, USA<br/>zworski@math.berkeley.edu</td></tr></table>

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ANALYSIS & PDE

Volume 16 No. 5 2023

<table>
  <tbody>
    <tr>
      <td>Nonuniform stability of damped contraction semigroups<br>RALPH CHILL, LASSI PAUNONEN, DAVID SEIFERT, REINHARD STAHN and YURI TOMILOV</td>
      <td>1089</td>
    </tr>
    <tr>
      <td>Long time solutions for quasilinear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori<br>ROBERTO FEOLA, BENOIT GRÉBERT and FELICE IANDOLI</td>
      <td>1133</td>
    </tr>
    <tr>
      <td>An extension problem, trace Hardy and Hardy's inequalities for the Ornstein-Uhlenbeck operator<br>PRITAM GANGULY, RAMESH MANNA and SUNDARAM THANGAVELU</td>
      <td>1205</td>
    </tr>
    <tr>
      <td>On the well-posedness problem for the derivative nonlinear Schrödinger equation<br>ROWAN KILLIP, MARIA NTEKOUME and MONICA VIŞAN</td>
      <td>1245</td>
    </tr>
    <tr>
      <td>Exponential integrability in Gauss space<br>PAATA IVANISVILI and RYAN RUSSELL</td>
      <td>1271</td>
    </tr>
  </tbody>
</table>