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Remarks on nonlinear dispersive PDEs with rough dispersion management

Tristan Robert

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REMARKS ON NONLINEAR DISPERSIVE PDES WITH ROUGH DISPERSION MANAGEMENT

TRISTAN ROBERT

ABSTRACT. In this work, we study the Cauchy problem for a class of dispersive PDEs where a rough time coefficient is present in front of the dispersion. Under minimal assumptions on the occupation measure of this coefficient, we show that for the large class of semilinear dispersive PDEs whose well-posedness theory relies on linear estimates of Strichartz or local smoothing type, one has the same well-posedness theory with or without the modulation. We also show a regularization by noise type of phenomenon for rough modulations, namely, large data global well-posedness in the focusing mass-critical case for the modulated equation. Under rougher assumptions on the modulation, we show that one can also transfer the well-posedness theory based on multilinear Fourier analysis from the original dispersive PDE to the modulated one. In the case of the NLS equation on $\mathbb{R}^d$ and $\mathbb{T}^d$, this covers all the sub-critical and critical regularities, thus completing and extending the various results currently available in the literature. At last, in the case of the periodic Wick-ordered cubic NLS, we show an even stronger form of regularization by noise, namely well-posedness in the critical Fourier-Lebesgue space for rough modulations.

CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Setting</td><td>2</td></tr><tr><td>1.2. Assumptions</td><td>3</td></tr><tr><td>1.3. Statement of the results</td><td>6</td></tr><tr><td>2. Preliminaries</td><td>13</td></tr><tr><td>2.1. Notations</td><td>13</td></tr><tr><td>2.2. Properties associated with the noise</td><td>13</td></tr><tr><td>2.3. Function spaces</td><td>13</td></tr><tr><td>3. Local well-posedness for general models: proof of Theorem 1.4</td><td>17</td></tr><tr><td>3.1. Case (i)</td><td>17</td></tr><tr><td>3.2. Case (ii)</td><td>20</td></tr><tr><td>3.3. Proofs of Example 1</td><td>21</td></tr><tr><td>4. The modulated periodic NLS equation at critical regularity</td><td>30</td></tr><tr><td>5. The periodic cubic Wick-ordered modulated NLS equation</td><td>38</td></tr><tr><td>References</td><td>41</td></tr></table>

<footnote n="2020">2020 Mathematics Subject Classification. 35Q55,35Q53,35Q60,60L50.</footnote>

<footnote>Key words and phrases. Nonlinear Schrödinger equation, dispersion management, regularization by noise.</footnote>
# 1. INTRODUCTION

## 1.1. Setting.
We consider abstract dispersive PDEs with modulated dispersion:

$$ egin{cases} \partial_t u + \frac{dW}{dt} \mathcal{L}u + \mathcal{N}(u) = 0, \\ u|_{t=0} = u_0 \end{cases} \quad u: (t, x) \in \mathbb{R} \times \mathcal{M} \to \mathbb{K}, \qquad (1.1) $$

where $\mathcal{M}$ is some boundaryless manifold to be specified, $\mathbb{K} \in \{\mathbb{R}; \mathbb{C}\}$, $\mathcal{L}$ is a skew-adjoint operator, and $\mathcal{N}$ is a nonlinear function of $u$ possibly depending also on its derivatives. The modulation function $W_t: \mathbb{R} \to \mathbb{R}$ in (1.1) is a given continuous but not necessarily differentiable function, and its time derivative in (1.1) above is in general interpreted in the sense of distribution. Actually, to avoid the issue of ill-defined products due to a potential lack of regularity in time, we will always work at the level of the mild (Duhamel) formulation of (1.1), namely

$$ u(t) = e^{-(W_t - W_0)\mathcal{L}} u_0 - \int_0^t e^{-(W_t - W_\tau)\mathcal{L}} \mathcal{N}(u(\tau)) d\tau, \qquad (1.2) $$

which indeed can be made sense of as soon as $W_t$ is continuous.

The model (1.1) has been widely studied in the particular case of the nonlinear Schrödinger (NLS) equation, namely $\mathcal{L} = -i\Delta$ and $\mathcal{N}(u) = |u|^{p-2}u$ for some $p \ge 2$. In this context, when $\mathcal{M} = \mathbb{R}$, the NLS equation with modulation (1.1) appears as a model for the propagation of a signal in an optical fibre with dispersion management; due to their fruitful applications to the improvement on the transmission rate in optical fibres, there is a wide physical literature on dispersion-managed solitons, see e.g. [31, 32, 58, 60, 61, 63, 67, 90], the reviews [84, 85], and references therein. In turn, this sparked more rigorous mathematical investigations [1, 16, 17, 18, 19, 25, 27, 49, 50, 59, 69, 70, 71]. In these works, $\frac{dW_t}{dt}$ is typically assumed to be regular enough, e.g. periodic and piece-wise constant with a finite number of discontinuities. In this context one is often interested in effective models as the period goes to zero, in either the so-called strong [16, 17, 19, 49, 50, 51, 59, 69, 70, 71] or fast [1, 12] dispersion regimes, depending if the amplitude of the modulation also blows-up as the period goes to zero.

In contrast, in [25, 27, 64], the case of a random dispersion is considered, and the effective model appearing as a scaling limit for optical fibres with random dispersion management is given by (1.1) with $W_t$ being a Brownian motion. This motivated the study of (1.1) with rough modulation $W_t$. Note that our abstract result in Theorem 1.4 (i) below applies to all three settings, i.e. the usual setting $W_t = t$, the case of $\frac{dW_t}{dt}$ periodic and piece-wise constant with a finite number of discontinuities, or the case of rougher $W_t$; see Remark 1.2 below.

Let us now describe the literature regarding the NLS equation with rough modulation. In [25], de Bouard and Debussche studied the modulated NLS equation (1.1) on $\mathcal{M} = \mathbb{R}^d$ and with Brownian modulation. Using stochastic properties of $W_t$, they established global well-posedness of (1.1) in $L^2(\mathbb{R}^d)$ for mass-subcritical nonlinearities $\mathcal{N}(u) = |u|^{2\sigma}u$, $0 < \sigma < \frac{2}{d}$. In dimension $d=1$, the mass-critical case $\sigma = 2$ was considered by Debussche and Tsutsumi [27], who proved a regularization by noise phenomenon: the mass-critical NLS equation
(1.1) with white noise dispersion is globally well-posed in $L^2(\mathbb{R})$ even for large data and focusing nonlinearity. This is in sharp contrast with the deterministic focusing mass-critical NLS equation (1.1), for which finite time blow-up occurs for data with mass bigger than that of the ground state [66]. Then Duboscq and Réveillac [29] proved a path-wise version of the results in [25, 27] by establishing path-wise dispersive and Hardy-Littlewood-Sobolev inequalities adapted to $W_t$ a (fractional) Brownian motion in place of $t$. Following the development of deterministic regularization by noise in the context of ODEs by Catellier and Gubinelli [13], Chouk and Gubinelli [20], and Galeati [34]<fnref n="1" /> established a deterministic version of the results in [25, 27, 29], under appropriate assumptions on the occupation measure of $W_t$; see Definition 1.1 and Example 1 (i) below for a comparison between these results and our abstract Theorem 1.4 below. Very recently, Tanaka [83] studied the modulated NLS equation on tori $M = \mathbb{T}^d$. See also [23] for a study of the modulated NLS equation on quantum graphs, and the very recent preprint [37] discussing Strichartz estimates for the Schrödinger equation with both a potential and white noise dispersion.

Numerical simulations of the modulated NLS equation (1.1) with Brownian modulation [4, 24, 64, 65] suggest that a stronger regularization by noise phenomenon should occur in this case. Namely, it seems like the 1d NLS equation with white noise dispersion is well-posed in $L^2(\mathbb{R})$ for mass-supercritical power nonlinearities $|u|^{2\sigma}u$, $\sigma \in (2; 4]$. However, in the case of the periodic quintic NLS equation (1.1) with Brownian modulation, Stewart [82] showed that the same semilinear ill-posedness in $L^2(\mathbb{T})$ as in the deterministic case (1.4) by Kishimoto [54] also holds. This indicates that the regularization by noise phenomenon numerically conjectured in [4, 24, 64, 65] is not systematic. A strong regularization by noise is observed by Chouk, Gubinelli, Li, Li, and Oh in [21] for KdV, Benjamin-Ono and the intermediate long wave equation. They showed well-posedness at *any* regularity provided that $W_t$ is a suitably *irregular* deterministic modulation. We generalize and extend this result in [77] by showing that this phenomenon is typical in *strongly non-resonant* dispersive PDEs. Thus these results are in sharp contrast with the counter-example provided by Stewart that we discussed above, and raise the question of observing such a phenomenon for models which are *not* strongly non-resonant, such as the quintic periodic NLS equation studied in [82]. We give such examples in Theorem 1.13 below and in [78]; see Remark 1.14 below. We point out that as a by-product of the proof of Theorem 1.4 below, we show that such regularization by noise can only come from *multilinear* estimates, since linear space-time estimates à la Strichartz for the modulated semi-group are bounded above and below by that of the deterministic semi-group; see Remark 3.2. We refer to the introduction of [77] and references therein for a further discussion and history on regularization by noise phenomena for ODEs and PDEs, in particular dispersive PDEs; see also [76] for an overview of these results.

## 1.2. Assumptions.
In the following, we will use different assumptions on the irregularity of $W_t$, expressed through the regularity of its occupation measure $\mu_{[0;T]}$ measured in some appropriate topology.

<footnote n="1">¹See the discussion before Theorem 5.37 in [34].</footnote>
**Definition 1.1.**

* Let $W: \mathbb{R} \to \mathbb{R}$ be a continuous path. For all time interval $I \subset \mathbb{R}$, its **occupation measure** is defined as the measure on Borel sets $A \in \mathcal{B}(\mathbb{R})$

$$ \mu_I(A) := \mathrm{Leb}(t \in I, W_t \in A) = \int_I \mathbf{1}_A(W_t)dt. \qquad (1.3) $$

* In the following we assume that $\mu_I \ll dz$ for all interval $I \subset \mathbb{R}$. Then $\frac{d\mu_I}{dz}$ is the **local time** of $W_t$ in $I$.

The relation between regularity properties of a deterministic or stochastic path $W_t$ and its occupation measure $\mu$ have been investigated since the 70s, and we refer to the review [42] by Geman and Horowitz on these earlier results. Different topologies have been used to measure the regularity of $\mu$: while Hölder regularity is discussed in [42], more recently the use of Fourier-Lebesgue regularity has been proved usefull by Catellier and Gubinelli [13] in the context of ODEs under rough additive perturbations $W_t$, or Chouk-Gubinelli [20], also with Li, Li and Oh [21] in the context of modulated dispersive PDEs. Even more recently, the work of Romito and Tolomeo [79] studies Besov regularity of $\mu$ under similar assumptions as in [42] on $W_t$. The common feature of these works is that regularity of $\mu$ implies local irregularity of $W_t$. Actually, the notion of prevalence allows to show some form of the converse, namely paths $W_t$ in some Hölder spaces satisfy “a.e.” a regularity property for their occupation measures; see the recent works [35, 36] by Galeati and Gubinelli where this is investigated.

Below, we will assume either Hölder or Fourier-Lebesgue regularity à la Catellier-Gubinelli for the occupation measure $\mu$ of the path $W_t$ in (1.1). Therefore we will make one of the following assumptions on the irregularity of $W_t$:

**Assumption (A0).** $W_t$ is a continuous function with local time

$$ \frac{d\mu_{[0;\cdot]}}{dz}(\cdot) \in L_{\mathrm{loc}}^{\infty}(\mathbb{R} \times \mathbb{R}). $$

**Assumption (A0)\*.** $W_t$ is a continuous function with local time

$$ \frac{d\mu_{[0;\cdot]}}{dz}(\cdot) \in C_t^0(\mathbb{R}; L_{\mathrm{loc}}^{\infty}(\mathbb{R})). $$

**Assumption (A0)\_($\rho, \gamma$).** $W_t$ is a continuous function with local time

$$ \frac{d\mu_{[0;\cdot]}}{dz}(\cdot) \in C_t^{\gamma}(\mathbb{R}; \mathcal{FL}^{\rho,\infty}(\mathbb{R})). $$

Before stating the assumptions on the model (1.1), let us give some comments on the set of assumptions above regarding the regularity of the occupation measure $\mu$ of $W_t$.

**Remark 1.2.**

(i) The definition of the Fourier-Lebesgue space $\mathcal{FL}^{\rho,\infty}(\mathbb{R})$ appearing in assumption
(A0)$_{(ρ,γ)}$ above is recalled in Section 2 below.

(ii) The assumption (A0)$_{(ρ,γ)}$ corresponds to the definition of $(ρ, γ)$-irregularity introduced in [13] and also used in [20, 21].

(iii) From the standard embedding $\mathcal{FL}^{ρ,∞}(\mathbb{R}) \hookrightarrow L^{∞}(\mathbb{R})$ when $ρ > 1$, we have the following implications between the different assumptions above:

$$ ((\mathbf{A}0)_{(ρ,γ)}, γ ≥ 0, ρ > 1) \implies (\mathbf{A}0)^* \implies (\mathbf{A}0). $$

(iv) When $W_t = t$, one has $\frac{dμ_{[0;t]}}{dz}(z) = \mathbf{1}_{[0;t]}(z)$, which clearly satisfies (A0) but not (A0)*. Moreover, from

$$ |\widehat{dμ_{[0;t]}}(ξ)| = |\frac{e^{itξ} - 1}{iξ}| \lesssim |t|^{γ}|ξ|^{1-γ} $$

for any $γ ∈ [0; 1]$, one finds that $W_t = t$ also satisfies (A0)$_{(ρ,γ)}$ if and only if $γ ∈ [0; 1]$ and $ρ ≤ 1 - γ$. The same holds for more regular one-to-one maps, see [20, Proposition 1.4].

(v) Similarly, when $\frac{dW_t}{dt}$ is T-periodic and piecewise constant with a finite number of discontinuities, i.e. $\frac{dW_t}{dt} = \sum_{k=1}^{K} \mathbf{1}_{(t_{k-1};t_k)}\dot{w}_{k-1}$ for some partition $\{t_k\}_{k=0}^K$ of $[0;T]$, provided that $\dot{w}_k \neq 0$ for all $k$, one finds for $t \in [0;T]$ that $\frac{dμ_{[0;t]}}{dz}(z) = \sum_{k=1}^{K} \mathbf{1}_{[w_{k-1};w_{k-1}+\dot{w}_{k-1}(t_k-t_{k-1}))\cap[0;t]}(z)$ for some $w_k \in \mathbb{R}$. Thus $W_t$ also satisfies (A0) in this case.

(vi) On the contrary, it follows from [42] that as soon as (A0)* is satisfied, $W_t$ is not locally Lipschitz continuous at any $t$. Similarly, for any $η ∈ (0; 1)$, the assumption (A0)$_{(ρ,γ)}$ for some $ρ > 1$ and $γ > 1 - η$ implies that $W_t$ is not $η$-Hölder continuous at any $t$.

We now turn to the dispersive properties of equation (1.1). We will make use of one of the following assumptions on the domain and the dispersion.

**Assumption (A1).** $\mathcal{M}$ is a connected abelian Lie group: $\mathcal{M} = \mathbb{R}^{d_1} \times \mathbb{T}^{d_2}$ for some $d_1, d_2 \in \mathbb{N}$, and $\tilde{\mathcal{M}} = \mathbb{R}^{d_1} \times \mathbb{Z}^{d_2}$ denotes its Pontryagin dual. Then $\mathcal{L}$ is a skew-adjoint operator given by a Fourier multiplier $\varphi \in C^1(\tilde{\mathcal{M}}; \mathbb{R})$ such that for any $u \in S(\mathcal{M})$ and $ξ \in \tilde{\mathcal{M}}$, $\tilde{\mathcal{L}}u(ξ) = i\varphi(ξ)\tilde{u}(ξ)$.

**Assumption (A1)\*.** $\mathcal{M}$ is a closed Riemannian manifold, $\mathcal{L}$ is a skew-adjoint operator with compact resolvent and discrete spectrum $\{iλ_j\}_{j \in \mathbb{N}}$, $λ_j \in \mathbb{R}$, $|λ_j| \to \infty$, and associated orthonormal basis of eigenfunctions $\{ψ_j\}_{j \in \mathbb{N}}$ of $L^2(\mathcal{M})$. Then $\tilde{\mathcal{M}} = \mathbb{N}$, and for any $u \in S(\mathcal{M})$ and $ξ \in \mathbb{N}$, $\tilde{u}(ξ) = \langle u, ψ_ξ \rangle_{L^2}$ and $\tilde{\mathcal{L}}u(ξ) = iλ_ξ \tilde{u}(ξ)$.

We will also eventually use a further assumption regarding the conservation laws associated with (1.4).

**Assumption (A2).** The mass $u \mapsto \|u\|_{L^2}^2$ is a conservation law for (1.4).

**Remark 1.3.** Even if the corresponding deterministic equation (1.4) is Hamiltonian with a conserved energy, the latter is not conserved anymore for (1.1). This is why we are only concerned with globalisation using invariance of the $L^2$ norm, hence assumption (A2).
1.3. **Statement of the results.** Equipped with the set of assumptions from the previous subsection, we can now state our results on the Cauchy problem for the abstract modulated equation (1.1).

1.3.1. **Well-posedness through space-time estimates.**

Informally, our first result shows that for any semilinear dispersive PDE treated by a fixed-point argument in appropriate space-time norms, the same well-posedness results hold for both (1.1) and

$$ \begin{cases} \partial_t u + \mathcal{L}u + \mathcal{N}(u) = 0, \\ u(t=0) = u_0. \end{cases} \qquad (1.4) $$

To state our result more precisely, we use the following notations: for a set of spatial functions $X(\mathcal{M})$ and $T > 0$, we define the sets of space-time functions

$$ S_T^{p,+}(X) := L^p([0;T]; X(\mathcal{M})) \quad \text{and} \quad S_T^{p,-}(X) = X(\mathcal{M}; L^p([0;T))). \qquad (1.5) $$

For a finite family of spaces $\{S^{p_j, t_j}(X_j)\}_{j}$ and a regularity exponent $s \in \mathbb{R}$, we define for $T > 0$ the Strichartz/local smoothing type space

$$ \mathbf{S}_T^s := C([0;T]; H^s(\mathcal{M})) \bigcap_j \langle D \rangle^{-s} S_T^{p_j, t_j}(X_j), $$

endowed with the norm

$$ \|u\|_{\mathbf{S}_T^s} := \|u\|_{L_T^\infty H^s} + \max_j \|\langle D \rangle^s u\|_{S_T^{p_j, t_j}(X_j)}. \qquad (1.6) $$

**Theorem 1.4.** Assume that either (A1) or (A1)* hold, and that the deterministic equation (1.4) is locally well-posed in $H^s(\mathcal{M})$ for some $s \in \mathbb{R}$ via (local in time) linear space-time (Strichartz/local smoothing) estimates. Namely, given $s \in \mathbb{R}$, there is a finite set $\mathcal{E} \subset (2; \infty) \times E \times \{\pm\}$ of admissible exponents and there are $n, C_0 > 0$ and a constant $\eta_{\mathcal{E}}(s) \ge 0$ such that for any $(p, \zeta, \iota) \in \mathcal{E}$, there are spatial function spaces $X_\zeta$ such that for any $T \in (0; 1]$, there is a constant $C_{p,\zeta,\iota}(T) > 0$ non-decreasing in $T$ satisfying

$$ \|e^{-t\mathcal{L}}u_0\|_{S_T^{p,\iota}(X_\zeta)} \le C_{p,\zeta,\iota}(T)\|u_0\|_{L^2} \qquad (1.7) $$

for any $u_0 \in L^2(\mathcal{M})$, and moreover the nonlinear estimate

$$ \left| \int_0^T \int_{\mathcal{M}} (\mathcal{N}(u) - \mathcal{N}(v)) \overline{w} dxdt \right| \le C_0 T^{\eta_{\mathcal{E}}(s)} \|u - v\|_{\mathbf{S}_T^s} (\|u\|_{\mathbf{S}_T^s} + \|v\|_{\mathbf{S}_T^s})^n \|w\|_{\mathbf{S}_T^{-s}} \qquad (1.8) $$

holds for any $T \in (0; 1]$ and any $u, v \in \mathbf{S}_T^s(\mathcal{M})$ and $w \in \mathbf{S}_T^{-s}(\mathcal{M})$. Then the following hold. (i) If (A0) holds, the corresponding modulated equation (1.1) is also locally well-posed in $H^s(\mathcal{M})$. More precisely, for any $u_0 \in H^s(\mathcal{M})$, there is $T = T(u_0) > 0$ with $T = T(\|u_0\|_{H^s})$ in case $\eta_{\mathcal{E}}(s) > 0$, such that there is a unique mild solution $u \in C([0;T]; H^s(\mathcal{M})) \cap X_T^s$ to (1.1) on $[0;T)$. Moreover, the flow map $u_0 \in H^s(\mathcal{M}) \mapsto u \in C([0;T]; H^s(\mathcal{M})) \cap X_T^s$ is locally Lipschitz continuous.

(ii) If (A2) holds, and either (A0)*, or (A0) with $\eta_{\mathcal{E}}(0) > 0$, also hold, then in case $s = 0$ the solution extends globally in time for any initial data in $L^2(\mathcal{M})$.
Before giving some examples of nonlinear dispersive PDEs satisfying the assumptions in Theorem 1.4, we first make some comments on these results.

**Remark 1.5.**

(i) Clearly, (1.7)-(1.8) allow to run a fixed point argument on the Duhamel formulation of (1.4) in a suitable ball of $S_T^s$ for $T > 0$ small enough depending on $u_0 \in H^s(\mathcal{M})$, thus showing that indeed the deterministic equation (1.4) is well-posed in $H^s(\mathcal{M})$.

(ii) Only (A0) is assumed in Theorem 1.4 (i). In particular, this contains the case $W_t = t$ as pointed out in Remark 1.2 (ii). Of course Theorem 1.4 (i) is a tautology in this case due to the previous remark. This also covers the case of periodic and piece-wise constant $\frac{dW_t}{dt}$, see Remark 1.2 (v), and the case of rougher $W_t$ in the sense of having a more regular local time. This means that Theorem 1.4 does not distinguish between $W_t = t$ and more irregular $W_t$, in sharp contrast to Theorems 1.10 and particularly 1.13 below. Note that this is however in some sense optimal: see Remark 3.2 below.

(iii) As pointed out in Remark 1.2 (i), (A0)* is less restrictive than (A0)$_{(\rho,\gamma)}$ for $\rho > 1$. For example, if $W_t$ is a sample path of a fractional Brownian motion with Hurst parameter $H \in (0; 1)$, then (A0)$_{(\rho,\gamma)}$ holds a.s. only for $\gamma \in [\frac{1}{2}; 1]$ and $\rho < \frac{1-\gamma}{H}$ (see [13, Theorem 1.4]), while (A0)* holds for any $H \in (0; 1)$, see [5, Theorem 8.1] or [88].

(iv) From Remark 1.2 (iii) and (iv) we see that the assumptions on $W_t$ in Theorem 1.4 (i) and (ii) are in some sense optimal.

**Remark 1.6.** Theorem 1.4 (ii) can be seen as a *noiseless regularization by noise* phenomenon. Indeed this result shows that due to the irregularity of $W_t$ in (1.1), expressed through the regularity assumption (A0)* on its occupation measure, we have that even for focusing mass-critical equations, one can get *large data* global well-posedness. This of course is false in general for the deterministic equation (1.4), and thus the assumption (A0)* in Theorem 1.4 (ii) is minimal to see this regularization effect. This phenomenon was first observed in [27] then [20, 29] in the context of the quintic NLS equation on $\mathcal{M} = \mathbb{R}$ with (fractional) Brownian motion $W_t$. Theorem 1.4 (ii) generalizes this observation to other mass-critical equations and to rough but deterministic modulations. See Examples 1 below for examples of dispersive PDEs where Theorem 1.4 (i) and (ii) apply. Similar regularization by noise effects, namely lack of blow-up under stochastic perturbations of dispersive PDEs, also appear for some models with multiplicative noise; see e.g. [3, 46, 47] and Remark 3.1 below.

Note at last that the special role of $s = 0$ in Theorem 1.4 (ii) is due to the fact that apart from the $L^2$ norm, we do not expect other conservation laws for (1.1) because of the time-dependent coefficient in front of the dispersion, as pointed out in Remark 1.3.

**Remark 1.7.** The proof of Theorem 1.4 relies on the transfer of the linear space-time estimate (1.7) from the equation (1.4) to the modulated equation (1.1), in the same spirit as in e.g. [89]. Note that under (A0), $W_t$ can be very rough. In particular, as far as Schrödinger equations with non constant rough *spatial* coefficients are concerned, it is known that there is a threshold of regularity for the spatial coefficients, below which
Strichartz estimates come with a loss of derivative [11, 30, 81]; see also the recent works [26, 28, 39, 68, 86] where perturbations of the Laplacian by rough random potentials are considered. It is interesting to note that time coefficients are allowed to be much more singular than what is possible for spatial coefficients, and that actually their roughness improves the behaviour of (local) space-time estimates; see (3.1)-(3.4)-(3.7) and Remark 3.1 below.

**Remark 1.8.** The local solution space $X_T^s$ in Theorem 1.4 (i) is a $U^2$-type space adapted to the modulated linear evolution $e^{-W_t \mathcal{L}}$; see Subsection 2.3 below for definitions and properties of these spaces. It replaces the space $D^W(H^s)$ used in [20, 21]. In particular the duality relation of Proposition 2.9 (iii) below can be seen as the endpoint case $\gamma = \frac{1}{2}$ of the nonlinear Young integral theory of [20, Theorem 2.3] for $f \in C_t^\gamma \text{Lip}_M(H^s)$ and $g \in C_t^\gamma H^s$; see also [21, 33] for further results based on nonlinear Young integration.

These spaces are nowadays heavily used to study the Cauchy problem for nonlinear dispersive PDEs at scaling-critical regularity. Here, even at sub-critical regularity, we crucially exploit their atomic structure, which is particularly compatible with assumptions (A0), (A0)* or (A0)$_{(\rho, \gamma)}$ on the occupation measure (1.3). On the contrary, it is not clear at all how one could use the standard $X^{s,b}$ spaces à la Bourgain when $W_t \not\equiv t$, due to a lack of informations on the Fourier transform of $\frac{d\mu_{[0;t]}}{dz}$ with respect to $t$. This choice of spaces also allows to track more efficiently the dependence on the time interval. Though this does not play such a big role in this work, it is crucial when performing a short-time analysis as in [77, 78].

**Remark 1.9.** It is also possible to extend the admissible space-time estimates (1.7) of Theorem 1.4 to an anisotropic setting. This would allow for example to take into account the local smoothing estimate for the Schrödinger equation in $\mathbb{R}^d$ when $d \ge 2$.

We give below some examples of semilinear dispersive PDEs of the form (1.4) satisfying the assumptions of Theorem 1.4 and we refer to Subsection 3.3 below for further discussion on the validity of these assumptions for each model.

**Example 1.**

(1) The modulated nonlinear Schrödinger (NLS) equation on $M = \mathbb{R}^d$:

Our first example is the modulated NLS equation

$$i \partial_t u - \frac{dW}{dt} \Delta u \pm |u|^{2m} u = 0 \qquad (1.9)$$

for<fnref n="2" /> some $m \in \mathbb{N}^*$. In this case, since the deterministic NLS equation is locally well-posed in $H^s(\mathbb{R}^d)$ in the whole (sub-) critical range $s \ge \max(s_c, 0)$, where $s_c = \frac{d}{2} - \frac{1}{m}$, via Strichartz estimates (1.7) with $S_T^{p,+}(X) = L_T^p L_x^q$, Theorem 1.4 shows local well-posedness of the modulated NLS equation (1.4) also in the whole (sub-) critical range $s \ge \max(s_c, 0)$. This recovers and improves the results of [25,

<footnote n="2">²The fact that we treat a nonlinearity $|u|^p u$ with $p$ restricted to be an even integer is not necessary and only made to simplify the presentation, but the same arguments as in [14] allow to treat a more general nonlinearity $f(u)$ with suitable polynomial growth and $f(0) = 0$.</footnote>
27, 20, 29, 34]. Indeed, compared to [25, 27, 29], here the assumption on $W_t$ is
completely non-random, similarly as what is explained by Galeati [34, Theorem 5.37]
regarding the result of Duboscq and Réveillac [29]. The assumption (A0) allows to
relax the assumptions in [25, 27, 29, 34] treating the case of (fractional) Brownian
modulations<fnref n="3" />, and the assumptions of Chouk and Gubinelli in [20, Theorems 1.9.1&
1.10] regarding the cases $d=1, m=1, 2$, and $d=2, m=1$. Moreover, Theorem 1.4
improves the result of [20, Theorem 1.9.1] regarding the case $d=2, m=1$ to
well-posedness in the whole (sub-) critical range $s \ge 0$.

At last, when $-\Delta$ is perturbed by a suitable potential as in [38, 37], then (the proof of) Theorem 1.4 recovers the results of [37] establishing the Strichartz estimate for the modulated equation from the one of [38] for the standard Schrödinger equation with potential.

(2) The modulated NLS equation on $\mathcal{M} = \mathbb{T}^d$:

We still consider (1.9), but with spatial domain $\mathcal{M} = \mathbb{T}^d$. This satisfies again assumptions (A1) and (A2). In this case, under assumption (A0), Theorem 1.4 recovers the sub-critical well-posedness of the periodic NLS equation (1.9) in $H^s(\mathbb{T}^d)$, $s > \max(s_c, 0)$. In [83], Tanaka studies the same model, but under assumption (A0)$_{(\rho,\gamma)}$ for some $\gamma, \rho \in (0; 1]$, and shows well-posedness in $H^s(\mathbb{T}^d)$ for $s > s(\rho) = \max(\frac{d}{2} - \frac{\rho}{m}; s_c)$. Note in particular that $s(\rho) > s_c$ if and only if $\rho < 1$. Since (A0)$_{(\rho,\gamma)}$ implies (A0) when $\rho > 1$ as mentioned in Remark 1.2 (iii), Theorem 1.4 thus recovers and improve the result of Tanaka in this regime. In case $0 < \rho \le 1$, assumptions (A0)$_{(\rho,\gamma)}$ and (A0) are not comparable anymore, but in the particular case where $W_t$ is a $H$-fBm, as mentioned in Remark 1.5 (iii), (A0) is satisfied for any $H \in (0; 1)$, and thus our Theorem 1.4 improves the result of Tanaka by treating the full sub-critical range of regularity.

Under only the minimal assumption (A0), our proof misses$^4$ the case $s = 0$ when $d = m = 1$. This is recovered if one assumes the slightly stronger assumption (A0)*, see Remark 3.3 after the proof of Theorem 1.4. In the particular case $d = 1, m = 2$, Theorem 1.4 also extends the result of Stewart [82, Theorem 1] by treating the case of deterministic modulations, including the case of a.e. paths of Brownian motion. In particular, this result is sharp in terms of range of regularity $s$ for the periodic quintic NLS in view of the counter-example provided by Stewart.

As for the scaling-critical case $s = \max(s_c, 0)$, the existing scaling-critical theory for the deterministic NLS equation on tori relies on multilinear Fourier analysis refining the simpler use of linear Strichartz estimates, and is thus not covered by Theorem 1.4. See Theorems 1.10 and 1.13 below where we address this issue.

<footnote n="3">³In particular, as mentioned in Remark 1.5, our results hold for a.e. path $W_t$ of a fractional Brownian motion of any Hurst parameter $H \in (0; 1)$.</footnote>

<footnote n="4">⁴Actually, our proof shows well-posedness when $s = 0$ and $d = m = 1$ but only for *small* initial data, even though the 1d cubic NLS is mass-subcritical.</footnote>
(3) The modulated NLS equation on closed Riemannian manifolds:

We now consider the case where $\mathcal{M}$ is a closed (smooth, compact, boundaryless) $d$-dimensional Riemannian manifold. Then (1.9) satisfies assumptions (A1)* and (A2). Under (A0), Theorem 1.4 provides local well-posedness of (1.9) in $H^s(\mathcal{M})$ for $s > \frac{d}{2} - \frac{1}{2m}$ similarly as in [8, 81] for the deterministic NLS equation on $\mathcal{M}$. Note that improvement on this range of regularity requires global informations on the geometry of $\mathcal{M}$, see e.g. [9, 10, 44], together with suitable multilinear Fourier type analysis going beyond the assumptions of Theorem 1.4. See Theorem 1.10 and Remark 4.5 below.

(4) The modulated quintic generalized Korteweg-de Vries (gKdV) equation on $\mathcal{M} = \mathbb{R}$

Our last example is the modulated quintic gKdV equation

$$ \partial_t u + \frac{dW}{dt} \partial_x^3 u \pm u^4 \partial_x u = 0. \qquad (1.10) $$

In this case $\mathcal{L} = \partial_x^3$, $\mathcal{N}(u) = \pm \frac{1}{5} \partial_x(u^5)$ and $\mathcal{M} = \mathbb{R}$. This satisfies the assumptions (A1)-(A2). The corresponding deterministic equation (1.4) is mass-critical and locally well-posed in $H^s(\mathbb{R})$ for any $s \ge 0$, and globally for small enough initial data [52], while finite-time blow-up occurs for large enough initial data in the focusing case [66]. Thus under assumption (A0) we also get local well-posedness in $H^s(\mathbb{R})$, $s \ge 0$, for the modulated quintic gKdV equation (1.10). Moreover, under (A0)*, this gives another example of mass-critical equation where we see large data global well-posedness for the modulated equation.

### 1.3.2. Further results for the NLS equation with noisy dispersion.

Theorem 1.4 above is only stated in a setting where the deterministic equation (1.4) is well-posed through linear space-time estimates. Thus we can wonder if equations whose well-posedness relies on multilinear refinements of space-time estimates (e.g. bilinear Strichartz estimates) also enjoy the transfer of multilinear estimates from the deterministic equation (1.4) to the noisy one (1.1). This is the purpose of our next result, specialized to the case of the modulated NLS equation on tori at scaling critical regularity.

**Theorem 1.10.** Assume that $\mathcal{M} = \mathbb{T}^d$ for some $d \ge 1$, $\mathcal{L} = i\Delta$, $\mathbb{K} = \mathbb{C}$, and $\mathcal{N}(u) = |u|^{2m}u$ for some $m \in \mathbb{N}^*$. Then, if $m > \frac{2}{d}$, and (A0)$_{(\rho, \gamma)}$ holds for some $\gamma > \frac{1}{2}$ and $\rho > 1$, the modulated periodic NLS (1.1) is locally well-posed in $H^s(\mathbb{T}^d)$ for any $s \ge s_c = \frac{d}{2} - \frac{1}{m}$. More precisely, for any $R > 0$, there exists $T(R) \sim \langle R \rangle^{-\frac{2m}{\gamma}} > 0$ such that for any $u_0 \in H^s(\mathbb{T}^d)$ with $\|u_0\|_{H^s} \le R$, there exists a unique mild solution $u \in C([0; T]; H^s(\mathbb{T}^d)) \cap X_T^s$ to (1.1) on $[0; T)$. Moreover, the flow map $u_0 \in H^s(\mathbb{T}^d) \mapsto u \in C([0; T]; H^s(\mathbb{T}^d)) \cap X_T^s$ is locally Lipschitz continuous.
**Remark 1.11.** Note that the local time of existence in Theorem 1.10 above only depends on the size of the initial data in $H^s(\mathbb{T}^d)$, even in the scaling-critical case $s = s_c$. This suggests that under $(\mathbf{A}0)_{(\rho,\gamma)}$ with $\gamma > \frac{1}{2}$ and $\rho > 1$, $H^{s_c}(\mathbb{T}^d)$ becomes sub-critical for the modulated periodic NLS equation (1.1).

**Remark 1.12.** The results of Theorem 1.10 for the modulated periodic NLS (1.1) are similar to the ones for the usual periodic NLS (1.4) in [48, 87]. From the restrictions on $(m, d)$, the only cases not covered by Theorem 1.10 are: $d=1, m=1,2$, and $d=2, m=1$. The cases $d=1, m=2$ and $d=2, m=1$ correspond to the mass-critical cases. Due to the $\varepsilon$ loss of derivatives in the Strichartz estimate (3.13) in the endpoint case $p = \frac{2(d+2)}{d}$, well-posedness in $L^2(\mathbb{T}^d)$ in these cases for the unmodulated equation (1.4) remains a challenging open problem; see [45] for a recent improvement on (3.13) giving a sharp logarithmic loss in the case $d=2, p=4$. The same loss of derivatives cannot be overcome using the modulation in (1.1); see again [82] for a counter-example in the case $d=1, m=2$, and the discussion in Remark 3.2. The case $d=m=1$ is treated in [20, Theorem 1.8] under less restrictive assumptions on $\mu$ than that of Theorem 1.10, due to the sub-critical nature of the 1d cubic NLS in $L^2(\mathbb{T})$. See also Theorem 1.13 below.

Theorem 1.10 also extends the results of [83] in the case $\rho > 1$ which only covered the sub-critical range of regularity for the well-posedness, missing the critical one covered by Theorem 1.10.

The result of Theorems 1.4 (i) and 1.10 show that one has the same range of regularity exponent $s$ where well-posedness holds for both the deterministic equation (1.4) and the modulated one (1.1), including the scaling critical case $s = s_c$. However, in view of the discussion in the introduction, we expect that in some cases the rough modulation in (1.1) can actually improve the well-posedness theory compared to that of the deterministic equation (1.4). Note that this is by no mean systematic, since there are counter-examples showing that for some models, it is not possible to improve on the well-posedness theory of (1.1) compared to that of (1.4); again we point to the work of Stewart [82] on this point. The point of Theorem 1.4 is to show that locally in time, the space-time estimates hold for both the deterministic equation and the modulated one with same range of integrability exponents $(p, q)$. And this is sharp since we can also show that the space-time norm for the modulated dispersion is bounded below by that of the standard dispersion; see Remark 3.2 below. This shows that the modulation cannot help to improve the range of admissible exponents $(p, q)$ in (1.7). Thus any improvement on the well-posedness theory (in terms of larger spaces for the initial data) can only come from multilinear estimates in $X^{s,b}$-type norms. Our last result provides such an example.

**Theorem 1.13.** Let $\mathcal{M} = \mathbb{T}$, $\mathcal{L} = i\partial_x^2$, $\mathbb{K} = \mathbb{C}$, and $\mathcal{N}(u) = \pm(|u|^2 - 2\|u\|_{L^2}^2)u$. Let $s \ge 0$ and $r \in [1; \infty]$. Assume that $(\mathbf{A}0)_{(\rho,\gamma)}$ holds for some $\gamma > \frac{1}{2}$ and $\rho > \frac{1}{r}$. Then the periodic Wick-ordered cubic modulated NLS (1.1) is locally well-posed in $\mathcal{FL}^{s,r}(\mathbb{T})$. More precisely, for any $R > 0$, there exists $T(R) \sim \langle R \rangle^{-\frac{2}{\gamma}} > 0$ such that for any $u_0 \in \mathcal{FL}^{s,r}(\mathbb{T})$ with $\|u_0\|_{\mathcal{FL}^{s,r}} \le R$, there exists a unique mild solution $u \in C([0; T]; \mathcal{FL}^{s,r}(\mathbb{T})) \cap X_T^{s,r}$ to (1.1)
on $[0; T)$. Moreover, the flow map $u_0 \in \mathcal{FL}^{s,r}(\mathbb{T}) \mapsto u \in C([0; T); \mathcal{FL}^{s,r}(\mathbb{T})) \cap X_T^{s,r}$ is locally Lipschitz continuous.

The definition of the Fourier-Lebesgue space $\mathcal{FL}^{s,r}(\mathbb{T})$ is recalled in Section 2 below. In this more general Banach scale, the scaling critical regularity is $s_c = -\frac{1}{r}$, $r \in [1; \infty]$. Thus Theorem 1.13 is in the same spirit as Theorem 1.10, showing that the modulated equation (1.1) is also well-posed at sub-critical and critical regularity. Note however that the usual cubic Wick-ordered periodic NLS (1.4) is only known to be locally well-posed in $\mathcal{FL}^{s,r}(\mathbb{T})$ for any $s \ge 0$ and $r \in [1; \infty)$ [41]; see also [73, 74]. Since $L^2(\mathbb{T}) \subsetneq \mathcal{FL}^{0,r}(\mathbb{T})$ when $r > 2$, the “Wick renormalization” $|u|^2u - 2\|u\|_{L^2}^2u$ of the usual cubic nonlinearity $|u|^2u$ is necessary to have existence of solutions for data outside of $L^2(\mathbb{T})$, as pointed out in [40], while in $L^2$ solutions of the usual cubic equation are converted to solution of the Wick-ordered equation via the gauge transform $u(t) \mapsto e^{it2\|u_0\|_{L^2}^2}u(t)$ due to the conservation of the $L^2$ norm. Thus, in the case $r=2$, Theorem 1.13 recovers [20, Theorem 1.8] and addresses the case $d=m=1$ missing in Theorem 1.10 above.

In the scale of Fourier-Lebesgue spaces, local well-posedness in the critical space $\mathcal{FL}^{0,\infty}(\mathbb{T})$ is still a challenging open problem for the deterministic equation (1.4). Thus Theorem 1.13 shows a strong regularization effect of the noisy dispersion in (1.1) compared to its deterministic counterpart (1.4) since the well-posedness result of Theorem 1.13 includes the critical space $\mathcal{FL}^{0,\infty}(\mathbb{T})$.

Note also that this improvement on the well-posedness for the modulated equation (1.1) compared to the deterministic one (1.4) is somehow surprising in view of the counter-example provided by Stewart [82] showing that such an improvement is not possible in the case of the quintic periodic NLS.

**Remark 1.14.** In [77], we show an even stronger regularization effect for modulated dispersive PDEs enjoying a strong non-resonant condition, similar to the result in [21]. Namely, for these models, well-posedness holds at *any* (in particular super-critical) regularity $s$ provided that $W_t$ is *irregular* enough (i.e. its occupation measure is regular enough). This in particular applies to the periodic Wick-ordered cubic fractional NLS with slightly more dispersion than the usual NLS, namely $\mathcal{L} = iD^\alpha$ for $\alpha > 2$ instead of $\alpha = 2$. Still, Theorem 1.13 shows a regularization by noise effect for a model which is *not* strongly non-resonant.

**Structure of the paper** After recalling the notations and definitions of the initial data and solution spaces in Section 2, we give the proof of Theorem 1.4 in Section 3, and in particular we detail how the various models of Example 1 fit into the framework of Theorem 1.4 in Subsection 3.3. Then we give the proofs of Theorems 1.10 and 1.13 in Sections 4 and 5, respectively.

**Acknowledgements.** The author would like to thank Anne de Bouard, Arnaud Debussche, Tadahiro Oh, Nikolay Tzvetkov, and Yuzhao Wang for helpful discussions. The author was partially supported by the ANR project Smooth ANR-22-CE40-0017 and the PEPS JCJC project BFC n°272076.
## 2. PRELIMINARIES

**2.1. Notations.** For complex numbers $a \in \mathbb{C}$, we write $a^{-} = \bar{a}$ and $a^{+} = a$. For positive numbers $A$ and $B$, we write $A \lesssim B$ if there is a constant $C > 0$, independent of the various parameters, such that $A \le CB$; $C = 10^{42}$ should work throughout this text. We also write $A \sim B$ if both $A \lesssim B$ and $B \lesssim A$. We also write $A \vee B = \max(A; B)$ and $A \wedge B = \min(A; B)$.

In the following, we use $N, N_j, M \in 2^{\mathbb{N}}$ to denote dyadic integers in $2^{\mathbb{N}} = \{2^m, m \in \mathbb{N}\}$. For $N \in 2^{\mathbb{N}}$ a dyadic integer, we first define $\mathbf{P}_{\le N}$ to be a smooth version of the Dirichlet projection onto the frequencies $\{|\xi| \le N\} \subset \tilde{\mathcal{M}}$. Namely, take a smooth even non-increasing cut-off $\psi_0 \in C_0^{\infty}(\mathbb{R})$ satisfying supp $\psi_0 \subset [-1, 1]$ and $\psi_0 \equiv 1$ on $[-1/2, 1/2]$, then $\mathbf{P}_N$ is defined as the Fourier multiplier with symbol $\psi_0(N^{-1}\xi)$, $\xi \in \tilde{\mathcal{M}}$. Then we define

$$ \mathbf{P}_N = \begin{cases} \mathbf{P}_{\le 1}, & N = 1, \\ \mathbf{P}_{\le N} - \mathbf{P}_{\le \frac{N}{2}}, & N \ge 2. \end{cases} $$

More generally, for $S \subset \tilde{\mathcal{M}}$ a set of frequencies, we define the projector $\mathbf{P}_S$ as the Fourier multiplier with symbol $\mathbf{1}_S(\xi)$, $\xi \in \tilde{\mathcal{M}}$.

**2.2. Properties associated with the noise.** In this whole manuscript, we assume that $W : \mathbb{R} \to \mathbb{R}$ is continuous, and we write $\mu$ for its occupation measure (1.3). We will always assume either (A0), (A0)* or (A0)$_{(\rho, \gamma)}$ holds, and in particular that $d\mu_{[0;T]}$ is absolutely continuous with respect to the Lebesgue measure $dz$ for any $T > 0$. We will make heavy use of the following **occupation time formula**; see [42, Theorem 6.4].

**Proposition 2.1.** For every non-negative measurable function $F : \mathbb{R} \to [0; \infty)$ and $T > 0$, it holds

$$ \int_0^T F(W_t)dt = \int_{\mathbb{R}} F(z) \frac{d\mu_{[0;T]}(z)}{dz}dz. \quad (2.1) $$

## 2.3. Function spaces.

**2.3.1. Spatial function spaces.** Hereafter, we will use the Sobolev and Besov spaces $W^{s,p}(M)$ and $B_{p,q}^s(M)$, $s \in \mathbb{R}$, $1 \le p, q \le \infty$, which are defined via the norms

$$ \|u\|_{W^{s,p}} \stackrel{\text{def}}{=} \|\langle D \rangle^s u\|_{L^p}, $$

where $\langle D \rangle^s$ is the Fourier multiplier with symbol $\langle \xi \rangle^s$, $\xi \in \tilde{\mathcal{M}}$, and

$$ \|u\|_{B_{p,q}^s} \stackrel{\text{def}}{=} \left( \sum_{N \ge 1} N^{sq} \|\mathbf{P}_N u\|_{L^p}^q \right)^{\frac{1}{q}}. $$

We recall the following properties of Besov spaces; see e.g. [2] in case $M = \mathbb{R}^d$ and [72] in case $M$ is a smooth closed Riemannian manifold.

**Lemma 2.2.** Let $M = \mathbb{R}^d$ or $M$ be any compact Riemannian manifold of dimension $d$ without boundary.
(i) For any $s \in \mathbb{R}$ we have $B_{2,2}^s(\mathcal{M}) = H^s(\mathcal{M})$, and more generally for any $2 \le p < \infty$ and $\varepsilon > 0$ we have

$$ \|u\|_{B_{p,\infty}^s(\mathcal{M})} \lesssim \|u\|_{W^{s,p}(\mathcal{M})} \lesssim \|u\|_{B_{p,2}^s(\mathcal{M})} \lesssim \|u\|_{B_{p,\infty}^{s+\varepsilon}(\mathcal{M})}. $$

(ii) Let $s \in \mathbb{R}$ and $1 \le p_1 \le p_2 \le \infty$ and $q \in [1, \infty]$. Then for any $f \in B_{p_1,q}^s(\mathcal{M})$ we have

$$ \|f\|_{B_{p_2,q}^{s-d}\left(\frac{1}{p_1}-\frac{1}{p_2}\right)(\mathcal{M})} \lesssim \|f\|_{B_{p_1,q}^s(\mathcal{M})}. $$

(iii) Let $\alpha, \beta \in \mathbb{R}$ with $\alpha + \beta > 0$ and $p_1, p_2, q_1, q_2 \in [1, \infty]$ with

$$ \frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2} \quad \text{and} \quad \frac{1}{q} = \frac{1}{q_1} + \frac{1}{q_2}. $$

Then for any $f \in B_{p_1,q_1}^\alpha(\mathcal{M})$ and $g \in B_{p_2,q_2}^\beta(\mathcal{M})$, we have $fg \in B_{p,q}^{\alpha\wedge\beta}(\mathcal{M})$, and moreover it holds

$$ \|fg\|_{B_{p,q}^{\alpha\wedge\beta}(\mathcal{M})} \lesssim \|f\|_{B_{p_1,q_1}^{\alpha}(\mathcal{M})} \|g\|_{B_{p_2,q_2}^{\beta}(\mathcal{M})}. $$

Finally, for $s \in \mathbb{R}$ and $r \in [1; \infty]$, we define the Fourier-Lebesgue space $\mathcal{FL}^{s,r}(\mathcal{M})$ via the norm

$$ \|u\|_{\mathcal{FL}^{s,r}} = \|\langle\xi\rangle^s \hat{u}\|_{L^r(\widehat{\mathcal{M}})}. $$

In particular $H^s(\mathcal{M}) = W^{s,2}(\mathcal{M}) = \mathcal{FL}^{s,2}(\mathcal{M})$.

### 2.3.2. Adapted function spaces.
In this subsection we recall the definition and properties of $U^2/V^2$-type spaces associated with the flow of the linear dispersive equation with modulated dispersion.

These spaces were first used in [55, 56] to study the Cauchy problem for some dispersive PDEs, and we refer to [57, 43, 48, 80] and in particular [22] for the proofs of their properties that we list below. We give the necessary adaptations to the case of Banach-valued functions as in [57] since we consider functions of $t$ with values in $\mathcal{FL}^{s,r}(\mathcal{M})$.

**Definition 2.3.** Let $s \in \mathbb{R}$, $r \in [1; \infty]$, $1 \le p < \infty$, and $I = [a; b) \subset \mathbb{R}$ with $-\infty \le a < b \le \infty$. Let then $\mathcal{Z}(I)$ be the collection of finite non-decreasing sequences $\{t_k\}_{k=0}^K$ in $I$ with $t_0 = a$ and $t_K = b$.

(i) We define $V^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ as the space of functions $u : I \to \mathcal{FL}^{s,r}(\mathcal{M})$ such that $u(t)$ has a limit as $t \searrow a$ and also $\lim_{t \searrow b} u(t) = 0$, endowed with the norm

$$ \|u\|_{V^p(\mathcal{FL}^{s,r})} = \sup_{\{t_k\}_{k=0}^K \in \mathcal{Z}(I)} \left( \sum_{k=1}^K \|u(t_k) - u(t_{k-1})\|_{\mathcal{FL}^{s,r}}^p \right)^{\frac{1}{p}}. $$

(ii) We define a $U^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$-atom to be a piecewise continuous function $A : I \to \mathcal{FL}^{s,r}(\mathcal{M})$ such that

$$ A(t) = \sum_{k=1}^{K} \mathbf{1}_{[t_{k-1}; t_k)} \phi_k $$
for some partition {$t_k$}$_{k=0}^K \in \mathcal{Z}(I)$ and collection {$\phi_k$}$_{k=1}^K \in \mathcal{FL}^{s,r}(\mathcal{M})^K$ such that
$$
\sum_{k=1}^K \|\phi_k\|_{\mathcal{FL}^{s,r}}^p \le 1. \quad \text{Note that } A \text{ is right-continuous with } \lim_{t \searrow a} A(t) = 0 \text{ and has a limit as } t \nearrow b.
$$

Then $U^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ is the space of functions $u: I \to \mathcal{FL}^{s,r}(\mathcal{M})$ such that

$$
u = \sum_{j=0}^{\infty} \lambda_j A_j,
$$

with {$\lambda_j$}$_{j \in l^1(\mathbb{N}; \mathbb{K})}$, and $A_j$ are $U^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$-atoms, endowed with the norm

$$
\|u\|_{U^p(I)\mathcal{FL}^{s,r}} = \inf \left\{ \sum_{j=0}^{\infty} |\lambda_j|, u = \sum_{j=0}^{\infty} \lambda_j A_j, \text{ A}_j \text{ are } U^p(I)\mathcal{FL}^{s,r}\text{-atoms} \right\}.
$$

These spaces enjoy the following properties; again, we refer to [57, 43, 48, 22, 80] for the
proofs.

**Proposition 2.4.** The following properties hold:

(i) $U^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ and $V^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ are Banach spaces.

(ii) If $u \in U^p([-\infty; \infty))\mathcal{FL}^{s,r}(\mathcal{M})$, $a > -\infty$, and $u(a) = 0$, then $\mathbf{1}_I(t)u \in U^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ and $\|u\|_{U^p(I)\mathcal{FL}^{s,r}} = \|\mathbf{1}_I u\|_{U^p([-\infty; \infty))\mathcal{FL}^{s,r}}$.

Similarly, if $u \in V^p([-\infty; \infty))\mathcal{FL}^{s,r}(\mathcal{M})$, $b < \infty$, and $u(b) = 0$, then $\mathbf{1}_I(t)u \in V^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ and $\|u\|_{V^p(I)\mathcal{FL}^{s,r}} + \|u(a)\|_{\mathcal{FL}^{s,r}} = \|\mathbf{1}_I u\|_{V^p([-\infty; \infty))\mathcal{FL}^{s,r}}$.

(iii) For any $1 \le p < q < \infty$, there are the continuous embeddings $U^p(I)\mathcal{FL}^{s,r}(\mathcal{M}) \subset U^q\mathcal{FL}^{s,r}(\mathcal{M}) \subset L^\infty(I; \mathcal{FL}^{s,r}(\mathcal{M}))$ and $V^p(I)\mathcal{FL}^{s,r}(\mathcal{M}) \subset V^q(I)\mathcal{FL}^{s,r}(\mathcal{M}) \subset L^\infty(I; \mathcal{FL}^{s,r}(\mathcal{M}))$.

(iv) If $u \in U^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ with $u(b) = 0$, then $u \in V^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ and $\|u\|_{V^p(I)\mathcal{FL}^{s,r}} \lesssim \|u\|_{U^p(I)\mathcal{FL}^{s,r}}$.

Similarly, for any $1 \le p < q < \infty$, if $u \in V^p(I)\mathcal{FL}^{s,r}(\mathcal{M})$ is right-continuous with $u(a) = 0$, then $u \in U^q(I)\mathcal{FL}^{s,r}(\mathcal{M})$ and $\|u\|_{U^q(I)\mathcal{FL}^{s,r}} \lesssim \|u\|_{V^q(I)\mathcal{FL}^{s,r}}$.

**Definition 2.5.** We define the Banach space

$$
DU^2(I)\mathcal{FL}^{s,r}(\mathcal{M}) = \{\partial_t u, u \in U^2(I)\mathcal{FL}^{s,r}(\mathcal{M})\},
$$

where the derivative is taken in the sense of distributions, endowed with the norm
$$
\|f\|_{DU^2(I)\mathcal{FL}^{s,r}} = \|u\|_{U^2(I)\mathcal{FL}^{s,r}}. \quad \text{The condition } u(a) = 0 \text{ for } u \in U^p(I)\mathcal{FL}^{s,r}(\mathcal{M}) \text{ guarantees that for } f \in DU^2(I)\mathcal{FL}^{s,r}(\mathcal{M}) \text{ there is a unique choice of } u \in U^2(I)\mathcal{FL}^{s,r}(\mathcal{M}) \text{ such that } f = \partial_t u.
$$

The main result regarding this space is the duality relation between $DU^2(I)\mathcal{FL}^{s,r}(\mathcal{M})$ and $V^2(I)\mathcal{FL}^{-s,r'}(\mathcal{M})$; see [57] for the Banach-valued version.

**Proposition 2.6.** For all $f \in L^1(I; \mathcal{FL}^{s,r}(\mathcal{M})) \subset DU^2(I)\mathcal{FL}^{s,r}(\mathcal{M})$, it holds

$$
\|f\|_{DU^2(I)\mathcal{FL}^{s,r}} = \sup_{\|v\|_{V^2(I)\mathcal{FL}^{-s,r'}} \le 1} \left| \iint_M f(t,x) \bar{v}(t,x) dx dt \right|.
$$
As mentioned in the introduction, the adapted solution space $X_T^{s,r}$ for $u$ solving (1.1) is the replacement of the solution space $D^W(H^s)$ used by Chouk and Gubinelli in the context of modulated Schrödinger and KdV equations; see [20, Definition 2.2].

**Definition 2.7.** Let $s \in \mathbb{R}$, $r \in [1; \infty]$, and $1 \le p < \infty$. We define

$$X^{s,p,r}(I) = \{u : I \times \mathcal{M} \to \mathbb{K} : e^{W_t \mathcal{L}} u \in U^p(I) \mathcal{F}L^{s,r}(\mathcal{M})\}$$

and

$$Y^{s,p,r}(I) = \{u : I \times \mathcal{M} \to \mathbb{K} : e^{W_t \mathcal{L}} u \in V^p(I) \mathcal{F}L^{s,r}(\mathcal{M})\}$$

with norms given by

$$\|u\|_{X^{s,p,r}(I)} = \|e^{W_t \mathcal{L}} u\|_{U^p(I) \mathcal{F}L^{s,r}} \text{ and } \|u\|_{Y^{s,p,r}(I)} = \|e^{W_t \mathcal{L}} u\|_{V^p(I) \mathcal{F}L^{s,r}}.$$

When $I = [0; T)$ for some $T > 0$, we simply write $X^{s,p,r}(I) = X_T^{s,p,r}$ and $Y^{s,p,r}(I) = Y_T^{s,p,r}$.

Finally, when $p = 2$, we simply write $X_T^{s,r}$ and $Y_T^{s,r}$, and when $r = 2$, we just write $X_T^s$, $Y_T^s$.

**Remark 2.8.** Since $U^2 \mathcal{F}L^{s,r}(\mathcal{M}) \subset L^\infty(\mathbb{R}; \mathcal{F}L^{s,r}(\mathcal{M}))$ continuously, we have that for any $T > 0$, $C([0; T); \mathcal{F}L^{s,r}(\mathcal{M})) \cap X_T^{s,r}$ is a closed subspace of $C([0; T); \mathcal{F}L^{s,r}(\mathcal{M}))$.

**Proposition 2.9.** Let $s \in \mathbb{R}$ and $r \in [1; \infty]$. Then the following hold:

(i) We have the continuous embeddings

$$X^{s,r}(I) \subset Y^{s,r}(I) \subset X^{s,p,r}(I)$$

for any $2 < p < \infty$.

(ii) For any $T > 0$ and $\phi \in \mathcal{F}L^{s,r}(\mathcal{M})$, we have $e^{-(W_t-W_0)\mathcal{L}}\phi \in X_T^{s,r}$ and

$$\|e^{-(W_t-W_0)\mathcal{L}}\phi\|_{X_T^{s,r}} \le \|\phi\|_{\mathcal{F}L^{s,r}}.$$

(iii) For any $T > 0$ and $F \in L^1([0; T); \mathcal{F}L^{s,r}(\mathcal{M}))$, we have $t \mapsto \mathbf{1}_{[0;\infty)}(t) \int_0^t e^{-(W_t-W_s)\mathcal{L}}F(s)ds \in X_T^{s,r}$ and

$$\left\| \int_0^t e^{-(W_t-W_s)\mathcal{L}} F(s) ds \right\|_{X_T^{s,r}} \le \sup_{\substack{w \in Y_T^{-s,r'} \\ \|w\|_{Y_T^{-s,r'}} \le 1}} \left| \int_0^T \int_{\mathcal{M}} F(t,x) \bar{w}(t,x) dx dt \right|.$$

(iv) (Transference principle) If for some spatial function space $X$, some $s \in \mathbb{R}$, $r \in [1; \infty]$, $p \in [1; \infty)$, $\iota \in \{\pm\}$, and any $T \in (0; 1]$, there is $C_{X,s,r,p,\iota}(T) > 0$ non-decreasing in $T$ such that

$$\|e^{-W_t \mathcal{L}} \phi\|_{S_T^{p,\iota}(X)} \le C_{X,s,r,p,\iota}(T) \|\phi\|_{\mathcal{F}L^{s,r}}$$

holds for any $\phi \in \mathcal{F}L^{s,r}(\mathcal{M})$, then

$$\|u\|_{S_T^{p,\iota}(X)} \le C_{X,s,r,p,\iota}(T) \|u\|_{U_{\mathcal{W}\mathcal{L}}^p([0;T))\mathcal{F}L^{s,r}} \quad (2.2)$$

for any $u \in U_{\mathcal{W}\mathcal{L}}^p([0; T); \mathcal{F}L^{s,r}(\mathcal{M}))$.
*Proof.* The proof of (i) is straightforward from the definition of $X_T^{s,r}$ and $Y_T^{s,r}$ together with the embedding $V^2(I) \subset U^p(I)$ (Proposition 2.4 (iii)).

Similarly, (ii) follows directly from the definition of the space $X_T^{s,r}$.

The property (iii) is fundamental and established e.g. in [43, 48] for Hilbert-valued functions of $t$. Here we give a straightforward adaptation of the proof to the case of $\mathcal{F}L^{s,r}(\mathcal{M})$-valued function. Indeed, extending $t \mapsto \int_0^t e^{W_s\mathcal{L}}F(s)ds$ by $\int_0^T e^{W_s\mathcal{L}}F(s)ds$ for $t \ge T$, since $t \mapsto e^{W_t\mathcal{L}}F(t) \in L^1([0; T]; \mathcal{F}L^{s,r}(\mathcal{M}))$, we have by a $\mathcal{F}L^{s,r}(\mathcal{M})$-valued Riemann sum approximation that $t \mapsto \int_0^t e^{W_s\mathcal{L}}F(s)ds$ is in $U^2([0; T])\mathcal{F}L^{s,r}(\mathcal{M})$, and so $e^{W_t\mathcal{L}}F(t) \in DU^2([0; T])\mathcal{F}L^{s,r}(\mathcal{M})$ with $e^{W_t\mathcal{L}}F(t) = \partial_t(\int_0^t e^{W_s\mathcal{L}}F(s)ds)$. This shows that

$$ \left\| \int_0^t e^{-(W_t-W_s)\mathcal{L}} F(s)ds \right\|_{X_T^{s,r}} = \left\| e^{W_t\mathcal{L}} F \right\|_{DU^2([0;T])\mathcal{F}L^{s,r}}, $$

and the conclusion of (iii) follows from Proposition 2.6.

As for (iv), we have from the atomic structure of $U^p\mathcal{F}L^{s,r}(\mathcal{M})$ (Definition 2.3) and the definition of $U_{W\mathcal{L}}^p\mathcal{F}L^{s,r}(\mathcal{M})$ (Definition 2.7) that if

$$ u = \sum_{j=0}^{\infty} \lambda_j a_j \text{ with } a_j(t) = \sum_{k=1}^{K_j} \mathbf{1}_{[t_{j,k-1};t_{j,k})} e^{-W_t \mathcal{L}} \phi_{j,k} $$

for some $\{t_{j,k}\}_{k=0}^{K_j-1} \in \mathcal{Z}(T)$ and $\{\phi_{j,k}\}_{k=1}^{K_j} \in \mathcal{F}L^{s,r}(\mathcal{M})^{K_j}$ such that $\sum_{k=1}^{K_j} \|\phi_{j,k}\|_{\mathcal{F}L^{s,r}}^p = 1$, and some $\{\lambda_j\}_{j \in \ell^1(\mathbb{N})}$, then

$$ \begin{aligned} \|u\|_{S^{p,+}(X)} & \le \sum_{j=0}^{\infty} |\lambda_j| \left( \sum_{k=1}^{K_j} \int_{t_{j,k-1}}^{t_{j,k}} \|e^{-W_t \mathcal{L}} \phi_{j,k} \|_X^p dt \right)^{\frac{1}{p}} \\ & \le C_{X,s,r,p,\ell}(T) \sum_{j=0}^{\infty} |\lambda_j| \left( \sum_{k=1}^{K_j} \|\phi_{j,k}\|_{\mathcal{F}L^{s,r}}^p \right)^{\frac{1}{p}} \\ & \le C_{X,s,r,p,\ell}(T) \sum_{j=0}^{\infty} |\lambda_j|. \end{aligned} $$

We have a similar bound if we replace $L^p([0; T]; X(\mathcal{M}))$ by $X(\mathcal{M}; L^p([0; T]))$. This proves (2.2) in view of Definition 2.7 and Definition 2.3 (ii). $\square$

### 3. LOCAL WELL-POSEDNESS FOR GENERAL MODELS: PROOF OF THEOREM 1.4

We now present the proof of Theorem 1.4.

3.1. **Case (i).** We assume that (A0) holds, and that we have the space-time linear estimate (1.7) together with the non-linear estimate (1.8). We first treat the case of Strichartz estimates, namely (1.7) holds for some $(p, \zeta, \iota) \in \mathcal{E}$ with $\iota = +$ (recall that $p < \infty$ by assumption). This implies that for any $u_0 \in L^2(\mathcal{M})$, the map $z \mapsto \|e^{z\mathcal{L}}u_0\|_{X_\zeta^p}$ is integrable
on $[0;T]$, in particular measurable and positive. Then we can use the occupation time formula (2.1) together with (A0) and (1.7). Indeed, letting $I_W(T) := [W_0 - 2\|W\|_{L_T^\infty}; W_0 + 2\|W\|_{L_T^\infty}]$, for all $t \in [0;T]$ it holds $W_t \in I_W(T)$, we thus get that for any $(p, \zeta, +) \in \mathcal{E}$:

$$
\begin{align*}
\|e^{-(W_t-W_0)\mathcal{L}} u_0\|_{L_T^p X_\zeta}^p &= \int_0^T \mathbf{1}_{|W_t-W_0| \le 2\|W\|_{L_T^\infty}} \|e^{-(W_t-W_0)\mathcal{L}} u_0\|_{X_\zeta}^p dt \\
&= \int_{I_W(T)} \|e^{-z\mathcal{L}} e^{W_0\mathcal{L}} u_0\|_{X_\zeta}^p d\mu_{[0;T]}(z) \\
&\le \left\| \frac{d\mu_{[0;\cdot]}(\cdot)}{dz} \right\|_{L^\infty([0;T]\times I_W(T))} \|e^{-z\mathcal{L}} u_0\|_{L_{I_W(T)}^p X_\zeta}^p \\
&\le \left\| \frac{d\mu_{[0;\cdot]}(\cdot)}{dz} \right\|_{L^\infty([0;T]\times I_W(T))} C_{p,\zeta,+}(|I_W(T)|)^p \|u_0\|_{L^2}^p.
\end{align*}
$$

Similarly, if $L_T^p X_\zeta$ is replaced by $X_\zeta L_T^p$, we get for any fixed $x \in \mathcal{M}$

$$
\begin{align*}
\|e^{-(W_t-W_0)\mathcal{L}} u_0(x)\|_{L_T^p}^p &= \int_0^T \mathbf{1}_{I_W(T)}(W_t) |e^{-(W_t-W_0)\mathcal{L}} u_0(x)|^p dt \\
&= \int_{I_W(T)} |e^{-z\mathcal{L}} e^{W_0\mathcal{L}} u_0(x)|^p d\mu_{[0;T]}(z) \\
&\le \left\| \frac{d\mu_{[0;\cdot]}(\cdot)}{dz} \right\|_{L^\infty([0;T]\times I_W(T))} \|e^{-z\mathcal{L}} u_0(x)\|_{L_{I_W(T)}^p}^p.
\end{align*}
$$

Taking the $\frac{1}{p}$-th power then the $X_\zeta$-norm, together with (1.7) this yields

$$
\begin{align*}
\|e^{-(W_t-W_0)\mathcal{L}} u_0(x)\|_{X_\zeta L_T^p} &\le \left\| \frac{d\mu_{[0;\cdot]}(\cdot)}{dz} \right\|_{L^\infty([0;T]\times I_W(T))}^{\frac{1}{p}} \|e^{-z\mathcal{L}} u_0(x)\|_{X_\zeta L_{I_W(T)}^p} \\
&\le \left\| \frac{d\mu_{[0;\cdot]}(\cdot)}{dz} \right\|_{L^\infty([0;T]\times I_W(T))}^{\frac{1}{p}} C_{p,\zeta,-}(|I_W(T)|) \|u_0\|_{L^2}.
\end{align*}
$$

All in all we obtain

$$
\| \langle D \rangle^s e^{-(W_t - W_0) \mathcal{L}} u_0 \|_{S_T^{p, \iota} X_\zeta} \le C_{p, \zeta, \iota}^W (T) \| u_0 \|_{H^s}, \quad (3.1)
$$

for all $(p, \zeta, \iota) \in \mathcal{E}$, where

$$
C_{p,\zeta,\iota}^{W}(T) = \left\| \frac{d\mu_{[0:\cdot]}(\cdot)}{dz} \right\|_{L^{\infty}([0:T]\times I_{W}(T))}^{\frac{1}{p}} C_{p,\zeta,\iota}(|I_{W}(T)|). \quad (3.2)
$$

From the definition (1.6) of $S_T^s$, the transference principle of Proposition 2.9 (iv) and the embeddings of Proposition 2.4 (ii), we deduce that

$$
\|u\|_{S_T^s} \le C_W(T) \|u\|_{U_{W\mathcal{L}}^p(T)H^s}, \quad (3.3)
$$

where

$$
C_W(T) = \max_{(p, \zeta, \iota) \in \mathcal{E}} C_{p, \zeta, \iota}^W(T), \quad (3.4)
$$

and

$$
p_{\mathcal{E}} = \min \{p > 2, \exists (\zeta, \iota) \in E \times \{\pm\}, (p, \zeta, \iota) \in \mathcal{E}\}.
$$

Recall that $p_{\mathcal{E}} > 2$ from our assumption that $\mathcal{E} \subset (2; \infty) \times E \times \{\pm\}$ is finite.
From the nonlinear estimate (1.8), the linear estimate (3.3), and the embeddings of Propositions 2.4 (ii) and 2.9 (iv), we get

$$
\begin{align}
& \left| \int_0^T \int_M (\mathcal{N}(u) - \mathcal{N}(v)) \overline{w} dx dt \right| \nonumber \\
& \leq CT^{\eta_{\varepsilon}(s)} \|u - v\|_{\mathbf{S}_T^s} (\|u\|_{\mathbf{S}_T^s} + \|v\|_{\mathbf{S}_T^s})^r \|w\|_{\mathbf{S}_T^{-s}} \nonumber \\
& \leq CT^{\eta_{\varepsilon}(s)} C_W(T)^{r+2} \|u - v\|_{U_{W\mathcal{L}}^{p_{\varepsilon}}(T)H^s} (\|u\|_{U_{W\mathcal{L}}^{p_{\varepsilon}}(T)H^s} + \|v\|_{U_{W\mathcal{L}}^{p_{\varepsilon}}(T)H^s})^r \|w\|_{U_{W\mathcal{L}}^{p_{\varepsilon}}(T)H^{-s}} \nonumber \\
& \leq CT^{\eta_{\varepsilon}(s)} C_W(T)^{r+2} \|u - v\|_{X_T^s} (\|u\|_{X_T^s} + \|v\|_{X_T^s})^r \|w\|_{Y_T^{-s}}. \tag{3.5}
\end{align}
$$

Note that in the last step we used that $p_{\varepsilon} > 2$ to estimate $\|w\|_{U_{W\mathcal{L}}^{p_{\varepsilon}}(T)H^{-s}} \lesssim \|w\|_{Y_T^{-s}}$.

Then, for $R, T > 0$, we define the ball

$$
B_R(T) := \{u \in C([0; T]; H^s(\mathcal{M})) \cap X_T^s, \|u\|_{X_T^s} \le R\},
$$

which is closed in $X_T^s$, hence a complete metric space when endowed with the metric
inherited from $\| \cdot \|_{X_T^s}$.

Given $u_0 \in H^s(\mathcal{M})$, we set $R = 2A\|u_0\|_{H^s}$ for some constant $A \ge \max(1, C_W(T))$ to be
chosen later, and consider the map

$$
\Phi_{u_0} : u \in B_R(T) \mapsto e^{-(W_t - W_0)\mathcal{L}} u_0 + \int_0^t e^{-(W_t - W_{t'})\mathcal{L}} \mathcal{N}(u)(t') dt'.
$$

Note that from (3.5) and the embedding $Y_T^{-s} \subset L^\infty([0; T]; H^{-s}(\mathcal{M}))$, we have that

$$
\begin{align*}
\| \mathcal{N}(u) - \mathcal{N}(v) \|_{L_T^1 H^s} &= \sup_{\|w\|_{L_T^\infty H^{-s}} \le 1} \left| \int_0^T \int_M (\mathcal{N}(u) - \mathcal{N}(v)) \overline{w} dx dt \right| \\
&\lesssim CT^{\eta_{\varepsilon}(s)} C_W(T)^{r+2} \|u - v\|_{X_T^s} (\|u\|_{X_T^s} + \|v\|_{X_T^s})^r \|w\|_{Y_T^{-s}} \\
&\lesssim CT^{\eta_{\varepsilon}(s)} C_W(T)^{r+2} \|u - v\|_{X_T^s} (\|u\|_{X_T^s} + \|v\|_{X_T^s})^r.
\end{align*}
$$

This shows that $\mathcal{N}(u)-\mathcal{N}(v) \in L^1([0;T]; H^s(\mathcal{M}))$ for any $u,v \in X_T^s$. By Proposition 2.9 (ii) and (iii), we thus have

$$
\| \Phi_{u_0}(u) - \Phi_{v_0}(v) \|_{X_T^s} \le \|u_0 - v_0\|_{H^s} + \sup_{\substack{w \in Y_T^s \\ \|w\|_{Y_T^{-s}} \le 1}} \left| \int_0^T \int_M (\mathcal{N}(u) - \mathcal{N}(v)) \overline{w} dx dt \right|.
$$

Thus, with (3.5) we find

$$
\| \Phi_{u_0}(u) - \Phi_{v_0}(v) \|_{X_T^s} \le \|u_0 - v_0\|_{H^s} + CT^{\eta_{\varepsilon}(s)} C_W(T)^{r+2} (2R)^r \|u - v\|_{X_T^s} \quad (3.6)
$$

for any $u, v \in B_R(T)$.

First, in case where the equation (1.4) is sub-critical, we have $\eta_{\varepsilon}(s) > 0$, so that we can choose $A$ such that $R \ge \max(1, \|u_0\|_{H^s})$, then $T \sim R^{-\frac{r}{\eta_{\varepsilon}(s)}} \in (0; 1)$ and $C_W(T) \le C_W(1)$, which guarantee that $\Phi_{u_0}$ maps $B_R(T)$ to $B_R(T)$. With our choice of $R$ and $T$ as above, we deduce that $\Phi_{u_0}$ is also a contraction on $B_R(T)$, thus having a unique fixed point $u \in B_R(T)$. If $v \in C([0; T]; H^s(\mathcal{M})) \cap X_T^s$ is another solution, with $R' = \max(R; \|v\|_{X_T^s})$ and $\tau = \tau(R') \le T$, the argument above shows that $v \equiv u$ on $[0; \tau]$. But then iterating this argument on $[\tau; 2\tau]$, etc., shows that $v \equiv u$ on the whole time interval where $u$ is defined,
showing uniqueness in the whole class $C([0; T]; H^s(M)) \cap X_T^s$. The estimate (3.6) also provides the Lipschitz dependence of the flow map on the initial data.

On the other hand, for critical equations, one has $\eta_\epsilon(s) = 0$, and thus neglecting again the term $C_W(T)$, which is only bounded under (A0), we have to take $R \ll 1$, which gives small data local well-posedness, similarly as for the deterministic equation (1.4).

**3.2. Case (ii).** If now (A0)* holds instead of (A0), the same discussion as in the previous subsection still holds and leads again to (3.6). However, from (3.2) and the fact that now $T \mapsto \|\frac{d\mu_{[0;T]}}{dz}\|_{L^\infty}$ is continuous and vanishes at $T = 0$, we see that

$$C_W(T) \underset{T \searrow 0}{\longrightarrow} 0. \qquad (3.7)$$

Thus, even if $\eta_\epsilon(s) = 0$ in the critical case, we can choose again A such that $R \ge \max(1, \|u_0\|_{H^s})$, and then $T = T(\|u_0\|_{H^s}) \in (0; 1)$ small enough guaranteeing again that $\Phi_{u_0}$ defines a contraction on $B_R(T)$.

Finally, if $s = 0$ and if (A2) also holds, then from the previous argument we have subcritical local well-posedness in the sense that for any $u_0 \in L^2(M)$, there is $T = T(\|u_0\|_{L^2}) > 0$ and a unique solution $u \in C([0; T]; L^2(M)) \cap X_T^0$. But since $\|u(T)\|_{L^2} = \|u_0\|_{L^2}$ from (A2), we can iterate the local well-posedness on $[T; 2T]$, then $[kT; (k+1)T]$ for any $k \in \mathbb{N}$. This shows global well-posedness of (1.1) in this case.

**Remark 3.1.** The gain (3.7) from the noise $W_t$ under assumption (A0)* is similar in spirit to the use of the refined rescaling transform used to study mass-critical equations with multiplicative noise in [3, 46]. While the latter relies heavily on stochastic properties of the noise, here this gain is entirely deterministic and only relies on the irregularity property of $W_t$.

**Remark 3.2.** In the proof of Theorem 1.4 above, the transfer of the linear space-time estimate (1.7) for the deterministic equation (1.4) to the modulated one (1.1) simply follows from the occupation time formula (2.1):

$$\int_0^T \|e^{-W_t \mathcal{L}} u\|_X^p dt = \int_{I_W(T)} \|e^{-z \mathcal{L}} u\|_X^p \frac{d\mu_{[0;T]}}{dz}(z) dz,$$

from which we got an upper bound in the proof of Theorem 1.4.

In case

$$\inf_{z \in I_W(T)} \frac{d\mu_{[0;T]}}{dz}(z) > 0,$$

this also provides a lower bound

$$\int_0^T \|e^{-W_t \mathcal{L}} u\|_X^p dt \gtrsim \int_{I_W(T)} \|e^{-z \mathcal{L}} u\|_X^p dz.$$

This shows in this case that one cannot get an improvement for the range of exponents in (1.7) for the modulated equation compared to the deterministic one. Note that for the standard Brownian motion, the local time at zero $\frac{d\mu_{[0;T]}}{dz}(0) = L_T(0)$ has the same law as $\frac{1}{2} \max_{[0;T]} W_t$ [75], which is a.s. positive. Thus the assumption above is satisfied for $T > 0$
small enough, which shows the sharpness of Theorem 1.4 in case of a Brownian modulation. In the case of the Schrödinger equation with white noise dispersion, this goes against the intuition that from the usual dispersion inequality $||e^{i(W_t-W_\tau)\Delta}u||_{L^\infty_x} \lesssim |W_t - W_\tau|^{-\frac{d}{2}}||u||_{L^1_x}$ and the scaling $W_t - W_\tau \sim \sqrt{t-\tau}$, we would have expected a better (i.e. more integrable in 0) time decay, locally. Thus, through the usual $TT^*$ argument via Hardy-Littlewood-Sobolev inequality, we would have expected a better range of exponents $(p, q)$ for the local in time Strichartz estimate with white noise dispersion.

### 3.3. Proofs of Example 1.
In this subsection we detail how the various models presented in Example 1 fit into the framework of Theorem 1.4.

#### 3.3.1. NLS equation on $\mathbb{R}^d$.
In this case, we have $M = \mathbb{R}^d$, $\mathcal{L} = i\Delta$, and $N(u) = \pm i|u|^{2m}u$ with $m \in \mathbb{N}^*$. In particular (1.9) satisfies assumptions (A1) and (A2). The deterministic NLS equation (1.9) is locally well-posed in $H^s(\mathbb{R}^d)$ for any $s \ge \max(s_c, 0) = \max(\frac{d}{2} - \frac{1}{m}, 0)$. Since the case $s > \frac{d}{2}$ can be treated simply using the algebra property of $H^s(\mathbb{R}^d)$ (i.e. there is no need for Strichartz estimates in this case), we focus on the case $\max(s_c, 0) \le s < \frac{d}{2}$. We mainly follow the arguments in [14, 62] for the sub-critical case, and [15] for the critical case.

Then (1.7) corresponds to the Strichartz estimate (see e.g. [14, Theorem 2.3.3] or [62, Theorem 4.2])

$$ \|e^{it\Delta}u_0\|_{L_t^p L_x^q} \le c\|u_0\|_{L^2} \quad (3.8) $$

for any $p, q \in [2; \infty]$ satisfying the admissibility condition

$$ \frac{2}{p} + \frac{d}{q} = \frac{d}{2} \quad \text{and} \quad \begin{cases} q \in [2; \infty], \ d = 1; \\ q \in [2; \infty), \ d = 2; \\ q \in [2; \frac{2d}{d-2}), \ d \ge 3. \end{cases} \quad (3.9) $$

Namely, we have $E = [2; \infty]$, $X_q = L_x^q$ for $q \in E$, so that $S_T^{p,+}X = L_T^p L_x^q$ and $\tilde{\mathcal{E}} = \{(p, q, +), (p, q) \text{ admissible}\}$. Actually, to deal with the scaling critical case, we use the refined space

$$ X_q(\mathbb{R}^d) = B_{q,2}^0(\mathbb{R}^d), \quad (3.10) $$

so that (1.7) still follows from (3.8) in this case:

$$ \begin{aligned} \|e^{it\Delta}u_0\|_{S_T^{p,+}X_q} &= \|e^{it\Delta}\mathbf{P}_N u_0\|_{L_T^p \ell_N^2 L_x^q} \\ &\le \|e^{it\Delta}\mathbf{P}_N u_0\|_{\ell_N^2 L_T^p L_x^q} \\ &\le c\|\mathbf{P}_N u_0\|_{\ell_N^2 L_x^2} \sim \|u_0\|_{L_x^2}, \end{aligned} $$

where we used Minkowski's inequality in the second step. Since we assumed $\mathcal{E}$ to be finite in Theorem 1.4, we will restrict $\tilde{\mathcal{E}}$ to a suitable finite set $\mathcal{E}$ in the following.
Since $\mathcal{N}$ is multilinear in our case (recall that $m \in \mathbb{N}^*$), showing (1.8) boils down to estimating

$$ \int_0^T \int_{\mathbb{R}^d} \prod_{j=0}^{2m+1} u_j^{\pm_j} dxdt = \int_0^T \int_{\mathbb{R}^d} \sum_{N_0, \dots, N_{2m+1}} \prod_{j=0}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm_j} dxdt $$

for any $u_0 \in \mathbf{S}_T^{-s}$ and $u_j \in \mathbf{S}_T^s$, $j = 1, \dots, 2m+1$, and any choice of signs $\pm_j$. Then, since our analysis does not depend on the signs $\pm_j$, by symmetry we can assume that $N_1 = \max(N_1, \dots, N_{2m+1})$. Note that from the assumption on $N_1$ and the convolution constraint when writing the product on the Fourier side using Plancherel theorem, we get the conditions $N_0 \lesssim N_1$ and $N_1 \sim \max(N_0, \max(N_2, \dots, N_{2m+1}))$. In particular, if $N_0 \ge \max(N_2, \dots, N_{2m+1})$, (1.8) is reduced to estimating

$$ \int_0^T \int_{\mathbb{R}^d} \sum_{N_1 \sim N_0} \left( \prod_{j=0}^1 \mathbf{P}_{N_j} u_j^{\pm_j} \right) \left( \prod_{j=2}^{2m+1} \mathbf{P}_{\le N_0} u_j^{\pm_j} \right) dxdt, \quad (3.11) $$

while in the case $N_0 \le \max(N_2, \dots, N_{2m+1})$ need to estimate

$$ \int_0^T \int_{\mathbb{R}^d} \sum_{N_1 \sim N_2} \left( \prod_{j=1}^2 \mathbf{P}_{N_j} u_j^{\pm_j} \right) \left( \mathbf{P}_{\le N_2} u_0^{\pm_0} \prod_{j=3}^{2m+1} \mathbf{P}_{\le N_2} u_j^{\pm_j} \right) dxdt. \quad (3.12) $$

We first treat the case $s > \max(s_c, 0)$. In this case, taking $0 < \delta < s - s_c < \frac{1}{m}$, we simply restrict $\tilde{\mathcal{E}}$ to

$$ \mathcal{E} = \left\{ \left( \frac{2m}{1 - m\delta}, \frac{2dm}{dm - 2 + 2m\delta}, + \right) \right\}, $$

so that $\frac{d}{q} = s_c + \delta < s$ and $p > 2m$. Then in the case $N_0 \ge N_2$, (1.8) follows from estimating (3.11) using Hölder, Sobolev and Cauchy-Schwarz inequalities:

$$
\begin{aligned}
& \left| \int_0^T \int_{\mathbb{R}^d} \sum_{N_0 \sim N_1} \left( \prod_{j=0}^1 \mathbf{P}_{N_j} u_j^{\pm_j} \right) \left( \prod_{j=2}^{2m+1} \mathbf{P}_{\le N_0} u_j^{\pm_j} \right) dxdt \right| \\
& \le \left\| \sum_{N_0 \sim N_1} \left( \prod_{j=0}^1 \| \mathbf{P}_{N_j} u_j \|_{L_x^2} \right) \left( \prod_{j=2}^{2m+1} \| \mathbf{P}_{\le N_0} u_j \|_{L_x^\infty} \right) \right\|_{L_T^1} \\
& \lesssim \left\| \| u_0 \|_{H_x^{-s}} \| u_1 \|_{H_x^s} \prod_{j=2}^{2m+1} \| u_j \|_{W_x^{s,q}} \right\|_{L_T^1} \\
& \lesssim \| u_0 \|_{L_T^\infty H_x^{-s}} \| u_1 \|_{L_T^\infty H_x^s} \prod_{j=2}^{2m+1} \| u_j \|_{L_T^{2m} W_x^{s,q}} \\
& \lesssim T^{m\delta} \| u_0 \|_{L_T^\infty H_x^{-s}} \| u_1 \|_{L_T^\infty H_x^s} \prod_{j=2}^{2m+1} \| u_j \|_{L_T^p B_{q,2}^s},
\end{aligned}
$$
where $(p, q, +) \in \mathcal{E}$, and in the last step we used Hölder's inequality and Lemma 2.2. Similarly, (3.12) is estimated by

$$
\begin{aligned}
& \left| \int_0^T \int_{\mathbb{R}^d} \sum_{N_1 \sim N_2} \left( \prod_{j=1}^2 \mathbf{P}_{N_j} u_j^{\pm_j} \right) \left( \mathbf{P}_{\le N_2} u_0^{\pm_0} \prod_{j=3}^{2m+1} \mathbf{P}_{\le N_2} u_j^{\pm_j} \right) dxdt \right| \\
& \lesssim \left\| \sum_{N_1 \sim N_2} \| \mathbf{P}_{N_1} u_1 \|_{L_x^2} \| \mathbf{P}_{N_2} u_2 \|_{L_x^\infty} N_2^s \| \mathbf{P}_{\le N_2} u_0 \|_{H^{-s}} \prod_{j=3}^{2m+1} \| u_j \|_{L_x^\infty} \right\|_{L_T^1} \\
& \lesssim \left\| \sum_{N_1 \sim N_2} N_1^s \| \mathbf{P}_{N_1} u_1 \|_{L_x^2} N_2^s \| \mathbf{P}_{N_2} u_2 \|_{L_x^q} \| u_0 \|_{L_T^\infty H^{-s}} \prod_{j=3}^{2m+1} \| u_j \|_{W_x^{s,q}} \right\|_{L_T^1} \\
& \lesssim T^{m\delta} \| u_1 \|_{L_T^\infty H_x^s} \| u_0 \|_{L_T^\infty H^{-s}} \prod_{j=2}^{2m+1} \| u_j \|_{L_T^p B_{q,2}^s}.
\end{aligned}
$$

This shows (1.8) with $\eta_{\mathcal{E}}(s) = m\delta$ in view of the definitions (1.5) and (3.10) of $S_T^{p,+}X_q$ and $X_q$, and (1.6) of $S_T^s$.

Next, we treat the critical case $s = s_c = \frac{d}{2} - \frac{1}{m} \ge 0$, and take instead

$$ \mathcal{E} = \left\{ (2m + 2, \frac{d(2m + 2)}{d(m + 1) - 2}, +) \right\} $$

so that for $(p, q, +) \in \mathcal{E}$, $(p, q)$ is admissible. Let also $\tilde{q} = dm(m + 1)$, so that

$$ d\left(\frac{1}{q} - \frac{1}{\tilde{q}}\right) = s_c \quad \text{and} \quad \frac{2}{q} + \frac{2m}{\tilde{q}} = 1. $$

Then we estimate (3.11) by

$$
\begin{aligned}
& \left| \int_0^T \int_{\mathbb{R}^d} \sum_{N_0 \sim N_1} \left( \prod_{j=0}^1 \mathbf{P}_{N_j} u_j^{\pm_j} \right) \left( \prod_{j=2}^{2m+1} \mathbf{P}_{\le N_0} u_j^{\pm_j} \right) dxdt \right| \\
& \le \left\| \sum_{N_0 \sim N_1} \left( \prod_{j=0}^1 \| \mathbf{P}_{N_j} u_j \|_{L_x^q} \right) \left( \prod_{j=2}^{2m+1} \| \mathbf{P}_{\le N_0} u_j \|_{L_x^{\tilde{q}}} \right) \right\|_{L_T^1} \\
& \lesssim \left\| \sum_{N_0 \sim N_1} \left( \prod_{j=0}^1 \| \mathbf{P}_{N_j} u_j \|_{L_x^q} \right) \left( \prod_{j=2}^{2m+1} \| u_j \|_{B_{q,2}^s} \right) \right\|_{L_T^1} \\
& \lesssim \| u_0 \|_{L_T^p B_{q,2}^{-s}} \prod_{j=1}^{2m+1} \| u_j \|_{L_T^p B_{q,2}^s}.
\end{aligned}
$$
We estimate (3.12) similarly by

$$
\begin{aligned}
& \left| \int_0^T \int_{\mathbb{R}^d} \sum_{N_1 \sim N_2} \left( \prod_{j=1}^2 \mathbf{P}_{N_j} u_j^{\pm_j} \right) \left( \mathbf{P}_{\le N_2} u_0^{\pm_0} \prod_{j=3}^{2m+1} \mathbf{P}_{\le N_2} u_j^{\pm_j} \right) dx dt \right| \\
& \lesssim \left\| \sum_{N_1 \sim N_2} \| \mathbf{P}_{N_1} u_1 \|_{L_x^q} \| \mathbf{P}_{N_2} u_2 \|_{L_x^{\tilde{q}}} \| \mathbf{P}_{\le N_2} u_0 \|_{L_x^q} \prod_{j=3}^{2m+1} \| u_j \|_{L_x^{\tilde{q}}} \right\|_{L_T^1} \\
& \lesssim \left\| \sum_{N_1 \sim N_2} N_1^s \| \mathbf{P}_{N_1} u_1 \|_{L_x^q} N_2^s \| \mathbf{P}_{N_2} u_2 \|_{L_x^q} \| u_0 \|_{B_{q,2}^{-s}} \prod_{j=3}^{2m+1} \| u_j \|_{B_{q,2}^s} \right\|_{L_T^1} \\
& \lesssim \| u_0 \|_{L_T^p B_{q,2}^{-s}} \prod_{j=1}^{2m+1} \| u_j \|_{L_T^p B_{q,2}^s}.
\end{aligned}
$$

This finally shows (1.8) in the case $s = s_c$, with $\eta_{\mathcal{E}}(s_c) = 0$.

The only remaining case is $s \ge 0 > s_c$, which corresponds only to the 1d-cubic case, i.e. $d=m=1$. In this case we take $\mathcal{E} = \{(8,4,+\)}$ since $(8,4)$ is admissible, and estimate exactly as above

$$
\begin{aligned}
& \left| \int_0^T \int_{\mathbb{R}^d} \sum_{N_0 \sim N_1} \left( \prod_{j=0}^1 \mathbf{P}_{N_j} u_j^{\pm_j} \right) \left( \prod_{j=2}^3 \mathbf{P}_{\le N_0} u_j^{\pm_j} \right) dx dt \right| \\
& \lesssim \left\| \sum_{N_0 \sim N_1} N_0^{-s} N_1^s \left( \prod_{j=0}^1 \| \mathbf{P}_{N_j} u_j \|_{L_x^4} \right) \left( \prod_{j=2}^3 \| u_j \|_{L_x^4} \right) \right\|_{L_T^1} \\
& \lesssim T^{\frac{1}{2}} \| u_0 \|_{L_T^8 B_{4,2}^{-s}} \| u_1 \|_{L_T^8 B_{4,2}^s} \prod_{j=2}^3 \| u_j \|_{L_T^8 B_{4,2}^s},
\end{aligned}
$$

and similarly

$$
\begin{aligned}
& \left| \int_0^T \int_{\mathbb{R}^d} \sum_{N_1 \sim N_2} \left( \prod_{j=1}^2 \mathbf{P}_{N_j} u_j^{\pm_j} \right) \left( \mathbf{P}_{\le N_2} u_0^{\pm_0} \mathbf{P}_{\le N_2} u_4^{\pm_4} \right) dx dt \right| \\
& \lesssim \left\| \sum_{N_1 \sim N_2} \| \mathbf{P}_{N_1} u_1 \|_{L_x^4} \| \mathbf{P}_{N_2} u_2 \|_{L_x^4} \| \mathbf{P}_{\le N_2} u_0 \|_{L_x^4} \| u_4 \|_{L_x^4} \right\|_{L_T^1} \\
& \lesssim \left\| \sum_{N_1 \sim N_2} N_1^s \| \mathbf{P}_{N_1} u_1 \|_{L_x^4} N_2^s \| \mathbf{P}_{N_2} u_2 \|_{L_x^4} \| u_0 \|_{B_{4,2}^{-s}} \| u_4 \|_{L_x^4} \right\|_{L_T^1} \\
& \lesssim T^{\frac{1}{2}} \| u_0 \|_{L_T^8 B_{4,2}^{-s}} \prod_{j=1}^3 \| u_j \|_{L_T^8 B_{4,2}^s}.
\end{aligned}
$$

3.3.2. **NLS equation on $\mathbb{T}^d$.** We now consider the case $\mathcal{M} = \mathbb{T}^d$, $\mathcal{L} = i\Delta$, and $\mathcal{N}(u) = \pm i|u|^{2m}u$ with $m \in \mathbb{N}^*$. We have again $s_c = \frac{d}{2} - \frac{1}{m}$ and (1.9) is well-posed in $H^s(\mathbb{T}^d)$ for any $s > \max(s_c, 0)$, and for $s = \max(s_c; 0)$ except in the cases $d=1, m=2$ and $d=2, m=1$ [48, 87]. In the following, we focus on the sub-critical case$^5 s > \max(s_c, 0)$.

<footnote>$^5$In the 1d-cubic case, this yields the restriction $s > 0$. See Remark 3.3 below.</footnote>
In this case, (1.7) follows from the localized periodic Strichartz estimate

$$ \|\Pi_N e^{it\Delta} u_0\|_{L_{t,x}^p(\mathbb{T}\times\mathbb{T}^d)} \lesssim \Lambda_N(d,p) \|\Pi_N u_0\|_{L_x^2}, \quad (3.13) $$

with

$$ \Lambda_N(d,p) = \begin{cases} 1, & d=1, p=4, \\ N^\varepsilon, & 2 \le p \le \frac{2(d+2)}{d}, \\ N^{\frac{d}{2}-\frac{d+2}{p}}, & p > \frac{2(d+2)}{d}. \end{cases} $$

Using number theoretic argument, Bourgain [6] first established part of (3.13) for some range of $p$ and $d$. This was then extended to the whole range in (3.13) using the break-through of Bourgain and Demeter [7] regarding the decoupling conjecture, and refined in [53] in the range $p > \frac{2(d+2)}{d}$.

For $\tilde{N}$ a dyadic integer, let $\mathcal{C}(\tilde{N}) = \{\tilde{n} + [-\tilde{N}; \tilde{N}]^d\}$ be a collection of cubes of side-length $\tilde{N}$ covering $\mathbb{R}^d$. Note that for $\mathcal{C} = \tilde{n} + [-\tilde{N}; \tilde{N}]^d \in \mathcal{C}(\tilde{N})$, we have

$$ \mathbf{P}_C u = e^{i\tilde{n}\cdot x} \mathbf{P}_{[-\tilde{N}; \tilde{N}]^d} (e^{-i\tilde{n}\cdot x} u) $$

and

$$ \mathbf{P}_C(e^{it\Delta}u)(t,x) = e^{i\tilde{n}\cdot x+t|\tilde{n}|^2} \mathbf{P}_{[-\tilde{N};\tilde{N}]^d}(e^{-i\tilde{n}\cdot x}u)(t,x-2t\tilde{n}), $$

so that (3.13) also gives for any $N, \tilde{N}$ dyadic integers and $\mathcal{C} \in \mathcal{C}(\tilde{N})$:

$$ \|\mathbf{P}_C \mathbf{P}_N e^{it\Delta} u_0\|_{L_{t,x}^p(\mathbb{T}\times\mathbb{T}^d)} \lesssim \Lambda_{\tilde{N}\wedge N}(d,p) \|\mathbf{P}_C \mathbf{P}_N u_0\|_{L_x^2}. \quad (3.14) $$

Thus, for $p \ge 2$ and $0 < \varepsilon \ll 1$, letting

$$ \sigma(p) = \begin{cases} 2\varepsilon, & 2 \le p \le \frac{2(d+2)}{d}, \\ \frac{d}{2} - \frac{d+2}{p} + \varepsilon, & p > \frac{2(d+2)}{d}, \end{cases} $$

and using Minkowski's inequality and (3.14), we get that for any $p \ge 2$ and $T \in (0; 2\pi)$:

$$ \begin{aligned} \|\mathbf{1}_{\tilde{N} \lesssim N} \tilde{N}^{-\sigma(p)} \mathbf{P}_C \mathbf{P}_N e^{it\Delta} u_0\|_{L_T^p \ell_N^2 \ell_N^{\frac{1}{2}} \ell^2(\mathcal{C}(\tilde{N})) L_x^p} & \\ \lesssim \|\mathbf{1}_{\tilde{N} \lesssim N} \tilde{N}^{-\sigma(p)} \mathbf{P}_C \mathbf{P}_N e^{it\Delta} u_0\|_{\ell_N^{\frac{1}{2}} \ell_N^{\frac{1}{2}} \ell^2(\mathcal{C}(\tilde{N})) L_{T,x}^p} & \\ \lesssim \|\mathbf{1}_{\tilde{N} \lesssim N} \tilde{N}^{-\varepsilon} \mathbf{P}_C \mathbf{P}_N u_0\|_{\ell_N^{\frac{1}{2}} \ell_N^{\frac{1}{2}} \ell^2(\mathcal{C}(\tilde{N})) L_x^2} & \\ \sim \sum_{\tilde{N}} \tilde{N}^{-\varepsilon} \|\mathbf{P}_{\gtrsim \tilde{N}} u_0\|_{L^2} \lesssim \|u_0\|_{L^2}. & \end{aligned} \quad (3.15) $$

To capture the refinement (3.14), we thus define the spaces $X_\zeta$ with a multiscale structure:

$$ \|u\|_{X_p} := \|\mathbf{1}_{\tilde{N} \lesssim N} \tilde{N}^{-\sigma(p)} \mathbf{P}_C \mathbf{P}_N u\|_{\ell_N^2 \ell_N^{\frac{1}{2}} \ell^2(\mathcal{C}(\tilde{N})) L_x^p}, \quad (3.16) $$

so that (1.7) follows from (3.15) for any $(p, q, +) \in \tilde{\mathcal{E}}$, with

$$ \tilde{\mathcal{E}} := \{(p, p, +), p \in [2; \infty]\}. $$

Since we have $s > \max(s_c, 0)$, taking $0 < \varepsilon \ll s - \max(s_c, 0)$, we set $p = (d+2)m + \varepsilon$ and $q = \frac{2(d+2)}{d}$ so that $\frac{2}{q} + \frac{2m}{(d+2)m} = 1$. Then with these choices of $p$ and $q$ we restrict $\tilde{\mathcal{E}}$ to

$$ \mathcal{E} = \{(p, p, +), (q, q, +)\}. $$
Note that with this choice, we have $p > \frac{2(d+2)}{d}$ unless $d = m = 1$, and $q \le \frac{2(d+2)}{d}$, and that in all cases $\sigma(q) = 2\epsilon$ and $\sigma(p) = \max(s_c, 0) + O(\epsilon)$.

To establish (1.8), we decompose again

$$ \int_0^T \int_{\mathbb{T}^d} \prod_{j=0}^{2m+1} u_j^{\pm_j} dxdt = \int_0^T \int_{\mathbb{T}^d} \sum_{N_0, \dots, N_{2m+1}} \prod_{j=0}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm_j} dxdt $$

with now $N_1 \ge N_2 \ge \dots \ge N_{2m+1}$ by symmetry. We have again $N_0 \lesssim N_1$ and $N_1 \sim \max(N_0, N_2)$. In this case, we make a further decomposition: for example, if $N_0 \ge N_2$, since $\mathbf{P}_{N_0}u_0$ has its Fourier support in $\{|n| \sim N_1\}$, we can cover this region by $O(N_1^d N_2^{-d})$ cubes $C_\ell \in C(N_2)$, $|\ell| \lesssim N_1^d N_2^{-d}$, of side length $N_2$ and centred around some $|n_\ell| \sim N_1$. This leads to the decomposition

$$ \sum_{N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}} \sum_{C_\ell, C_{\tilde{\ell}} \in C(N_2)} \int_0^T \int_{\mathbb{T}^d} \mathbf{P}_{C_\ell} \mathbf{P}_{N_0} u_0^{\pm_0} \mathbf{P}_{C_{\tilde{\ell}}} \mathbf{P}_{N_1} u_1^{\pm_1} \prod_{j=2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm_j} dxdt. $$

Now, using again the convolution constraint when writing the product on the Fourier side, we see that for any fixed $|\ell| \lesssim N_1^d N_2^{-d}$, there are at most $O(1)$ values of $\tilde{\ell}$ such that the Fourier supports of $\mathbf{P}_{C_\ell} \mathbf{P}_{N_0} u_0^{\pm_0}$ and $\mathbf{P}_{C_{\tilde{\ell}}} \mathbf{P}_{N_1} u_1^{\pm_1} \prod_{j=2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm_j}$ are over-lapping, i.e. the integral on $\mathbb{T}^d$ is non-zero. Namely, similarly as in (3.11)-(3.12), we are left with estimating

$$ \sum_{N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}} \sum_{C_\ell, C_{\tilde{\ell}} \in C(N_2) \atop |\ell - \tilde{\ell}| \le 1} \int_0^T \int_{\mathbb{T}^d} \mathbf{P}_{C_\ell} \mathbf{P}_{N_0} u_0^{\pm_0} \mathbf{P}_{C_{\tilde{\ell}}} \mathbf{P}_{N_1} u_1^{\pm_1} \prod_{j=2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm_j} dxdt \quad (3.17) $$

and

$$ \sum_{N_2 \sim N_1 \ge N_0 \ge \dots \ge N_{2m+1}} \sum_{C_\ell, C_{\tilde{\ell}} \in C(N_0) \atop |\ell - \tilde{\ell}| \le 1} \int_0^T \int_{\mathbb{T}^d} \mathbf{P}_{C_\ell} \mathbf{P}_{N_2} u_2^{\pm_2} \mathbf{P}_{C_{\tilde{\ell}}} \mathbf{P}_{N_1} u_1^{\pm_1} \prod_{j=0,2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm_j} dxdt. \quad (3.18) $$

For $T \in (0; 2\pi)$, we then estimate (3.17) with the help of Hölder's inequality:

$$ \left| \sum_{N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}} \sum_{C_\ell, C_{\tilde{\ell}} \in C(N_2) \atop |\ell - \tilde{\ell}| \le 1} \int_{\mathbb{T}^d} \mathbf{P}_{C_\ell(N_2)} \mathbf{P}_{N_0} u_0^{\pm_0} \mathbf{P}_{C_{\tilde{\ell}}(N_2)} \mathbf{P}_{N_1} u_1^{\pm_1} \prod_{j=2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm_j} dx \right| \\ \lesssim \sum_{N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}} \sum_{C_\ell, C_{\tilde{\ell}} \in C(N_2) \atop |\ell - \tilde{\ell}| \le 1} \|\mathbf{P}_{C_\ell} \mathbf{P}_{N_0} u_0\|_{L_x^q} \|\mathbf{P}_{C_{\tilde{\ell}}} \mathbf{P}_{N_1} u_1\|_{L_x^q} \prod_{j=2}^{2m+1} \|\mathbf{P}_{N_j} u_j\|_{L_x^{(d+2)m}}. $$
Now, using Hölder's inequality with the compactness of $\mathbb{T}^d$ and the definition of $p$, and the definition (3.16) of $X_p$, we have from using Cauchy-Schwartz inequality:

$$
\begin{align*}
\sum_N \|\mathbf{P}_N u\|_{L_x^{(d+2)m}} &\lesssim \sum_N \|\mathbf{P}_N u\|_{L_x^p} \sim \|\mathbf{1}_{\tilde{N}\sim N} \mathbf{P}_C \mathbf{P}_N u\|_{\ell_N^\infty \ell_N^1 \ell_{C(\tilde{N})}^2 L_x^p} \\
&\lesssim \|\tilde{N}^{\sigma(p)} N^{-s} \mathbf{1}_{\tilde{N}\lesssim N} \tilde{N}^{-\sigma(p)} N^s \mathbf{P}_C \mathbf{P}_N u\|_{\ell_N^1 \ell_N^1 \ell_{C(\tilde{N})}^2 L_x^p} \\
&\lesssim \|\langle D \rangle^s u\|_{X_p} \|\mathbf{1}_{\tilde{N}\lesssim N} \tilde{N}^{\sigma(p)} N^{-s}\|_{\ell_N^\infty \ell_N^2} \lesssim \|\langle D \rangle^s u\|_{X_p},
\end{align*}
$$

where we used in the second step that $\#C(\tilde{N}) \cap \text{supp } \hat{\mathbf{P}}_N = O(\langle \frac{N}{\tilde{N}} \rangle^d) = O(1)$ when $\tilde{N} \sim N$, and in the last step that $\sigma(p) = \max(s_c, 0) + O(\varepsilon) < s$ with our choice of $p$ and $0 < \varepsilon \ll s - \max(s_c, 0)$.

Similarly, using Cauchy-Schwarz to sum on both $N_0 \sim N_1$ and $\ell, \tilde{\ell}$, we end up with

$$
\begin{align*}
\left| \sum_{N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}} \sum_{\substack{C_\ell, C_{\tilde{\ell}} \in C(N_2) \\ |\tilde{\ell} - \ell| \lesssim 1}} \int_{\mathbb{T}^d} \mathbf{P}_{C_\ell(N_2)} \mathbf{P}_{N_0} u_0^{\pm 0} \mathbf{P}_{C_{\tilde{\ell}}(N_2)} \mathbf{P}_{N_1} u_1^{\pm 1} \prod_{j=2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm j} dx \right| \\
&\lesssim \left( \prod_{j=3}^{2m+1} \|\langle D \rangle^s u_j\|_{X_p} \right) \sum_{N_2} \left( N_2^{2\sigma(q)} \|\mathbf{P}_{N_2} u_2\|_{L_x^p} \right. \\
&\qquad \left. \times \|N_2^{-\sigma(q)} N_0^{-s} \mathbf{P}_C \mathbf{P}_{N_0} u_0\|_{\ell_{N_0}^2 \ell^2(C(N_2))} L_x^q \|N_2^{-\sigma(q)} N_1^s \mathbf{P}_C \mathbf{P}_{N_1} u_1\|_{\ell_{N_1}^2 \ell^2(C(N_2))} L_x^q \right) \\
&\lesssim \|\langle D \rangle^{-s} u_0\|_{X_q} \|\langle D \rangle^s u_1\|_{X_p} \left( \prod_{j=3}^{2m+1} \|\langle D \rangle^s u_j\|_{X_p} \right) \sum_{N_2} N_2^{2\sigma(q)} \|\mathbf{P}_{N_2} u_2\|_{L_x^p}.
\end{align*}
$$

Since, with our choice of $q$, we have $\sigma(q) = 2\varepsilon$, we can finally sum as above

$$
\sum_{N_2} N_2^{2\sigma(q)} \|\mathbf{P}_{N_2} u_2\|_{L_x^p} \lesssim \|\langle D \rangle^s u_2\|_{X_p} \|\mathbf{1}_{\tilde{N} \lesssim N_2} \tilde{N}^{\sigma(p)} N_2^{4\varepsilon-s} \|_{\ell_N^\infty \ell_N^2} \lesssim \|\langle D \rangle^s u_2\|_{X_p},
$$

using again that $\sigma(p) = \max(s_c, 0) + O(\varepsilon) < s - 4\varepsilon$ with our choice of $p$ and $0 < \varepsilon \ll s - s_c$.

All in all, taking the $L_T^1$ norm of the previous computations, using Hölder's inequality with the choice of $q$ and $p$, we end up with the following estimate on (3.17):

$$
\begin{align*}
\left| \sum_{N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}} \sum_{\substack{C_\ell, C_{\tilde{\ell}} \in C(N_2) \\ |\tilde{\ell} - \ell| \lesssim 1}} \int_0^T \int_{\mathbb{T}^d} \mathbf{P}_{C_\ell(N_2)} \mathbf{P}_{N_0} u_0^{\pm 0} \mathbf{P}_{C_{\tilde{\ell}}(N_2)} \mathbf{P}_{N_1} u_1^{\pm 1} \prod_{j=2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm j} dx dt \right| \\
&\lesssim \left\| \|\langle D \rangle^{-s} u_0\|_{X_q} \|\langle D \rangle^s u_1\|_{X_q} \prod_{j=2}^{2m+1} \|\langle D \rangle^s u_j\|_{X_p} \right\|_{L_T^1} \\
&\lesssim \|\langle D \rangle^{-s} u_0\|_{L_T^q X_q} \|\langle D \rangle^s u_1\|_{L_T^q X_q} \prod_{j=2}^{2m+1} \|\langle D \rangle^s u_j\|_{L_T^{(d+2)m} X_p} \\
&\lesssim T^{2m\varepsilon} \|\langle D \rangle^{-s} u_0\|_{L_T^q X_q} \|\langle D \rangle^s u_1\|_{L_T^q X_q} \prod_{j=2}^{2m+1} \|\langle D \rangle^s u_j\|_{L_T^p X_p}.
\end{align*}
$$
Proceeding as above, we find a similar estimate for (3.18). This finally shows (1.8) with $\eta_{\mathcal{E}}(s) = 2m\varepsilon$.

**Remark 3.3.** Note that in the case $d = m = 1$, the argument above only shows well-posedness of (1.1) for $s > 0$, while Bourgain [6] proved global well-posedness in $L^2(\mathbb{T})$ for (1.9) using the lossless periodic Strichartz estimate

$$ \|e^{it\partial_x^2} u\|_{L_{t,x}^4(\mathbb{T}\times\mathbb{T})} \lesssim \|u\|_{L_x^2}. \qquad (3.19) $$

In particular, in case $d = m = 1$ and $s = 0$, the estimate (1.7) with $(p, \zeta, \iota) = (4, 4, +)$ and $X_4 = L_x^4$ follows from (3.19), while (1.8) follows from Hölder's inequality through

$$ \left| \int_0^T \int_{\mathbb{T}} \prod_{j=0}^2 u_j^{\pm j} dx dt \right| \lesssim \prod_{j=0}^3 \|u_j\|_{L_{T,x}^4} $$

for any $T \in (0; 2\pi)$, but provides only $\eta_{\mathcal{E}}(s) = 0$. This can only be used for a small data well-posedness result. For the deterministic cubic NLS in $L^2(\mathbb{T})$, this is overcome in [6] by using a refined $X^{s,b}$ version of (3.19), namely

$$ \|u\|_{L_{T,x}^4} \lesssim \|u\|_{X_T^{0, \frac{3}{8}+}} \qquad (3.20) $$

for $T \in (0; 2\pi)$, allowing to get $\eta_{\mathcal{E}} = \frac{1}{2}-$.

As far as the modulated periodic cubic NLS (1.4) is concerned, it seems possible to capture the refined $L^4$ estimate (3.20) only through the use of the assumption (A0)$_{(\rho,\gamma)}$ for some $\gamma > \frac{1}{2}$ and $\rho > \frac{1}{2}$ in the spirit of the proof of Theorem 1.10 below; see [20, Theorem 1.8] and the proof of Theorem 1.13 below. However, under assumption (A0)*, we can again use the smallness of $C_W(T)$ in (3.4) to recover large data global well-posedness in $L^2(\mathbb{T})$. Note that both assumptions (A0)* and (A0)$_{(\rho,\gamma)}$ with $\gamma, \rho > \frac{1}{2}$ barely miss the case $W_t = t$; see Remark 1.2.

**3.3.3. NLS equation on compact manifolds.** In the case where $\mathcal{M}$ is a general closed, boundaryless, smooth $d$-dimensional Riemannian manifold, [8, 81] established the Strichartz estimate

$$ \|e^{it\Delta} u\|_{L_T^p L_x^q} \lesssim \|u\|_{H^\frac{1}{p}} \qquad (3.21) $$

for any $T \in (0; 1]$ and admissible $(p, q)$ satisfying (3.9). In particular, this implies (1.7) with $\zeta = (p, q) \in [2; \infty]^2$, $X_{p,q} = W^{-\frac{1}{p},q}(\mathcal{M})$, and

$$ \tilde{\mathcal{E}} = \{(p, p, q, +), (p, q) \text{ admissible}\}. $$

Let $s > \frac{d}{2} - \frac{1}{2m}$. We then restrict $\tilde{\mathcal{E}}$ to $\mathcal{E} = \{(p, p, q, +)\}$ where $p = 2m + \varepsilon$ and $q$ given by $\frac{2}{p} + \frac{d}{q} = \frac{d}{2}$, for some $0 < \varepsilon \ll 1$ so that $s > \frac{d}{2} - \frac{1}{p} = \frac{d}{q} - \frac{1}{p}$.
Then from the proof of Proposition 3.1 in [8], it holds

$$
\begin{align*}
\left| \int_0^T \int_M \prod_{j=0}^{2m+1} u_j dx dt \right| &\lesssim \|u_0\|_{L_T^\infty H^{-s}} \left\| \prod_{j=1}^{2m+1} u_j \right\|_{L_T^1 H^s} \\
&\lesssim \|u_0\|_{L_T^\infty H^{-s}} \sum_{j=1}^{2m+1} \left\| \|u_j\|_{H^s} \prod_{\ell \neq j} \|u_j\|_{L_x^\infty} \right\|_{L_T^1} \\
&\lesssim \|u_0\|_{L_T^\infty H^{-s}} \sum_{j=1}^{2m+1} \|u_j\|_{L_T^\infty H^s} \prod_{\ell \neq j} \|u_j\|_{L_T^{2m} W^{s-\frac{1}{p},q}} \\
&\lesssim T^{2m\epsilon} \|u_0\|_{L_T^\infty H^{-s}} \sum_{j=1}^{2m+1} \|u_j\|_{L_T^\infty H^s} \prod_{\ell \neq j} \|u_j\|_{L_T^p W^{s-\frac{1}{p},q}},
\end{align*}
$$

where we used the fractional Leibniz rule, the Sobolev embedding $W^{s-\frac{1}{p},q}(M) \subset L^\infty(M)$ from our choice of $p$ and $q$, and Hölder's inequality. This shows (1.8) with $\eta_\varepsilon(s) = 2m\varepsilon$.

3.3.4. **Mass-critical gKdV equation on $\mathbb{R}$**. In the case of (1.10) we have that (1.7) holds with $\zeta = (q, \theta) \in [2; \infty) \times \mathbb{R}$ and $X_{q,\theta} = D^{-\theta}L^q(\mathbb{R})$. Indeed, from [62, Section 7.1], we have the local smoothing estimate

$$
\|D^{\theta} e^{t\partial_x^3} u\|_{L_x^q L_T^p} \lesssim \|u\|_{L^2}
$$

for $(p, q, \theta, \iota) \in \{(2, \infty, 1, -); (\infty, 4, -\frac{1}{4}, -)\}$, uniformly in $T > 0$. By interpolation, we deduce that it holds for any $(p, q, \theta, \iota) \in \tilde{\mathcal{E}}$, where

$$
\tilde{\mathcal{E}} = \bigcap_{\delta \in (0;1)} \left\{ \begin{array}{l} (p, q, \theta, \iota) \in (2; \infty) \times (4; \infty) \times \mathbb{R} \times \{\pm\}, \\ \iota = -, \ p = \frac{2}{\delta}, \ q = \frac{4}{1-\delta}, \ \theta = \delta - \frac{1-\delta}{4} \end{array} \right\}.
$$

Since we assumed $\mathcal{E}$ to be finite in Theorem 1.4, we restrict $\tilde{\mathcal{E}}$ by setting

$$
\mathcal{E} = \left\{ \left( \frac{10}{3}, 10, \frac{1}{2}, - \right), \left( 10, 5, 0, - \right) \right\}.
$$

Then, we can estimate with Hölder's inequality and the fractional Leibniz rule (see [62, Lemma 7.6]):

$$
\begin{align*}
\left| \int_0^T \int_{\mathbb{R}} \partial_x u_0 \prod_{j=1}^5 u_j dx dt \right| &\lesssim \|D^{\frac{1}{2}} u_0\|_{L_x^{10} L_T^{\frac{10}{3}}} \|D^{-\frac{1}{2}} \partial_x \prod_{j=1}^5 u_j\|_{L_x^{\frac{10}{3}} L_T^{\frac{10}{3}}} \\
&\lesssim \|D^{\frac{1}{2}} u_0\|_{L_x^{10} L_T^{\frac{10}{3}}} \sum_{j=1}^5 \|D^{\frac{1}{2}} u_j\|_{L_x^{10} L_T^{\frac{10}{3}}} \left\| \prod_{\ell \neq j} u_\ell \right\|_{L_x^{\frac{5}{4}} L_T^{\frac{5}{2}}} \\
&\lesssim \|D^{\frac{1}{2}} u_0\|_{L_x^{10} L_T^{\frac{10}{3}}} \sum_{j=1}^5 \|D^{\frac{1}{2}} u_j\|_{L_x^{10} L_T^{\frac{10}{3}}} \prod_{\ell \neq j} u_\ell \|_{L_x^5 L_T^{10}}
\end{align*}
$$

This shows (1.8) when $s = 0$ for this model, with $\eta_\varepsilon(s) = 0$ in this case, showing that it is indeed a mass-critical model. The case $s > 0$ follows from a straightforward adaptation.
4. THE MODULATED PERIODIC NLS EQUATION AT CRITICAL REGULARITY

We now turn to the proof of Theorem 1.10. Since, under $(\mathbf{A}0)_{(\rho,\gamma)}$ for $\gamma > \frac{1}{2}$ and $\rho > 1$, local well-posedness for $s > \max(s_c; 0) = \max(\frac{d}{2} - \frac{1}{m}; 0)$ follows from Theorem 1.4 as discussed in Subsection 3.3.2 and Remark 1.2 (iii), we focus on the critical case $s = s_c = \frac{d}{2} - \frac{1}{m} > 0$; recall the assumption $m > \frac{2}{d}$ in Theorem 1.10.

We start by recasting the Strichartz estimates (3.13)-(3.14) as a multilinear restriction estimate.

**Lemma 4.1.** Let $\frac{2(d+2)}{d} < p < m(d+2)$ and $q$ such that $\frac{2}{q} + \frac{2m}{p} = 1$. Then for any $N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}$, any $C, \tilde{C} \in C(N_2)$, and any $U_0, \dots, U_{2m+1} \in L^2(\mathbb{T}^d)$ with non-negative Fourier coefficients, it holds

$$ \sup_{\nu \in \mathbb{Z}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, \ n_0 \in C_\ell, \ n_1 \in C_{\tilde{\ell}} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\ \lesssim \|\mathbf{P}_{C_\ell} \mathbf{P}_{N_0} U_0\|_{L^2} \|\mathbf{P}_{C_{\tilde{\ell}}} \mathbf{P}_{N_1} U_1\|_{L^2} \\ \times N_2^{s_c+(2m-1)\varepsilon} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_j^{s_c-\varepsilon} \|\mathbf{P}_{N_j} U_j\|_{L^2} $$

with $\varepsilon = \frac{d+2}{p} - \frac{1}{m} > 0$.

The same estimate holds exchanging the roles of $N_0$ and $N_2$.

*Proof.* We observe that for any $\nu \in \mathbb{Z}$,

$$ \begin{align*} & \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, \ n_0 \in C_\ell, \ n_1 \in C_{\tilde{\ell}} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\ &= \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, \ n_0 \in C_\ell, \ n_1 \in C_{\tilde{\ell}}}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \int_0^{2\pi} e^{it(\sum_{j=0}^{2m+1} \pm_j |n_j|^2 - \nu)} dt \\ &= \int_0^{2\pi} \int_{\mathbb{T}^d} e^{-it\nu} (e^{it\Delta \mathbf{P}_C \mathbf{P}_{N_0} U_0})^{\pm_0} (e^{it\Delta \hat{\mathbf{P}}_C \mathbf{P}_{N_1} U_1})^{\pm_1} \prod_{j=2}^{2m+1} (e^{it\Delta \mathbf{P}_{N_j} U_j})^{\pm_j} dxdt. \end{align*} $$
Thus, since we have $p, q > \frac{2(d+2)}{d}$ from our choice of $p$ and $q$, we have from (3.13)-(3.14) that

$$
\begin{align*}
\sup_{\nu \in \mathbb{Z}} \sum_{\substack{n_0, \dots, n_5 \in \mathbb{Z}^3 \\ \sum_{j=0}^5 \pm_j n_j = 0 \\ |n_j| \sim N_j, \ n_0 \in C_\ell, \ n_1 \in C_{\tilde{\ell}} \\ \sum_{j=0}^5 \pm_j n_j|^2 = \nu}} \prod_{j=0}^5 \hat{U}_j(n_j) & \\
& \lesssim \|e^{it\Delta} \mathbf{P}_C \mathbf{P}_{N_0} U_0\|_{L^q(\mathbb{T} \times \mathbb{T}^3)} \|e^{it\Delta} \hat{\mathbf{P}}_{\tilde{C}} \mathbf{P}_{N_1} U_1\|_{L^q(\mathbb{T} \times \mathbb{T}^3)} \prod_{j=2}^5 \|e^{it\Delta} \mathbf{P}_{N_j} U_j\|_{L^p(\mathbb{T} \times \mathbb{T}^3)} \\
& \lesssim \|\mathbf{P}_C \mathbf{P}_{N_0} U_0\|_{L^2} \|\hat{\mathbf{P}}_{\tilde{C}} \mathbf{P}_{N_1} U_1\|_{L^2} \\
& \qquad \times N_2^{2\left(\frac{d}{2} - \frac{d+2}{q}\right) + \left(\frac{d}{2} - \frac{d+1}{p}\right)} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_j^{\frac{d}{2} - \frac{d+2}{p}} \|\mathbf{P}_{N_j} U_j\|_{L^2}.
\end{align*}
$$

The case $N_1 \sim N_2 \gtrsim N_0$ is estimated similarly. This proves Lemma 4.1 since $\frac{d}{2} - \frac{d+2}{p} = s_c - (\frac{d+2}{p} - \frac{1}{m})$ and $2(\frac{d}{2} - \frac{d+2}{q}) + (\frac{d}{2} - \frac{d+1}{p}) = s_c + (2m-1)(\frac{d+2}{p} - \frac{1}{m})$ by the definition of $s_c = \frac{d}{2} - \frac{1}{m}$ and our choice of $q$ such that $\frac{2}{q} + \frac{2m}{p} = 1$. $\square$

We now establish a refined version of Lemma 4.1 in the spirit of [48, Corollary 3.4].

**Lemma 4.2.** Let $p > m(d + 2)$ and $\frac{2(d + 2)}{d} < q < \frac{3m(d + 2)}{md + 1}$ be such that $\frac{3}{q} + \frac{2m-1}{p} = 1$. Then there exists $0 < \theta < 1 - \frac{2(d+2)}{dq}$ such that for any $N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}$, any $C_\ell, C_{\tilde{\ell}} \in C(N_2)$, let $M = (N_2^2 N_1^{-1}) \vee 1$ and $\{\mathcal{R}_\alpha\}$, $|\alpha| \sim M^{-1}N_1$, be a partition of $C_\ell$ into almost disjoint strips of width $M$ orthogonal to the center $n_\ell$ of $C_\ell$:

$$
\mathcal{R}_{\alpha} = \{n \in C_{\ell}, n \cdot n_{\ell} \in [\alpha|n_{\ell}|M; (\alpha + 1)|n_{\ell}|M)\},
$$

and similarly for $C_{\tilde{\ell}}$. Then for any $U_0, \dots, U_{2m+1} \in L^2(\mathbb{T}^d)$ with non-negative Fourier coefficients, it holds

$$
\begin{align*}
\sup_{\nu \in \mathbb{Z}} \sum_{\substack{|\alpha|,|\beta| \sim M^{-1}N_1 \\ |\pm_0 \alpha \pm_1 \beta| \lesssim 1}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, \ n_0 \in \mathcal{R}_\alpha, \ n_1 \in \mathcal{R}_\beta \\ \sum_{j=0}^{2m+1} \pm_j n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) & \\
& \lesssim \|\mathbf{P}_{C_\ell} \mathbf{P}_{N_0} U_0\|_{L^2} \|\mathbf{P}_{C_{\tilde{\ell}}} \mathbf{P}_{N_1} U_1\|_{L^2} \\
& \qquad \times N_2^{s_c - (2m-1)\varepsilon} \left(\frac{1}{N_2} + \frac{N_2}{N_1}\right)^{\theta} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_j^{s_c + \varepsilon} \|\mathbf{P}_{N_j} U_j\|_{L^2}
\end{align*}
$$

with $\varepsilon := \frac{1}{m} - \frac{d+2}{p} > 0$. The same estimate holds exchanging the roles of $N_0$ and $N_2$.

*Proof.* We proceed as in Lemma 4.1, but on top of (3.13)-(3.14) we also use the refined Strichartz estimate from [48, Corollary 3.4]. Namely, using Bernstein inequality, we have

$$
\| \mathbf{P}_{\mathcal{R}_{\alpha}} e^{it\Delta} \phi \|_{L^{\infty}(\mathbb{T} \times \mathbb{T}^d)} \lesssim M^{\frac{1}{2}} N_2^{\frac{d-1}{2}} \| \mathbf{P}_{\mathcal{R}_{\alpha}} \phi \|_{L_x^2}. \quad (4.1)
$$
Letting $p, q$ be as in the statement of Lemma 4.2 and taking $\frac{2(d+2)}{d} < q_0 < q$ we can interpolate between (4.1) and (3.14) for $q_0$ to get

$$ \begin{aligned} \|\mathbf{P}_{\mathcal{R}_\alpha} e^{it\Delta} \phi\|_{L^q(\mathbb{T}\times\mathbb{T}^d)} &\lesssim N_2^{\frac{q_0}{q}\left(\frac{d}{2}-\frac{d+2}{q_0}\right)+\frac{d-1}{2}\left(1-\frac{q_0}{q}\right)} M^{\frac{1}{2}\left(1-\frac{q_0}{q}\right)} \|\mathbf{P}_{\mathcal{R}_\alpha} \phi\|_{L_x^2} \\ &\sim N_2^{\frac{d}{2}-\frac{d+2}{q}} \left(\frac{1}{N_2} + \frac{N_2}{N_1}\right)^{\frac{1}{2}\left(1-\frac{q_0}{q}\right)} \|\mathbf{P}_{\mathcal{R}_\alpha} \phi\|_{L_x^2}. \end{aligned} \quad (4.2) $$

Thus, with $p$ and $q$ as in the statement, proceeding as in the proof of Lemma 4.1, we estimate using (3.13)-(4.2) and then Cauchy-Schwarz inequality in $\alpha, \beta$:

$$ \begin{aligned} & \sup_{\nu \in \mathbb{Z}} \sum_{\substack{|\alpha|,|\beta| \sim M^{-1}N_1 \\ |\pm 0 \alpha \pm 1 \beta| \lesssim 1}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{R}_\alpha, n_1 \in \mathcal{R}_\beta \\ \sum_{j=0}^{2m+1} \pm j |n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\ & \lesssim \sum_{\substack{|\alpha|,|\beta| \sim M^{-1}N_1 \\ |\pm 0 \alpha \pm 1 \beta| \lesssim 1}} \|e^{it\Delta} \mathbf{P}_{\mathcal{R}_\alpha} \mathbf{P}_{N_0} U_0\|_{L^q(\mathbb{T}\times\mathbb{T}^d)} \|e^{it\Delta} \mathbf{P}_{\mathcal{R}_\beta} \mathbf{P}_{N_1} U_1\|_{L^p(\mathbb{T}\times\mathbb{T}^d)} \\ & \qquad \times \|e^{it\Delta} \mathbf{P}_{N_2} U_2\|_{L^q(\mathbb{T}\times\mathbb{T}^d)} \prod_{j=3}^{2m+1} \|e^{it\Delta} \mathbf{P}_{N_j} U_j\|_{L^p(\mathbb{T}\times\mathbb{T}^d)} \\ & \lesssim \sum_{\substack{|\alpha|,|\beta| \sim M^{-1}N_1 \\ |\pm 0 \alpha \pm 1 \beta| \lesssim 1}} \|\mathbf{P}_{\mathcal{R}_m} \mathbf{P}_{N_0} U_0\|_{L^2} \|\mathbf{P}_{\mathcal{R}_{\bar{m}}} \mathbf{P}_{N_1} U_1\|_{L^2} \\ & \qquad \times N_2^{3\left(\frac{d}{2}-\frac{d+2}{q}\right)} \left(\frac{1}{N_2} + \frac{N_2}{N_1}\right)^{1-\frac{q_0}{q}} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_j^{\frac{d}{2}-\frac{d+2}{p}} \|\mathbf{P}_{N_j} U_j\|_{L^2} \\ & \lesssim \|\mathbf{P}_{C_\ell} \mathbf{P}_{N_0} U_0\|_{L^2} \|\mathbf{P}_{C_\ell} \mathbf{P}_{N_1} U_1\|_{L^2} \\ & \qquad \times N_2^{3\left(\frac{d}{2}-\frac{d+2}{q}\right)} \left(\frac{1}{N_2} + \frac{N_2}{N_1}\right)^{1-\frac{q_0}{q}} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_j^{\frac{d}{2}-\frac{d+2}{p}} \|\mathbf{P}_{N_j} U_j\|_{L^2}. \end{aligned} $$

The case $N_1 \sim N_2 \gtrsim N_0$ is treated similarly. This proves Lemma 4.2 with $\theta = 1 - \frac{q_0}{q}$. $\square$

We can now recast [48, Proposition 3.5] and [87, Theorem 3.4] also as a suitable multi-linear restriction estimate.

**Proposition 4.3.** For any $U_0 \in H^{-s_c}(\mathbb{T}^d)$ and $U_1, \dots, U_{2m+1} \in H^{s_c}(\mathbb{T}^d)$ with non-negative Fourier coefficients, it holds

$$ \begin{aligned} & \sum_{\substack{N_0 \sim N_1 \gtrsim N_2 \gtrsim \dots \gtrsim N_{2m+1} \\ C_\ell, C_{\tilde{\ell}} \in C(N_2) \\ |\ell - \tilde{\ell}| \lesssim 1}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm j n_j = 0 \\ |n_j| \sim N_j, n_0 \in C_\ell, n_1 \in C_{\tilde{\ell}}}} \langle \sum_{j=0}^{2m+1} \pm j |n_j|^2 \rangle^{-\rho} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\ & \lesssim \|U_0\|_{H^{-s_c}} \prod_{j=1}^{2m+1} \|U_j\|_{H^{s_c}} \end{aligned} $$
and

$$
\begin{align*}
& \sum_{\substack{N_1 \sim N_2 \gtrsim N_0, N_2 \ge \dots \ge N_{2m+1} \\ \mathcal{C}_\ell, \mathcal{C}_\tilde{\ell} \in \mathcal{C}(N_0) \\ |\ell - \tilde{\ell}| \lesssim 1}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_1, n_2 \in \mathcal{C}_\ell}} \langle \sum_{j=0}^{2m+1} \pm_j |n_j|^2 \rangle^{-\rho} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\
& \lesssim \|U_0\|_{H^{-sc}} \prod_{j=1}^{2m+1} \|U_j\|_{H^{sc}}.
\end{align*}
$$

*Proof.* We start by rewriting

$$
\sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{C}_\ell, n_1 \in \mathcal{C}_\tilde{\ell} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \langle \sum_{j=0}^{2m+1} \pm_j |n_j|^2 \rangle^{-\rho} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) = \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{C}_\ell, n_1 \in \mathcal{C}_\tilde{\ell} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \langle \nu \rangle^{-\rho} \prod_{j=0}^{2m+1} \hat{U}_j(n_j).
$$

Since $\rho > 1$, taking $0 < \delta < \rho - 1$ and $p, q$ as in Lemma 4.1 with $p$ close enough to $m(d+2)$ such that $\varepsilon := \frac{d+2}{p} - \frac{1}{m}$ satisfies $(2m-1)\varepsilon < 2\delta$, we can estimate the sum on $|\nu| \gtrsim N_2^2$ by using Lemma 4.1:

$$
\begin{align*}
& \sum_{|\nu| \gtrsim N_2^2} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{C}_\ell, n_1 \in \mathcal{C}_\tilde{\ell} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \langle \nu \rangle^{-\rho} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\
& \lesssim N_2^{-2\delta} \sup_{\nu \in \mathbb{Z}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{C}_\ell, n_1 \in \mathcal{C}_\tilde{\ell} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\
& \lesssim \|\mathbf{P}_{\mathcal{C}_\ell} \mathbf{P}_{N_0} U_0\|_{L^2} \|\mathbf{P}_{\mathcal{C}_\tilde{\ell}} \mathbf{P}_{N_1} U_1\|_{L^2} N_2^{s_c + (2m-1)\varepsilon - 2\delta} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_j^{s_c - \varepsilon} \|\mathbf{P}_{N_j} U_j\|_{L^2}.
\end{align*}
$$

With our choice of $\varepsilon, \delta$ we can thus use Cauchy-Schwarz inequality to sum on $\ell, \tilde{\ell}$ then
$N_0 \sim N_1 \gtrsim N_2 \ge \dots \ge N_{2m+1}$ to get

$$
\begin{align*}
& \sum_{\substack{N_0 \sim N_1 \gtrsim N_2 \ge \dots \ge N_{2m+1} \\ \mathcal{C}_\ell, \mathcal{C}_\tilde{\ell} \in \mathcal{C}(N_2) \\ |\ell - \tilde{\ell}| \lesssim 1}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{C}_\ell, n_1 \in \mathcal{C}_\tilde{\ell} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \langle \nu \rangle^{-\rho} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\
& \lesssim \sum_{\substack{N_0 \sim N_1 \gtrsim N_2 \ge \dots \ge N_{2m+1} \\ \mathcal{C}_\ell, \mathcal{C}_\tilde{\ell} \in \mathcal{C}(N_2) \\ |\ell - \tilde{\ell}| \lesssim 1}} \|\mathbf{P}_{\mathcal{C}_\ell} \mathbf{P}_{N_0} U_0\|_{L^2} \|\mathbf{P}_{\mathcal{C}_\tilde{\ell}} \mathbf{P}_{N_1} U_1\|_{L^2}
\end{align*}
$$
$$
\begin{align*}
& \times N_2^{1+15\varepsilon-2\delta} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{5} N_j^{1-5\varepsilon} \|\mathbf{P}_{N_j} U_j\|_{L^2} \\
& \lesssim \sum_{N_0 \sim N_1 \gtrsim N_2 \ge \dots \ge N_{2m+1}} N_0^{-s_c} \|\mathbf{P}_{N_0} U_0\|_{L^2} N_1^{s_c} \|\mathbf{P}_{N_1} U_1\|_{L^2} \\
& \quad \times N_2^{s_c+(2m-1)\varepsilon-2\delta} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_j^{s_c-\varepsilon} \|\mathbf{P}_{N_j} U_j\|_{L^2} \\
& \lesssim \|U_0\|_{H^{-s_c}} \prod_{j=1}^{2m+1} \|U_j\|_{H^{s_c}}.
\end{align*}
$$

It remains to estimate the contribution from $|\nu| \ll N_2^2$. Following [48], in order to use Lemma 4.2, for $C_\ell \in C(N_2)$, we write $C_\ell = n_\ell + [-N_2; N_2]^d$ for some $|n_\ell| \sim N_1$, and for

$M = (N_2^2 N_1^{-1}) \lor 1,$

we let $\{\mathcal{R}_\alpha\}$, $|\alpha| \sim M^{-1}N_1$, be a partition of $C_\ell$ into almost disjoint strips orthogonal to $n_\ell$ of width $M$:

$$
\mathcal{R}_{\alpha} = \{n \in C_{\ell}, n \cdot n_{\ell} \in [\alpha|n_{\ell}|M; (\alpha + 1)|n_{\ell}|M)\},
$$

and similarly for a partition $\{\mathcal{R}_\beta\}$ of $C_\ell$.
For $|\nu| \ll N_2^2$, we can thus decompose

$$
\sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in C_\ell, n_1 \in C_\ell \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) = \sum_{\alpha, \beta} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{R}_\alpha, n_1 \in \mathcal{R}_\beta \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j).
$$

Now, the point is that, as in [48, Proposition 3.5] and [87, Theorem 3.4], for a given $\alpha$, there
are at most $O(1)$ values of $\beta$ for which the sum above is non empty. Indeed, for $n_0 \in \mathcal{R}_\alpha$,
it holds

$$
\begin{align*}
|n_0|^2 &= |n_0 - n_\ell|^2 + 2n_0 \cdot n_\ell + |n_\ell|^2 \\
&= \underbrace{|n_0 - n_\ell|^2}_{O(N_2^2)} + \underbrace{\frac{1}{|n_\ell|^2} (n_0 \cdot n_\ell)^2}_{=\alpha^2 M^2 + O(\alpha M^2)} - \underbrace{\frac{1}{|n_\ell|^2} ((n_0 - n_\ell) \cdot n_\ell)^2}_{=O(N_2^2)} \\
&= \alpha^2 M^2 + O(\alpha M^2)
\end{align*}
$$

since $N_2^2 \lesssim |\alpha|M^2$ with our choice of $M$ and that $|\alpha| \sim M^{-1}N_1$. Similarly, we have
$|n_1|^2 = \beta^2 M^2 + O(\beta M^2)$ for $n_1 \in \mathcal{R}_\beta$, and thus from the constraint $\sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu$ with
$|\nu| \ll N_2^2$, we see that we must have $\pm_0|n_0|^2 \pm_1 |n_1|^2 = O(N_2^2)$, i.e.

$$
(\pm_0 \alpha^2 \pm_1 \beta^2) M^2 = O((|\alpha| + |\beta|)M).
$$
Thus $|\pm_0 \alpha \pm_1 \beta| \lesssim 1$. This yields

$$ \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{C}_\ell, n_1 \in \mathcal{C}_\tilde{\ell} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) = \sum_{\substack{|\alpha|, |\beta| \sim M^{-1} N_1 \\ |\pm_0 \alpha \pm_1 \beta| \lesssim 1}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{R}_\alpha, n_1 \in \mathcal{R}_\beta \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \prod_{j=0}^{2m+1} \hat{U}_j(n_j). $$

Thus, taking $p, q, \varepsilon, \theta$ as in Lemma 4.2, we can use Lemma 4.2 and then Cauchy-Schwarz inequality to sum on $N_j$:

$$ \begin{aligned} & \sum_{N_0 \sim N_1 \gtrsim N_2 \ge \dots \ge N_{2m+1}} \sum_{\substack{|\nu| \ll N_2^2 \\ C_\ell, C_\tilde{\ell} \in \mathcal{C}(N_2) \\ |\ell - \tilde{\ell}| \lesssim 1}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in \mathcal{C}_\ell, n_1 \in \mathcal{C}_\tilde{\ell} \\ \sum_{j=0}^{2m+1} \pm_j |n_j|^2 = \nu}} \langle \nu \rangle^{-\rho} \prod_{j=0}^{2m+1} \hat{U}_j(n_j) \\ & \lesssim \sum_{N_0 \sim N_1 \gtrsim N_2 \ge \dots \ge N_{2m+1}} \|\mathbf{P}_{N_0} U_0\|_{L^2} \|\mathbf{P}_{N_1} U_1\|_{L^2} \\ & \qquad \times N_2^{s_c-(2m-1)\varepsilon} \left(\frac{1}{N_2} + \frac{N_2}{N_1}\right)^{\theta} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_j^{s_c+\varepsilon} \|\mathbf{P}_{N_j} U_j\|_{L^2} \\ & \lesssim \sum_{N_0 \sim N_1 \gtrsim N_2} N_0^{-s_c} \|\mathbf{P}_{N_0} U_0\|_{L^2} N_1^{s_c} \|\mathbf{P}_{N_1} U_1\|_{L^2} \\ & \qquad \times N_2^{s_c-(2m-1)\varepsilon} \left(\frac{1}{N_2} + \frac{N_2}{N_1}\right)^{\theta} \|\mathbf{P}_{N_2} U_2\|_{L^2} \prod_{j=3}^{2m+1} N_2^{\varepsilon} \|U_j\|_{H^{s_c}} \\ & \lesssim \|U_0\|_{H^{-s_c}} \prod_{j=1}^{2m+1} \|U_j\|_{H^{s_c}}. \end{aligned} $$

The case $N_1 \sim N_2 \gtrsim N_0$ is treated similarly. This proves Proposition 4.3. $\square$

<figure><img src="image_3.png" /></figure>

We can now state our main multilinear estimate.

**Proposition 4.4.** For any $T > 0$, $u_j \in X_T^{s_c}$, $j = 1, \dots, 2m+1$, and $u_0 \in Y_T^{-s_c}$, and signs $\pm_j$, it holds

$$ \left| \int_0^T \int_{\mathbb{T}^d} \prod_{j=0}^{2m+1} u_j^{\pm_j} dxdt \right| \lesssim T^\gamma \|u_0\|_{Y_T^{-s_c}} \prod_{j=1}^{2m+1} \|u_j\|_{X_T^{s_c}}. \quad (4.3) $$

*Proof.* Following the argument in Subsection 3.3.2 it suffices to estimate

$$ \sum_{N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1}} \sum_{\substack{C_\ell, C_\tilde{\ell} \in \mathcal{C}(N_2) \\ |\tilde{\ell} - \ell| \lesssim 1}} \int_0^T \int_{\mathbb{T}^d} \mathbf{P}_{C_\ell} \mathbf{P}_{N_0} u_0^{\pm_0} \mathbf{P}_{C_\tilde{\ell}} \mathbf{P}_{N_1} u_1^{\pm_1} \prod_{j=2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm_j} dxdt \quad (4.4) $$
and

$$ \sum_{N_2 \sim N_1 \ge N_0 \ge \dots \ge N_{2m+1}} \sum_{\substack{C_\ell, C_\tilde{\ell} \in \mathcal{C}(N_0) \\ |\tilde{\ell} - \ell| \le 1}} \int_0^T \int_{\mathbb{T}^d} \mathbf{P}_{C_\ell} \mathbf{P}_{N_2} u_2^{\pm 2} \mathbf{P}_{C_\tilde{\ell}} \mathbf{P}_{N_1} u_1^{\pm 1} \prod_{j=0,2}^{2m+1} \mathbf{P}_{N_j} u_j^{\pm j} dxdt. \quad (4.5) $$

From the embedding $Y_T^{-sc} \subset U_{iW\Delta H^s_c}(\mathbb{T}^d)$ with $2 < b$ such that $b' < \frac{1}{\gamma}$, and the atomic structure of $U^2$ and $U^b$, it suffices to consider the case where for $j = 1, \dots, 2m+1$, $u_j$ are $U_{iW\Delta H^s_c}$-atoms and $u_0$ is a $U_{iW\Delta H^{-sc}}$-atom:

$$ u_j(t, x) = \sum_{k=1}^{K^{(j)}} \mathbf{1}_{[t_{k-1}^{(j)}; t_k^{(j)}}} e^{iW_t \Delta \phi_k^{(j)}}, \quad \{t_k^{(j)}\}_k \in \mathcal{Z}(T), \quad \|\phi_k^{(j)}\|_{\ell_k^2 H^{sc}} \le 1 $$

for any $j = 1, \dots, 2m+1$, and

$$ u_0(t, x) = \sum_{k=1}^{K^{(0)}} \mathbf{1}_{[t_{k-1}^{(0)}; t_k^{(0)}}} e^{iW_t \Delta \phi_k^{(0)}}, \quad \{t_k^{(0)}\}_k \in \mathcal{Z}(T), \quad \|\phi_k^{(0)}\|_{\ell_k^b H^{-sc}} \le 1. $$

Then, for given $k_0, \dots, k_{2m+1}$, letting

$$ I_{k_0, \dots, k_{2m+1}} = \bigcap_{j=0}^{2m+1} [t_{k_j-1}^{(j)}; t_{k_j}^{(j)}), $$

using Plancherel theorem and the occupation time formula (2.1), we estimate for any $N_0, \dots, N_{2m+1}, \ell, \tilde{\ell}$:

$$ \begin{aligned} & \left| \int_{I_{k_0, \dots, k_{2m+1}}} \int_{\mathbb{T}^d} (\mathbf{P}_{C_\ell} \mathbf{P}_{N_0} e^{iW_t \Delta \phi_{k_0}^{(0)}})^{\pm 0} (\mathbf{P}_{C_\tilde{\ell}} \mathbf{P}_{N_1} e^{iW_t \Delta \phi_{k_1}^{(1)}})^{\pm 1} \prod_{j=2}^{2m+1} (\mathbf{P}_{N_j} e^{iW_t \Delta \phi_{k_j}^{(j)}})^{\pm j} dxdt \right| \\ &= \left| \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0}} \int_{I_{k_0, \dots, k_{2m+1}}} e^{iW_t \sum_{j=0}^{2m+1} \pm_j n_j|^2} dt \right. \\ & \qquad \left. \times \widehat{\mathbf{P}_{C_\ell} \mathbf{P}_{N_0} \phi_{k_0}^{(0)}} (n_0)^{\pm 0} \widehat{\mathbf{P}_{C_\tilde{\ell}} \mathbf{P}_{N_1} \phi_{k_1}^{(1)}} (n_1)^{\pm 1} \prod_{j=2}^{2m+1} \widehat{\mathbf{P}_{N_j} \phi_{k_j}^{(j)}} (n_j)^{\pm j} \right| \\ &= \left| \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0}} \frac{d\mu_{\widehat{I}_{k_0, \dots, k_{2m+1}}}}{dz} \left( \sum_{j=0}^{2m+1} \pm_j n_j \right)^2 \right. \\ & \qquad \left. \times \widehat{\mathbf{P}_{C_\ell} \mathbf{P}_{N_0} \phi_{k_0}^{(0)}} (n_0)^{\pm 0} \widehat{\mathbf{P}_{C_\tilde{\ell}} \mathbf{P}_{N_1} \phi_{k_1}^{(1)}} (n_1)^{\pm 1} \prod_{j=2}^{2m+1} \widehat{\mathbf{P}_{N_j} \phi_{k_j}^{(j)}} (n_j)^{\pm j} \right| \\ & \lesssim |I_{k_0, \dots, k_{2m+1}}|^\gamma \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, \ n_0 \in C_\ell, \ n_1 \in C_\tilde{\ell}}} \langle \sum_{j=0}^{2m+1} \pm_j |n_j|^2 \rangle^{-\rho} \prod_{j=0}^{2m+1} |\widehat{\phi}_{k_j}^{(j)}(n_j)|, \end{aligned} $$
where the last step follows from assumption (A0)$_{(\rho,\gamma)}$. Now, using Proposition 4.3, we can sum on $N_j$ and $\ell, \tilde{\ell}$, to get

$$
\sum_{\substack{N_0 \sim N_1 \ge N_2 \ge \dots \ge N_{2m+1} \\ C_\ell, C_{\tilde{\ell}} \in C(N_2) \\ |\tilde{\ell} - \ell| \lesssim 1}} \sum_{\substack{n_0, \dots, n_{2m+1} \in \mathbb{Z}^d \\ \sum_{j=0}^{2m+1} \pm_j n_j = 0 \\ |n_j| \sim N_j, n_0 \in C_\ell, n_1 \in C_{\tilde{\ell}}}} \langle \sum_{j=0}^{2m+1} \pm_j |n_j|^2 \rangle^{-\rho} \prod_{j=0}^{2m+1} |\hat{\phi}_{k_j}^{(j)}(n_j)| \\
\lesssim \|\phi_{k_0}^{(0)}\|_{H^{-sc}} \prod_{j=1}^{2m+1} \|\phi_{k_j}^{(j)}\|_{H^{sc}},
$$

and similarly for the case $N_1 \sim N_2 \gtrsim N_0$. It remains to sum on $k_j$ using Hölder's inequality to get

$$
\begin{aligned}
\left| \int_0^T \int_{\mathbb{T}^d} \prod_{j=0}^{2m+1} u_j^{\pm_j} dx dt \right| & \lesssim \sum_{k_0, \dots, k_{2m+1}} |I_{k_0, \dots, k_{2m+1}}|^\gamma \|\phi_{k_0}^{(0)}\|_{H^{-sc}} \prod_{j=1}^{2m+1} \|\phi_{k_j}^{(j)}\|_{H^{sc}} \\
& \lesssim \left\| |I_{k_0, \dots, k_{2m+1}}|^\gamma \right\|_{\ell_{k_1, \dots, k_{2m+1}}^2 \ell_{k_0}^{\ell'_b} \ell_{k_0}^{(0)}} \|\phi_{k_0}^{(0)}\|_{\ell_{k_0}^b H^{-sc}} \prod_{j=1}^{2m+1} \|\phi_{k_j}^{(j)}\|_{\ell_{k_j}^2 H^{sc}} \\
& \lesssim \left\| |I_{k_0, \dots, k_{2m+1}}|^\gamma \right\|_{\ell_{k_0, \dots, k_{2m+1}}^\gamma} = T^\gamma.
\end{aligned}
$$

This finally proves (4.3) in view of the definition of the atomic spaces $X_T^{sc}$ and $U_{iW\Delta}^b H^{-sc}$.

<figure><img src="image_4.png" /></figure>

We can now finish the proof of Theorem 1.10.

*Proof of Theorem 1.10.* For $u_0 \in H^{sc}(\mathbb{T}^d)$, $R = 2\|u_0\|_{H^{sc}}$, and $T \in (0; 1]$, define

$$
B_R(T) := \{u \in C([0;T]; H^{sc}(\mathbb{T}^d)) \cap X_T^{sc}, \|u\|_{X_T^{sc}} \le R\}
$$

which is closed in $X_T^{sc}$, and

$$
\Phi_{u_0} : u \in X_T^{sc} \mapsto e^{i(W_t - W_0)\Delta} u_0 \pm i \int_0^t e^{i(W_t - W_\tau)\Delta} |u|^4 u(\tau) d\tau.
$$

We will show that $\Phi_{u_0}$ defines a contraction on $B_R(T)$. From Proposition 2.9, the multi-linearity of $\mathcal{N}(u) = |u|^{2m}u$, and Proposition 4.4, we have for any $u, v \in B_R(T)$:

$$
\begin{aligned}
\|\Phi_{u_0}(u) - \Phi_{v_0}(v)\|_{X_T^{sc}} &\le \|u_0 - v_0\|_{H^{sc}} + \sup_{\|w\|_{Y_T^{-sc}} \le 1} \left| \int_0^T \int_{\mathbb{T}^d} (|u|^{2m}u - |v|^{2m}v) \bar{w} dx dt \right| \\
&\le \|u_0 - v_0\|_{H^{sc}} + CT^\gamma \|w\|_{Y_T^{-sc}} \|u - v\|_{X_T^{sc}} (\|u\|_{X_T^{sc}}^4 + \|v\|_{X_T^{sc}}^4).
\end{aligned}
$$

Thus, taking $T(R) \sim \langle R \rangle^{-\frac{2m}{\gamma}}$ ensures that $\Phi_{u_0}$ defines a contraction on $B_R(T)$, thus providing a mild solution $u \in C([0; T]; H^{sc}(\mathbb{T}^d))$ to (1.1) on $[0; T)$, unique in $B_R(T)$. The uniqueness in the whole space $C([0; T]; H^{sc}(\mathbb{T}^d)) \cap X_T^{sc}$ follows from iterating the fixed point argument as in the proof of Theorem 1.4, and the estimate above shows the Lipschitz continuity of the flow map. This concludes the proof of Theorem 1.10. $\square$

<figure><img src="image_5.png" /></figure>
**Remark 4.5.** A straightforward adaptation of the arguments in [9, 44] following the proof of Theorem 1.10 above allows to extend Theorem 1.10 to the cases $\mathcal{M} = \mathbb{S}^2$, $m = 1$, $s > \frac{1}{4}$; and $\mathcal{M} = \mathbb{S}^3$, $m = 2$, $s \ge s_c = 1$, respectively.

## 5. THE PERIODIC CUBIC WICK-ORDERED MODULATED NLS EQUATION

In this last section we give the proof of Theorem 1.13. First, as usual [41, 40], we rewrite the Wick-ordered cubic nonlinearity in Fourier as

$$ \mathcal{N}(u) := (|u|^2 - 2\|u\|_{L^2}^2)u = \sum_{n_0 \in \mathbb{Z}} e^{in_0x} \left\{ \sum_{\substack{n_1-n_2+n_3=n_0 \\ n_2 \neq n_1, n_3}} \hat{u}_{n_1} \overline{\hat{u}_{n_2}} \hat{u}_{n_3} - |\hat{u}_n|^2 \hat{u}_n \right\} \\ =: \mathcal{N}_0(u) + \mathcal{R}(u). $$

Abusing notations, we also write $\mathcal{N}_0(u, v, w)$ and $\mathcal{R}(u, v, w)$ for the trilinear forms such that $\mathcal{N}_0(u) = \mathcal{N}_0(u, u, u)$ and $\mathcal{R}(u) = \mathcal{R}(u, u, u)$.

Let then $s \ge 0$ and $r \in [1; \infty]$. As in the proof of Theorem 1.10, using assumption (A0)$_{(\rho, \gamma)}$ with $\gamma > \frac{1}{2}$ and $\rho > \frac{1}{r}$, we get that for any interval $I \subset \mathbb{R}$, any $u_j \in \mathcal{FL}^{s,r}(\mathbb{T})$, $j = 1, 2, 3$, and $u_0 \in \mathcal{FL}^{-s,r'}(\mathbb{T})$, it holds

$$ \left| \int_I \int_{\mathbb{T}} \mathcal{N}_0(e^{iW_t \partial_x^2} u_1, e^{iW_t \partial_x^2} u_2, e^{iW_t \partial_x^2} u_3) \overline{e^{iW_t \partial_x^2} u_0} dx dt \right| \\ = \left| \sum_{\substack{n_0, \dots, n_3 \in \mathbb{Z} \\ \sum_{j=0}^3 \pm_j n_j = 0 \\ n_2 \neq n_1, n_3}} \frac{d\mu_I}{dz} \left( \sum_{j=0}^3 \pm_j n_j^2 \right) \prod_{j=0}^3 \hat{u}_j(n_j)^{\pm_j} \right| \\ \lesssim |I|^\gamma \sum_{\substack{n_0, \dots, n_3 \in \mathbb{Z} \\ \sum_{j=0}^3 \pm_j n_j = 0 \\ n_2 \neq n_1, n_3}} \langle (n_1 - n_2)(n_0 - n_2) \rangle^{-\rho} \prod_{j=0}^3 |\hat{u}_j(n_j)| $$

where we used the exact factorization of $\sum_{j=0}^3 \pm_j n_j^2$ under the convolution constraint $\sum_{j=0}^3 \pm_j n_j = 0$. Using now Hölder and Young inequalities together with $\langle n_0 \rangle^s \lesssim \sum_{j=1}^3 \langle n_j \rangle^s$ under the convolution constraint since $s \ge 0$, we can continue with

$$ \begin{aligned} & \lesssim |I|^\gamma \|u_0\|_{\mathcal{FL}^{-s,r'}} \left\| \langle n_0 \rangle^s \sum_{\substack{n_1,n_2,n_3 \in \mathbb{Z} \\ n_1-n_2+n_3=n_0 \\ n_2 \neq n_1, n_3}} \langle (n_1-n_2)(n_0-n_2) \rangle^{-\rho} \prod_{j=1}^3 \hat{u}_j(n_j) \right\|_{\ell_{n_0}^r}^2 \\ & \lesssim |I|^\gamma \|u_0\|_{\mathcal{FL}^{-s,r'}} \left( \sum_{j=1}^3 \|u_j\|_{\mathcal{FL}^{s,r}} \prod_{\substack{\ell=1,2,3 \\ \ell \neq j}} \|u_\ell\|_{\mathcal{FL}^{0,r}} \right) \end{aligned} $$
$$ \times \left\{ \sup_{n_0 \in \mathbb{Z}} \sum_{\substack{n_1, n_2, n_3 \in \mathbb{Z} \\ n_1 - n_2 + n_3 = n_0 \\ n_2 \neq n_1, n_3}} \langle (n_1 - n_2)(n_0 - n_2) \rangle^{-r'\rho} \right\}^{\frac{1}{r'}}. $$

Thus it remains to bound uniformly in $n_0 \in \mathbb{Z}$:

$$ \sum_{\substack{n_1, n_2, n_3 \in \mathbb{Z} \\ n_1 - n_2 + n_3 = n_0 \\ n_2 \neq n_1, n_3}} \langle (n_1 - n_2)(n_0 - n_2) \rangle^{-r'\rho} = \sum_{m_1, m_2 \neq 0} \langle m_1 m_2 \rangle^{-r'\rho} \lesssim 1 $$

from our assumption that $\rho > \frac{1}{r'}$. This shows that

$$ \left| \int_I \int_{\mathbb{T}} \mathcal{N}_0(e^{iW_t \partial_x^2} u_1, e^{iW_t \partial_x^2} u_2, e^{iW_t \partial_x^2} u_3) \overline{e^{iW_t \partial_x^2} u_0} dx dt \right| \\ \lesssim |I|^\gamma \|u_0\|_{\mathcal{F}L^{-s, p'}} \left( \sum_{j=1}^3 \|u_j\|_{\mathcal{F}L^{s, p}} \prod_{\substack{l=1,2,3 \\ l \neq j}} \|u_l\|_{\mathcal{F}L^{0, p}} \right). \quad (5.1) $$

We estimate the contribution of $\mathcal{R}$ similarly (note that $\sum_{j=0}^3 \pm_j n_j^2 = 0$ on its support):

$$ \left| \int_I \int_{\mathbb{T}} \mathcal{R} (e^{iW_t \partial_x^2} u_1, e^{iW_t \partial_x^2} u_2, e^{iW_t \partial_x^2} u_3) \overline{e^{iW_t \partial_x^2} u_0} dx dt \right| \\ \lesssim |I|^\gamma \sum_{n_0 \in \mathbb{Z}} \prod_{j=0}^3 |\hat{u}_j(n_0)| \lesssim |I|^\gamma \|u_0\|_{\mathcal{F}L^{-s, r'}} \left( \sum_{j=1}^3 \|u_j\|_{\mathcal{F}L^{s, r}} \prod_{\substack{l=1,2,3 \\ l \neq j}} \|u_l\|_{\mathcal{F}L^{0, r'}} \right). \quad (5.2) $$

Now, for $R > 0$ and $T \in (0; 1]$, we again consider

$$ B_R(T) := \{ u \in C([0; T]; \mathcal{F}L^{s,r}(\mathbb{T})) \cap X_T^{s,r}, \|u\|_{X_T^{s,r}} \le R \} $$

which is closed in $X_T^{s,r}$.

Given $u_0 \in \mathcal{F}L^{s,r}(\mathbb{T})$, we set $R = 2\|u_0\|_{\mathcal{F}L^{s,r}}$, and consider the map

$$ \Phi_{u_0} : u \in B_R(T) \mapsto e^{i(W_t - W_0)} \partial_x^2 u_0 \pm i \int_0^t e^{i(W_t - W_{t'})} \partial_x^2 \mathcal{N}(u)(t') dt'. $$

By Proposition 2.9 (ii) and (iii), we have

$$ \| \Phi_{u_0}(u) - \Phi_{v_0}(v) \|_{X_T^{s,r}} \le \|u_0 - v_0\|_{\mathcal{F}L^{s,r}} + \sup_{\substack{w \in Y_T^{-s,r'} \\ \|w\|_{Y_T^{-s,r'}} \le 1}} \left| \int_0^T \int_{\mathcal{M}} (\mathcal{N}(u) - \mathcal{N}(v)) \overline{w} dx dt \right|. $$

Since $u, v \in X_T^{s,r} = U_{W\mathcal{L}}^2([0; T))\mathcal{F}L^{s,r}(\mathbb{T})$ and from the embedding of Proposition 2.9 (i), $w \in Y_T^{-s,r'} = V_{iW\partial_x^2}^2([0; T))\mathcal{F}L^{-s,r'}(\mathbb{T}) \subset U_{iW\Delta}^b([0; T))\mathcal{F}L^{-s,r'}(\mathbb{T})$ for $b = (\frac{1}{\gamma})' > 2$, using the atomic structure of $U^2$ and $U^b$ and the multilinearity of $\mathcal{N}$, we can assume that $u, v$ (respectively $w$) are $U^2$ (respectively $U^b$) atoms:

$$ u = \sum_{k=1}^{K^{(1)}} \mathbf{1}_{[t_{k-1}^{(1)}; t_k^{(1)})} e^{iW_t \partial_x^2} \phi_k^{(1)} \quad \text{with} \quad \|\phi_k^{(1)}\|_{\ell_k^2 \mathcal{F}L^{s,r}} \le 1, $$
$$v = \sum_{k=1}^{K^{(2)}} \mathbf{1}_{[t_{k-1}^{(2)}; t_k^{(2)})} e^{iW_t \partial_x^2 \phi_k^{(2)}} \text{ with } \|\phi_k^{(2)}\|_{\ell_k^2 \mathcal{F}L^{s,r}} \le 1,$$

and

$$w = \sum_{k=1}^{K^{(0)}} \mathbf{1}_{[t_{k-1}^{(0)}; t_k^{(0)})} e^{iW_t \partial_x^2 \phi_k^{(0)}} \text{ with } \|\phi_k^{(0)}\|_{\ell_k^b \mathcal{F}L^{-s,r'}} \le 1,$$

We can also assume

$$u - v = \sum_{k=1}^{K^{(3)}} \mathbf{1}_{[t_{k-1}^{(3)}; t_k^{(3)})} e^{iW_t \partial_x^2 \phi_k^{(3)}} \text{ with } \|\phi_k^{(3)}\|_{\ell_k^2 \mathcal{F}L^{s,r}} \le 1.$$

Then, writing the nonlinearity as

$$\mathcal{N}(u) - \mathcal{N}(v) = \mathcal{N}(u - v, u, u) + \mathcal{N}(v, u - v, u) + \mathcal{N}(v, v, u - v)$$

and plugging the atomic decompositions into the Duhamel formula and using (5.1)-(5.2) together with Hölder's inequality, we find

$$\begin{aligned} & \left| \int_0^T \int_{\mathcal{M}} \mathcal{N}(u - v, u, u) \overline{w} dx dt \right| \\ & \le \sum_{k_0, k_1, k'_1, k_3} \left| \int_{I_{k_0, k_1, k'_1, k_3}} \int_{\mathcal{M}} \mathcal{N} \left( e^{iW_t \partial_x^2 \phi_{k_3}^{(3)}}, e^{iW_t \partial_x^2 \phi_{k_1}^{(1)}}, e^{iW_t \partial_x^2 \phi_{k'_1}^{(1)}} \right) \overline{e^{iW_t \partial_x^2 \phi_{k_0}^{(0)}}} dx dt \right| \\ & \lesssim \sum_{k_0, k_1, k'_1, k_3} \left| I_{k_0, k_1, k'_1, k_3} \right|^\gamma \|\phi_{k_0}^{(0)}\|_{\mathcal{F}L^{-s,r'}} \|\phi_{k_3}^{(3)}\|_{\mathcal{F}L^{s,r}} \|\phi_{k_1}^{(1)}\|_{\mathcal{F}L^{s,r}} \|\phi_{k'_1}^{(1)}\|_{\mathcal{F}L^{s,r}} \\ & \lesssim \left\| I_{k_0, k_1, k'_1, k_3} \right\|^\gamma \|\phi_{k_0}^{(0)}\|_{\ell_{k_1, k'_1, k_3}^2} \|\phi_{k_0}^{(3)}\|_{\ell_{k_1, k'_1, k_3}^b} \end{aligned}$$

where

$$I_{k_0, k_1, k'_1, k_3} = [t_{k'_1-1}^{(1)}; t_{k'_1}^{(1)}] \cap [t_{k_j-1}^{(j)}; t_{k_j}^{(j)}].$$

Since $\{t_{k_j}^{(j)}\} \in \mathcal{Z}(T)$, with our choice of $b = (\frac{1}{\gamma})' > 2$, we have

$$\left\| I_{k_0, k_1, k'_1, k_3} \right\|^\gamma \|\phi_{k_0}^{(1)}\|_{\ell_{k_1, k'_1, k_3}^2} \le \left\| I_{k_0, k_1, k'_1, k_3} \right\|^\gamma \|\phi_{k_0}^{(j)}\|_{\ell_{k_0, k_1, k'_1, k_3}^b} = T^\gamma.$$

This finally leads to

$$\left| \int_0^T \int_{\mathcal{M}} (\mathcal{N}(u) - \mathcal{N}(v)) \overline{w} dx dt \right| \lesssim T^\gamma \|u - v\|_{X_T^{s,r}} (\|u\|_{X_T^{s,r}} + \|v\|_{X_T^{s,r}})^2 \|w\|_{Y_T^{-s,r'}},$$

hence

$$\begin{aligned} \|\Phi_{u_0}(u) - \Phi_{v_0}(v)\|_{X_T^{s,r}} & \le \|u_0 - v_0\|_{\mathcal{F}L^{s,r}} + CT^\gamma \|u - v\|_{X_T^{s,r}} (\|u\|_{X_T^{s,r}} + \|v\|_{X_T^{s,r}})^2 \\ & \le \|u_0 - v_0\|_{\mathcal{F}L^{s,r}} + CT^\gamma (2R)^2 \|u - v\|_{X_T^{s,r}} \end{aligned}$$

as $u, v \in B_R(T)$. Taking $T = T(R) = T(\|u_0\|_{\mathcal{F}L^{s,r}}) \sim (1 + \|u_0\|_{\mathcal{F}L^{s,r}})^{-\frac{2}{\gamma}}$, the estimate above shows that $\Phi_{u_0}$ defines a contraction on $B_R(T)$, thus having a unique fixed point in $B_R(T)$ which solves the mild formulation (1.2). Uniqueness of the solution in the whole
class $X_T^{s,r}$ follows from a usual argument as in the proof of Theorem 1.4. At last, the estimate above shows the Lipschitz continuity of the flow with respect to initial data.

## REFERENCES

[1] P. Antonelli, J.-C. Saut, C. Sparber, *Well-posedness and averaging of NLS with time-periodic dispersion management*, Adv. Differential Equations 18, (2013) no. 1-2, 49–68.

[2] H. Bahouri, J.-Y. Chemin, R. Danchin, *Fourier analysis and nonlinear partial differential equations*, Grundlehren der mathematischen Wissenschaften 343. Springer, Heidelberg, 2011. xvi+523 pp.

[3] V. Barbu, M; Röckner, D. Zhang, *Stochastic nonlinear Schrödinger equations: no blow-up in the non-conservative case*, J. Differential Equations 263(2017), no.11, 7919–7940.

[4] R. Belaouar, A. de Bouard, A. Debussche, *Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion*, Stoch. Partial Differ. Equ. Anal. Comput.3(2015), no.1, 103–132.

[5] S. M. Berman, *Local nondeterminism and local times of Gaussian processes*, Indiana Univ. Math. J. 23 (1973), 69–94.

[6] J. Bourgain, *Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations*, Geom. Funct. Anal. 3 (1993), no. 2, 107–56.

[7] J. Bourgain, C. Demeter, *The proof of the ℓ² decoupling conjecture*, Ann. of Math. 182 (2015), no. 1, 351–89.

[8] N. Burq, P. Gérard, N. Tzvetkov, *Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds*, Amer. J. Math. 126 (2004), no. 3, 569–605.

[9] N. Burq, P. Gérard, N. Tzvetkov, *Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces*, Invent. Math. 159 (2005), no. 1, 187–223.

[10] N. Burq, P. Gérard, N. Tzvetkov, *Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations*, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 2, 255–301.

[11] N. Burq, F. Planchon, *Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications*, J. Funct. Anal.236(2006), no.1, 265–298.

[12] L. Campos, J. Murphy, T. Van Hoose, *Averaging for the 2d dispersion-managed NLS*, to appear in Commun. Contemp. Math.

[13] R. Catellier, M. Gubinelli, *Averaging along irregular curves and regularisation of ODEs*, Stochastic Process. Appl. 126 (2016), no. 8, 2323–2366.

[14] T. Cazenave, *Semilinear Schrödinger equations*, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp.

[15] T. Cazenave, F. B. Weissler, *The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal.14(1990), no.10, 807–836.*

[16] M. R. Choi, Y. Hong, Y. R. Lee, *Global existence versus finite time blowup dichotomy for the dispersion managed NLS*, arXiv:2311.02905.

[17] M. R. Choi, D. Hundertmark, Y. R. Lee, *Well-posedness of dispersion managed non-linear Schrödinger equations*, J. Math. Anal. Appl. 522 (2023), no. 1, 37pp.

[18] M. R. Choi, K. Lee, Y.R. Lee, *Scattering for the dispersion-managed nonlinear Schrödinger equation*, arXiv:2401.16717.

[19] M. R. Choi, Y. R. Lee, *Averaging of dispersion managed nonlinear Schrödinger equations*, Nonlinearity 35 (2022), no. 4, 2121–2133.

[20] K. Chouk, M. Gubinelli, *Nonlinear PDEs with modulated dispersion I: Nonlinear Schrödinger equations*, Comm. Partial Differential Equations 40 (2015), no. 11, 2047–2081.

[21] K. Chouk, M. Gubinelli, G. Li, J. Li, T. Oh, *Nonlinear PDEs with modulated dispersion II: Korteweg–de Vries equation*, arXiv:1406.7675 [math.AP].

[22] M. Christ, J. Holmer, D. Tataru, *Low regularity a priori bounds for the modified Korteweg-de Vries equation*, Lib. Math. (N.S.)32(2012), no.1, 51–75.

[23] I. Cîmpean, A. Grecu, *The nonlinear Schrödinger equation with white noise dispersion on quantum graphs*, Commun. Math. Sci. 19 (2021), no. 2, 405–435.

[24] J. Cui, G. Maierhofer, personal communication.

[25] A. de Bouard, A. Debussche, *The nonlinear Schrödinger equation with white noise dispersion*, J. Funct. Anal. 259 (2010), no. 5, 1300–1321.
[26] A. Debussche, A. Mouzard, Periodic nonlinear Schrödinger equation with distributional potential and invariant measures, arXiv:2405.00583 [math.AP].

[27] A. Debussche, Y. Tsutsumi, 1D quintic nonlinear Schrödinger equation with white noise dispersion, J. Math. Pures Appl. (9) 96 (2011), no. 4, 363-376.

[28] A. Debussche, H. Weber, The Schrödinger equation with spatial white noise potential, Electron. J. Probab., 23 (2018), Paper No. 28, 16pp.

[29] R. Duboscq; A. Réveillac, On a stochastic Hardy-Littlewood-Sobolev inequality with application to Strichartz estimates for a noisy dispersion, Ann. H. Lebesgue 5 (2022), 263-274.

[30] D. Frey, R. Schippa, Strichartz estimates for equations with structured Lipschitz coefficients, J. Evol. Equ. 23, 45 (2023).

[31] I. Gabitov, S.K. Turitsyn, Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation, Opt. Lett. 21 (1996), 327-329.

[32] I. Gabitov, S.K. Turitsyn, Breathing solitons in optical fiber links, JETP Lett. 63 (1996) 861.

[33] L. Galeati, *Nonlinear Young Differential Equations: A Review*, J. Dyn. Diff. Equ. (2021).

[34] L. Galeati, Pathwise methods in regularisation by noise, PhD Thesis.

[35] L. Galeati, M. Gubinelli, Noiseless regularisation by noise, Rev. Mat. Iberoam. 38 (2022), no. 2, 433-502.

[36] L. Galeati, M. Gubinelli, Prevalence of $\rho$-irregularity and related properties, to appear in Ann. Inst. H. Poincaré Probab. Statist.

[37] A. Goel, Strichartz Estimates for One Dimensional Stochastic Schrodinger Equation with Potential, arXiv:2410.05148 [math.AP].

[38] M. Goldberg, W. Schlag, Dispersive Estimates for Schrödinger Operators in Dimensions One and Three, Commun. Math. Phys. 251, 157-178 (2004).

[39] M. Gubinelli, B. Ugurcan, I. Zachhuber, Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions, Stoch. Partial Differ. Equ. Anal. Comput., 8 (2020), pp. 82-149.

[40] Z. Guo, T. Oh, Non-existence of solutions for the periodic cubic NLS below $L^2$, Int. Math. Res. Not. IMRN (2018), no.6, 1656-1729.

[41] A. Grünrock, S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal. 39(2008), no.6, 1890-1920.

[42] D. Geman, J. Horowitz, Occupation densities, Ann. Probab. 8 (1980), no. 1, 1-67.

[43] M. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 3, 917-941.

[44] S. Herr, The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds, Amer. J. Math. 135 (2013), no. 5, 1271-1290.

[45] S. Herr, B. Kwak, Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^2$, arXiv:2309.14275 [math.AP].

[46] S. Herr, M. Röckner, M. Spitz, D. Zhang, The three dimensional stochastic Zakharov system, to appear in Ann. Probab.

[47] S. Herr, M. Röckner, D. Zhang, Scattering for stochastic nonlinear Schrödinger equations, Comm. Math. Phys. 368 (2019), no.2, 843-884.

[48] S. Herr, D. Tataru, N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\mathbb{T}^3)$, Duke Math. J. 159(2011), no.2, 329-349.

[49] D. Hundertmark, D., Y.-R. Lee, Decay estimates and smoothness for solutions of the dispersion managed non-linear Schrödinger equation, Comm. Math. Phys. 286 (2009), 851-873.

[50] D. Hundertmark, D., Y.-R. Lee, Super-exponential decay of diffraction managed solitons, Comm. Math. Phys. 309 (2012), 1-21.

[51] J. Kawakami, J. Murphy, Small and large data scattering for the dispersion-managed NLS, arXiv:2407.11151 [math.AP].

[52] C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle, Comm. Pure Appl. Math. 46 (1993), 527-20.

[53] R. Killip, M. Visan, Scale invariant Strichartz estimates on tori and applications, Math. Res. Lett. 23 (2016), no. 2, 445-72

[54] N. Kishimoto, Remark on the periodic mass critical nonlinear Schrödinger equation, Proc. Amer. Math. Soc.142(2014), no.8, 2649-2660.

[55] H. Koch, D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58 (2) (2005) 217-284.
[56] H. Koch, D. Tataru, *A priori bounds for the 1D cubic NLS in negative Sobolev spaces*, Internat. Math. Res. Notices 2007 (2007), article ID rnm053, 36 pages.

[57] H. Koch, D. Tataru, M. Višan, *Dispersive equations and nonlinear waves. Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps*, Oberwolfach Semin., 45 Birkhäuser/Springer, Basel, 2014. xii+312 pp.

[58] S. Kumar, A. Hasegawa, *Quasi-soliton propagation in dispersion-managed optical fibers*, Opt. Lett. 22 (1997), 372-374.

[59] M. Kunze, J. Moeser, V. Zharnitsky, *Ground states for the higher order dispersion managed NLS equation in the absence of average dispersion*, J. Diff. Eqs. 209 (2005), 77-100.

[60] C. Kurtzke, *Suppression of fiber nonlinearities by appropriate dispersion management*, IEEE Phot. Tech. Lett. 5 (1993), 1250-1253.

[61] T. Lakoba, D.J. Kaup, *Shape of the stationary pulse in the strong dispersion management regime*, Electron. Lett. 34 (1998), 1124-1125.

[62] F. Linares, G. Ponce, *Introduction to nonlinear dispersive equations. Second edition*, Universitext. Springer, New York, 2015. xiv+301 pp.

[63] P.V. Mamyshev, N.A. Mamysheva, *Pulseoverlapped dispersion-managed data transmission and intra-channel four-wave mixing*, Opt. Lett. 24 (1999), 1454-1456.

[64] R. Marty, *On a splitting scheme for the nonlinear Schrödinger equation in a random medium*, Commun. Math. Sci. 4 (2006), no. 4, 679-705.

[65] R. Marty, *Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion*, Commun. Math. Sci. 19 (2021), no. 4, 1051-1069.

[66] F. Merle, *Existence of blow-up solutions in the energy space for the critical generalized KdV equation*, J. Amer. Math. Soc. 14(2001), no.3, 555-578.

[67] L.F. Mollenauer, P.V. Mamyshev, J. Gripp, M.J. Neubelt, N. Mamysheva, L. Grüner-Nielsen, T. Veng, *Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion-managed solitons*, Opt. Lett. 25 (1999), 704-706.

[68] A. Mouzard, I. Zachhuber, *Strichartz inequalities with white noise potential on compact surfaces*, Analysis & PDE 17-2 (2024), 421-454.

[69] J. Murphy, *A note on averaging for the dispersion-managed NLS*, arXiv:2403.00945 [math.AP].

[70] J. Murphy, T. Van Hoose, *Modified scattering for a dispersion-managed nonlinear Schrödinger equation*, NoDEA Nonlinear Differential Equations Appl. 29 (2022), no. 1, 11pp.

[71] J. Murphy, T. Van Hoose, *Well-posedness and blowup for the dispersion-managed non-linear Schrödinger equation*, Proc. Amer. Math. Soc. 151 (2023), 2489-2502.

[72] T. Oh, T. Robert, N. Tzvetkov, *Stochastic nonlinear wave dynamics on compact surfaces*, Annales Henri Lebesgue, Volume 6 (2023) p. 161-223.

[73] T. Oh, Y. Wang, *Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces*, J. Anal. Math. 143(2021), no.2, 723-762.

[74] T. Oh, Y. Wang, *Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces*, J. Differential Equations 269(2020), no.1, 612-640.

[75] D. Revuz, M. Yor. *Continuous martingales and Brownian motion*. Vol. 293. Springer Science & Business Media, 2013.

[76] T. Robert, *Regularization by noise for some nonlinear dispersive PDEs*, Journées équations aux dérivées partielles (2023), Exposé no. 9, 12 p.

[77] T. Robert, *Regularization by noise for some strongly non-resonant modulated dispersive PDEs*, preprint.

[78] T. Robert, *On the Kadomtsev-Petviashvili equations with rough dispersion management*, in preparation.

[79] M. Romito, L. Tolomeo, *Yet another notion of irregularity through small ball estimates*, arXiv:2207.02716 [math.PR].

[80] R. Schippa, *Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations*, PhD dissertation (2019).

[81] G. Staffilani, D. Tataru, *Strichartz estimates for a Schrödinger operator with nonsmooth coefficients*, Comm. Partial Differential Equations 27 (2002), 1337-1372.

[82] G. Stewart, *On the well-posedness for periodic nonlinear Schrödinger equations with white noise dispersion*, Stoch. PDE: Anal. Comp. (2023).

[83] T. Tanaka, *Remark on the local well-posedness for NLS with the modulated dispersion*, arXiv:2407.20005 [math.AP].
[84] S. Turitsyn, B. Bale, M. Fedoruk, *Dispersion-managed solitons in fibre systems and lasers*, Physics Reports 521 (2012), 135-203.

[85] S. K. Turitsyn, E. G. Shapiro, S. B. Medvedev, M. P. Fedoruk, V. K. Mezentsev, *Physics and mathematics of dispersion-managed optical solitons*, Comptes Rendus Physique, Académie des sciences, Éditions scientifiques et médicales 4 (2003), 145-161.

[86] N. Tzvetkov, N. Visciglia, *Global Dynamics of the 2d NLS with White Noise Potential and Generic Polynomial Nonlinearity*, Commun. Math. Phys. 401 (2023), 3109-3121.

[87] Y. Wang, *Periodic nonlinear Schrödinger equation in critical $H^s(T^n)$ spaces*, SIAM J. Math. Anal.45(2013), no.3, 1691-1703.

[88] Y. Xiao, *Hölder conditions for the local times and the hausdorff measure of the level sets of gaussian random fields*, Probability theory and related fields, 109 (1997), no.1, 129-157.

[89] D. Zhang, *Strichartz and local smoothing estimates for stochastic dispersive equations with linear multiplicative noise*, SIAM J. Math. Anal.54 (2022), no.6, 5981-6017.

[90] V. Zharnitsky, E. Grenier, C.K.R.T. Jones, S.K. Turitsyn, *Stabilizing effects of dispersion management*, Physica D 152-153 (2001), 794-817.

UNIVERSITÉ DE LORRAINE, CNRS, IECL, F-54000 NANCY, FRANCE

Email address: tristan.robert@univ-lorraine.fr