1. Introduction
In this paper, we consider the generalized nonlinear Schrödinger equation posed on the one dimensional torus 
$\mathbb{T} := \mathbb{R} / 2\pi \mathbb{Z}$,
\begin{align}
\begin{cases}
i\partial_t u + \partial_x^2 u = |u|^{p-1}u, (t,x) \in \mathbb{R} \times \mathbb{T} \\
u|_{t=0} = u_0,
\end{cases}
\end{align} for every 
$p \geq 5$ odd, and with initial data taken in the Sobolev spaces (
$\sigma \in \mathbb{R}$):
\begin{align*}
H^{\sigma}(\mathbb{T}) := \{u_0 : \mathbb{T} \to \mathbb{C} : \sum_{n \in \mathbb{Z}} \langle n \rangle^{2\sigma} |\widehat{u_0}(n)|^2 \lt +\infty \},
\end{align*} where 
$\langle n \rangle := (1+|n|^2)^{\frac{1}{2}}$ and 
$\widehat{u_0}$ refers to the Fourier transform. More precisely, we will be interested by regularities below the “energy level”, which means that σ < 1. Let us start by reviewing known results concerning the well-posedness of (pNLS).
Local well-posedness : To obtain local solutions to (pNLS), fixed point methods have been applied. It consists in working with the equivalent integral formulation of the equation, introducing for any initial data 
$u_0 \in H^{\sigma}(\mathbb{T})$ the Duhamel map 
$\Gamma_{u_0}$, defined by:
\begin{align}
\Gamma_{u_0} [u] (t) = e^{it\partial_x^2} u_0 - i \int_0^t e^{i(t-t') \partial_x^2} \big[u(t')|u(t')|^{p-1}\big] dt',
\end{align} where 
$e^{it\partial_x^2}$ is the propagator of the linear Schrödinger equation for which a fixed point is a solution, local in general. Following this approach, Bourgain proved in [Reference Bourgain6] that (pNLS) is locally well-posed for all 
$\sigma \gt \frac{1}{2} - \frac{2}{p-1}$. In this work, dispersive technologies are used. The fixed point argument is performed with the so-called Bourgain spaces 
$X^{\sigma,b}_\delta$ (for small time δ > 0 and 
$b \gt \frac{1}{2}$), which are in particular spaces that embed into the space of continuous functions 
$\mathcal C([-\delta,\delta],H^{\sigma})$, and suitable inequalities are established through Strichartz estimates.
However, since the full regime 
$\sigma \gt \frac{1}{2} - \frac{2}{p-1}$ is out of the reach of this paper, it is sufficient here to consider the regime 
$\sigma \gt \frac{1}{2}$, where 
$H^{\sigma}(\mathbb{T})$ is known to be an algebra, thereby making a more direct local theory available using only the standard space 
$\mathcal C([-\delta,\delta],H^{\sigma})$, see Proposition 3.1. In particular, this local theory does not incorporate dispersive effects.
The fixed point procedure allows us to consider for every initial data 
$u_0 \in H^{\sigma}(\mathbb{T})$ the interval:
\begin{align}
I_{max}(u_0) := \bigcup_{I \in \mathfrak{I}} , I,
\end{align} where 
$\mathfrak{I}$ is the set formed by all the intervals I containing 0 such that there exists a solution 
$u \in \mathcal C(I,H^{\sigma}(\mathbb{T}))$ to (pNLS) with initial data u 0. The local theory (see, here, Proposition 3.1) ensures that 
$\mathfrak{I}$ is non-empty, and that 
$I_{max}(u_0)$ is open. It is the maximal interval on which the solution (emanating from u 0) exists.
Deterministic globalization arguments : Once the local Cauchy has been elaborated, one naturally wonders whether the solutions obtained are global (defined on 
$\mathbb{R}$). Proving that the equation is (deterministically) globally well-posed in 
$H^{\sigma}(\mathbb{T})$ consists in establishing that:
\begin{align*}
\forall u_0 \in H^{\sigma}(\mathbb{T}), I_{max}(u_0) = \mathbb{R}.
\end{align*}Again by the fixed point algorithm, we know that there is the alternative:
\begin{align*}
&\text{If}\ I_{max}(u_0) \neq \mathbb{R},\ \text{then:} & & \| u(t) \|_{H^{\sigma}} \to + \infty, \text{as}\ t \to \partial I_{max}(u_0),
\end{align*} where 
$u \in \mathcal C(I_{max}(u_0), H^{\sigma})$ is the (maximal) solution emanating from u 0. In other words, the 
$H^{\sigma}(\mathbb{T})$-norm of non-global solutions blow up in finite time. As a consequence, if one is able to prove that:
\begin{align*}
\forall t \in I_{max}(u_0), \| u(t) \|_{H^{\sigma}} \leq f(t),
\end{align*} for a certain continuous function 
$f : \mathbb{R} \to \mathbb{R}_+$, then one can conclude that 
$I_{max}(u_0) = \mathbb{R}$, meaning that u exists for every time.
In the range of regularity 
$\sigma \geq 1$, one is able to establish such a bound using the conservation of the mass and the Hamiltonian:
\begin{align*}
M(u) &:= \int_{\mathbb{T}} |u|^2 dx, \qquad\qquad H(u) := \frac{1}{2} \int_{\mathbb{T}} |\partial_x u|^2 dx + \frac{1}{p+1} \int_{\mathbb{T}} |u|^{p+1} dx.
\end{align*} Indeed, these conservation laws imply that the 
$H^1(\mathbb{T})$-norm of the solutions stays bounded, which, by a Grönwall argument using the Duhamel formulation (1.1), yields an exponential bound for the 
$H^{\sigma}(\mathbb{T})$-norm of the solutions. Hence, from the criteria above, this means that the solution of (pNLS) are global. It is also worth noting that this exponential bound can be improved into a polynomial one using more sophisticated method such as normal form reduction and the (upside-down) I-method (see [Reference Bourgain8, Reference Colliander, Kwon and Oh13, Reference Sohinger32, Reference Staffilani33]). In the range of regularity 
$\frac{1}{2} - \frac{2}{p-1} \lt \sigma \lt 1$, this standard globalization procedure is no longer applicable, because even though the mass is still finite and conserved, the full energy:
\begin{align}
E(u) := \frac{1}{2}\int_{\mathbb{T}} |u|^2 dx + \frac{1}{2} \int_{\mathbb{T}} |\partial_x u|^2 dx + \frac{1}{p+1} \int_{\mathbb{T}} |u|^{p+1} dx,
\end{align} may be infinite (when 
$u_0 \in H^{\sigma}(\mathbb{T}) \setminus H^1(\mathbb{T})$), thereby making any hope of controlling the 
$H^1(\mathbb{T})$-norm illusory. To overcome this obstacle, a technique—the I-method—has been developed two decades ago by Colliander, Keel, Staffilani, Takaoka and Tao (see for instance in [Reference Colliander, Keel, Staffilani, Takaoka and Tao12]). The key idea is that even for σ < 1, the energy (1.3) plays a role. The method consists in considering a specific smoothing (Fourier multiplier) operator I of order 
$1-\sigma$, which maps the 
$H^{\sigma}(\mathbb{T})$-solutions into 
$H^1(\mathbb{T})$, and then use the (finite) modified energy E(Iu), acting like an “almost conservation law”, to obtain a bound on the 
$H^{\sigma}(\mathbb{T})$-norm of the solutions. In the end, the application of the I-method would allow to prove the global well-posedness for (pNLS) up to 
$\sigma \gt \sigma_p$, for some 
$\sigma_p \in [\frac{1}{2}- \frac{2}{p-1},1)$. Moreover, it is worth noting that, in principle, this threshold σp could be lowered by incorporating dispersive technologies (normal form reduction, Strichartz estimates, resonant decomposition) as it was performed for example in [Reference Bernier, Grébert and Robert3, Reference Bourgain9, Reference Li, Wu and Xu25, Reference Schippa30]. For a pleasant introduction to the I-method, we refer to [Reference Erdoğan and Tzirakis15] and [Reference Tao35].
Probabilistic globalization argument : In this paper, we explore an other direction to globalize the local solutions of (pNLS) adapted to random Gaussian initial data. Here the initial data take the form of the following random Fourier series (
$s\in \mathbb{R}$):
\begin{align}
u_0^\omega := \sum_{n \in \mathbb{Z}} \frac{g_n(\omega)}{\langle n \rangle^s} e^{inx},
\end{align} where 
$(g_n)_{n\in \mathbb{Z}}$ are independent complex standard Gaussian random variables on a probability space 
$(\Omega, \mathcal F, \mathbb{P})$. For 
$\sigma \lt s - \frac{1}{2}$, this series converges in the space 
$L^2(\Omega, H^{\sigma})$, therefore defining a random variable valued in 
$H^{\sigma}(\mathbb{T})$ whose law, denoted µs, defined a Gaussian (probability) measure on 
$H^{\sigma}(\mathbb{T})$:
\begin{align*}
\mu_s := \text{law}(u_0^\omega) = (u_0^\omega)_\# \mathbb{P},
\end{align*} where the symbol 
$\#$ refers to the push-forward measure. For details about Gaussian measures, we refer to [Reference Kuo24] and [Reference Bogachev4]. It is well-know that almost surely:
\begin{align*}
u_0^\omega &\in \bigcap_{r \lt s -\frac{1}{2}} H^r(\mathbb{T}) & &\text{and} & u_0^\omega & \notin H^{s-\frac{1}{2}}(\mathbb{T}).
\end{align*} Since we are interested in initial data below the energy level, we take 
$s \leq \frac{3}{2}$, so that, again, the standard (deterministic) globalization argument by conservation of the energy is not applicable. Also, we take s > 1 to be able to apply the local theory from Proposition 3.1. Since we aim to construct a global flow to (pNLS), we are interested in the global well-posedness set:
\begin{align}
\mathcal G := \big\{u_0 \in H^{\sigma}(\mathbb{T}) : I_{max}(u_0) = \mathbb{R} \big\},
\end{align} where we recall that 
$I_{max}(u_0)$ is the maximal interval of existence of the solution generated by u 0, defined in (1.2). Let us briefly justify that 
$\mathcal G$ is a Borel set. It is even a 
$\mathcal G_\delta$ set. To see this, we first write:
\begin{align*}
\mathcal G = \bigcap_{N \in \mathbb{N}} \{u_0 \in H^{\sigma}(\mathbb{T}) : [-N,N] \subset I_{max}(u_0) \},
\end{align*} and then, we observe that every set in the intersection is open. Indeed, by continuous dependence on the initial data (which is a consequence of the fixed point procedure to construct the solution), if u 0 generates a solution that lives on 
$[-N,N]$, then so do the solutions generated by initial data close enough to u 0.
We recall that for 
$\sigma \geq 1$, we have 
$\mathcal G = H^{\sigma}(\mathbb{T})$. Below the energy level, we prove as a first result that:
Theorem 1.1 Let 
$p \geq 5$ an odd integer. Let 
$\frac{3}{2} \geq s \gt s_p$, where sp is defined by:
\begin{align}
s_p = \frac{3}{2} - \frac{p}{4}(1 - \sqrt{1-\frac{8}{p^2}}),
\end{align} Then, for 
$\sigma \lt s - \frac{1}{2}$ close to 
$s-\frac{1}{2}$, we have
\begin{align*}
\mu_s(\mathcal G^c) = 0.
\end{align*} The pioneering work [Reference Bourgain7] initiated a general strategy—the so-called Bourgain’s invariant measure argument—to prove analogous results with respect to invariant measures, notably the Gibbs measure; and it has since been applied for various models. Here, this method cannot be followed directly, since the Gaussian measures µs are not expected to be invariant. However, we will prove Theorem 1.1 in a similar spirit, but relying instead on the quasi-invariance of these Gaussian measures (which will be combined with a deterministic Cauchy theory). In [Reference Forlano and Tolomeo18], this idea was explored in a more involved setting for the cubic fractional NLS, with a probabilistic Cauchy theory for low regularity Gaussian initial data (below 
$L^2(\mathbb{T})$).
The study of quasi-invariance of Gaussian measures was initiated by Tzvetkov in [Reference Tzvetkov38]. Let us recall that considering the flow:
\begin{align}
\Phi : (t,u_0) \longmapsto \Phi_t(u_0):=\text{the solution of~(pNLS) with initial data}\ u_0\ \text{at time}\ t,
\end{align} we say that µs is quasi-invariant under the flow Φ if for every time 
$t\in \mathbb{R}$:
\begin{align*}
\text{law}(\Phi_t(u_0^\omega)) &\ll \text{law}(u_0^\omega), & & \text{that is,} & (\Phi_t)_\# \mu_s &\ll \mu_s,
\end{align*} where 
$\ll$ denotes absolute continuity of measures, meaning that for all Borel set A in 
$H^{\sigma}(\mathbb{T})$:
\begin{align*}
\mu_s(A) = 0 \implies \mu_s( \Phi_t^{-1}A) = 0.
\end{align*} Here, we encounter an obstruction because the knowledge of the local theory for (pNLS) does not allow us a priori to consider, for a fixed time 
$t \in \mathbb{R}$, the object 
$\Phi_t(u)$ for µs-almost every u. It then appears that the idea of using the quasi-invariance of µs precisely to construct the flow is a circular argument. For this reason, we consider first a truncated version of (pNLS) for 
$N \in \mathbb{N}$:
\begin{align}
\begin{cases}
i\partial_t u + \partial_x^2 u = \pi_N \left(|\pi_Nu|^{p-1}\pi_Nu \right), (t,x) \in \mathbb{R} \times \mathbb{T} \\
u|_{t=0} = u_0,
\end{cases}
\end{align} where πN is the sharp Fourier projector onto frequencies of absolute value 
$\leq N$. We denote by 
$\Phi^N$ its flow (similarly defined as in (1.7)), and we refer to it as the truncated flow. This equation is interesting for few reasons: the solutions are global, they conserve the truncated energy:
\begin{align}
E_N(u) := M(u)+ H(\pi_Nu),
\end{align} they share a common local Cauchy theory with (pNLS) (see Proposition (3.1)), and they approximate the solutions of (pNLS) (see Proposition 3.2). Moreover, the Gaussian measures µs are quasi-invariant under the truncated flow (for every 
$s \in \mathbb{R}$), with an explicit formula for the Radon-Nikodym derivatives; more precisely, for every 
$t\in \mathbb{R}$:
\begin{align}
(\Phi^N_t)_\# \mu_s &= g_{s,N,t} d\mu_s, & g_{s,N,t} &:= \exp \big( -\frac{1}{2}( \| \pi_N \Phi^N_{-t}(u)\|_{H^s(\mathbb{T})}^2 - \| \pi_N u\|_{H^s(\mathbb{T})}^2 )\big).
\end{align}These properties are not proven here; however, the proofs can be found in [Reference Knezevitch22, Reference Knezevitch23]. With these ingredients, the crucial point to globalize the local solutions up to an arbitrary time T > 0, consists in establishing a proper flow tail estimate (see later in Lemma 3.7) which take the form:
\begin{align*}
\mu_s\big( u_0 : \sup_{t \in [0,T]} \| \Phi_t^Nu_0 \|_{H^{\sigma}} \gt M \big) \leq f_s(T,M),
\end{align*} where 
$f_s(T,M)$ is independent of N and decays sufficiently fast to 0 as 
$M \to \infty$. This estimate is obtained through an iterative argument (see proof of Lemma 3.7)—in 
$\lfloor \frac{T}{\delta} \rfloor$ steps, with 
$\delta \sim M^{1-p}$ the appropriate local existence time—which, at every step k, yields the quantity:
\begin{align*}
(\Phi^N_{k \delta})_\# \mu_s \big( \| u_0 \|_{H^{\sigma}} \gt M \big) &= g_{s,N,k \delta} d\mu_s \big( \| u_0 \|_{H^{\sigma}} \gt M \big) \\
& \leq \| g_{s,N,k \delta} \|_{L^q(d\mu_s)} \mu_s \big( \| u_0 \|_{H^{\sigma}} \gt M \big)^{1-\frac{1}{q}},
\end{align*} estimated with the Hölder inequality (
$1 \lt q \lt \infty$). Unfortunately, here, the (uniform in N) 
$L^q(d\mu_s)$-integrability for the Randon-Nikodym derivatives (1.10) is out of reach, and to obtain such a bound, we need to add a suitable cutoff to µs. This is an other central point in our analysis. Using ideas from [Reference Forlano and Tolomeo18] and [Reference Tzvetkov and Visciglia39], we choose a renormalized-energy cutoff. Indeed, since the standard energy E, for which in our regime 
$s \leq \frac{3}{2}$ we have:
\begin{align*}
E(u_0) = + \infty \mu_s-\text{almost-surely},
\end{align*} cannot be considered directly, we introduce instead a renormalization of it, which consists in removing a divergent term. More precisely, introducing the truncated renormalizations (for 
$N \in \mathbb{N}$):
\begin{align}
\mathcal E_N (u) &:= M(u) + \big| H(\pi_N u) - \sigma_N \big| & & \text{with:} & \sigma_N := \frac{1}{2} \sum_{|n| \leq N} \frac{|n|^2}{\langle n \rangle^{2s}},
\end{align} that are conserved by 
$\Phi^N$, since 
$\Phi^N$ conserves EN in (1.9), we have for 
$\frac{5}{4} \lt s \leq \frac{3}{2}$ and every 
$q \in [1,\infty)$ (see Proposition 5.1):
\begin{align*}
& (\mathcal E_N)_{N \in \mathbb{N}}\ \text{is Cauchy in}\ L^q(d\mu_s),\ \text{and therefore:} & & \exists \mathcal E \in L^q(d\mu_s), \mathcal E_N \underset{N}{\longrightarrow} \mathcal E \ \text{in}\ L^q(d\mu_s).
\end{align*}Hence, we will work with the cutoff Gaussian measures:
\begin{align}
\mu_{s,R,N} &:= \chi_R(\mathcal E_N(u)) d\mu_s, & \mu_{s,R} &:= \chi_R(\mathcal E(u)) d\mu_s,
\end{align} for 
$s \in (\frac{5}{4}, \frac{3}{2}]$, R > 0, and where 
$\chi_R = \chi(\frac{.}{R})$, with 
$\chi : \mathbb{R} \to [0,1]$ smooth such that 
$\chi \equiv 1$ on 
$[-\frac{1}{2},\frac{1}{2}]$ and 
$\chi \equiv 0$ on 
$[-1,1]^c$. We will see the benefits of these cutoffs in Proposition 5.4. It is however worth noting that we are limited here to the values of s in 
$(\frac{5}{4},\frac{3}{2}]$. The main tools to obtain Lq-bounds (with respect to these measures) for the Radon-Nikodym derivatives will be a Poincaré-Dulac normal form reduction, suitable bounds on the resonant function and the symmetrized derivative (see Definition 2.2), and the Boué-Dupuis variational formula.
Properties of the resulting flow : Thanks to Theorem 1.1, (recalling that 
$\mathcal G$ is defined in (1.5)) we have for every time 
$t \in \mathbb{R}$ a well-defined flow to (pNLS):
\begin{align*}
\Phi_t : \mathcal G \longrightarrow \mathcal G,
\end{align*}for which we prove the following properties:
Theorem 1.2 (Quantitative quasi-invariance)
Let 
$\frac{3}{2} \geq s \gt s_p$. Then, for every 
$t\in \mathbb{R}$, there exists a non-negative measurable function 
$g_{s,t}$ such that:
\begin{align*}
(\Phi_t )_\# \mu_s = g_{s,t} \mu_s.
\end{align*} Furthermore, the renormalized energy 
$\mathcal E$ is invariant under the flow Φ, meaning that for every 
$t \in \mathbb{R}$,
\begin{align}
\mathcal E(\Phi_t u) = \mathcal E(u), \mu_s-\textrm{almost everywhere}
\end{align} Moreover, for every R > 0, the cutoff Gaussian measure 
$\mu_{s,R}$ is transported by the flow as:
\begin{align*}
(\Phi_t )_\# \mu_{s,R} = g_{s,t} \mu_{s,R},
\end{align*} and, for every 
$q \in [1,+\infty)$, there exists a constant 
$C_{s,R,q} \gt 0$ such that:
\begin{align}
\| g_{s,t}\|_{L^q(d\mu_{s,R})} \leq \exp \big( C_{s,R,q}(1+|t|)^A \big),
\end{align}where A > 0 is independent of q and R > 0.
Remark 1.3. In the theorem we see all the measures on 
$H^{\sigma}(\mathbb{T})$. For example, if B is a Borel set of 
$H^{\sigma}(\mathbb{T})$,
\begin{align*}
(\Phi_t )_\# \mu_s(B) := (\Phi_t )_\# \mu_s(B \cap \mathcal G) = \mu_s(\Phi_t^{-1}(B \cap \mathcal G)),
\end{align*}Concerning (1.13), one can also prove the following:
\begin{align}
\mathcal E(\Phi_t u) = \mathcal E(u), \text{for}\ \mathcal L^1 \otimes \mu_s-\text{almost every} (t,u) \in \mathbb{R} \times H^{\sigma},
\end{align} where 
$\mathcal L^1$ is the Lebesgue measure on 
$\mathbb{R}$. For a more detailed discussion see Remark 4.2.
This theorem provides an improved version of the quasi-invariance of the Gaussian measures µs, as it includes 
$L^q(d\mu_{s,R})$-bounds of the Radon-Nikodym derivatives. Similar results were obtained in [Reference Debussche and Tsutsumi14, Reference Forlano16, Reference Forlano and Seong17, Reference Genovese, Lucà and Tzvetkov20, Reference Knezevitch23, Reference Planchon, Tzvetkov and Visciglia29]. We can derive from these bounds quantitative results on the solutions. For example, we are able to obtain a polynomial bound on the Sobolev norm of the solutions:
Corollary 1.4. Let 
$\frac{3}{2} \geq s \gt s_p$. There exists A > 0 such that for every 
$t \in \mathbb{R}$, we have for µs-almost every u 0:
\begin{align}
\| \Phi_t u_0 \|_{H^{\sigma}(\mathbb{T})} \lesssim_{s,u_0} (1+|t|)^A,
\end{align} where 
$\sigma \lt s -\frac{1}{2}$ is closed to 
$s -\frac{1}{2}$.
Further remarks and comments: The threshold 
$s \gt s_p$ is technical, it comes from the proof of the Lq-bounds on the Radon-Nikodym derivatives in Proposition 3.6. Even though, the formula for sp is complicated, we have:
\begin{align*}
s_p & \lt \frac{3}{2} - \frac{1}{p} & &\text{and asymptotically,} & s_p & \sim_{p \to \infty} \frac{3}{2} - \frac{1}{p}.
\end{align*} It is not excluded that this threshold could be improved. However, it is also important to note that there is still the limitation 
$s \gt \frac{5}{4}$ appearing in Proposition 5.1 to consider the renormalized energy, which is crucial in our analysis.
Theorem 1.1 ensures that Gaussian initial data of regularity 
$H^{s_p - \frac{1}{2}+}(\mathbb{T})$ generate global solutions of (pNLS). On the other hand, a deterministic approach using the I-method would in principle allow to prove the global-wellposedness in 
$H^{\sigma}(\mathbb{T})$, with 
$\sigma \gt \sigma_p$, for some 
$\sigma_p \lt 1$. For example in the case p = 5, the (deterministic) global well-posedness was obtained for all 
$\sigma \gt \frac{2}{5}$ in [Reference Li, Wu and Xu25] (see also [Reference Bernier, Grébert and Robert3]), which is beyond the scope of this paper. Instead of considering these two approaches independently, it could be interesting to explore whether the strengths of both methods can be leveraged together.
In the case p = 5, Theorem 1.2 completes a result established in [Reference Knezevitch23], where the quasi-invariance with Lq bounds on the Radon-Nikodym derivatives with respect to (standard) energy-cutoff Gaussian measures is proven for the full range 
$s \gt \frac{3}{2}$.
2. Preliminaries
In this section, we gather some preliminaries. First, we state a lemma which explains how the condition on s in the theorems will appear. Second, we introduce quantities that will play a role later. Third, we construct our probabilistic toolbox.
Lemma 2.1. Let s > 0 and 
$p \geq 5$. Then,
\begin{align*}
(3-2s)(2s + p - 3) \lt 2 \iff s \gt s_p,
\end{align*}where sp is defined in (1.6).
Proof. We have,
\begin{align}
(3-2s)(2s + p - 3) \lt 2 \iff (3-2s)^2 -(3-2s) p + 2 \gt 0.
\end{align} Moreover, the polynomial 
$x \mapsto x^2 - px + 2$ is positive exactly when 
$x \lt \frac{1}{2}(p-\sqrt{p^2-8})$ or 
$x \gt \frac{1}{2}(p+\sqrt{p^2-8})$, so (2.1) is satisfied if and only if:
\begin{align*}
s & \gt \frac{3}{2} - \frac{1}{4}(p - \sqrt{p^2-8}) = s_p & & \text{or} & s & \lt \frac{3}{2} - \frac{1}{4}(p + \sqrt{p^2-8}).
\end{align*} Since 
$p \geq 5$, we have 
$\frac{3}{2} - \frac{1}{4}(p + \sqrt{p^2-8}) \lt 0$, and since we consider positive values of s, the condition on the right is never satisfied. This completes the proof.
2.1. The resonant function and the symmetrized derivatives
In this paragraph, we introduce and study quantities that emerge in the Poincaré-Dulac normal reduction of 
$\S$ 6. In particular, we provide a convenient lower bound on the resonant function. Here we recall that 
$p \geq 5$ is an odd integer.
Definition 2.2. For 
$\vec{k}=(k_1,...,k_{p+1})\in \mathbb{Z}^{p+1}$, we define the quantities:
\begin{align*}
\psi_{2s}(\vec{k}) &:= \sum_{j=1}^{p+1} (-1)^{j-1}|k_j|^{2s}, & \Omega(\vec{k}) &:= \sum_{j=1}^{p+1} (-1)^{j-1} k_j^2,
\end{align*}called respectively the symmetrized derivative (of order 2s) and the resonant function.
Moreover, we consider:
\begin{align*}
\Psi_{2s}^{(0)}(\vec{k}) &:= \mathbb{1}_{\Omega(\vec{k}) = 0} \psi_{2s}(\vec{k}), & \Psi_{2s}^{(1)}(\vec{k}) &:= \mathbb{1}_{\Omega(\vec{k}) \neq 0} \frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}.
\end{align*}For the two following lemmas we use this notation:
Notation 2.3. Given a set of frequencies 
$k_1,...,k_m \in \mathbb{Z}$, we denote by 
$k_{(1)},...,k_{(m)}$ a rearrangement of the 
$k_j's$ such that
\begin{align*}
|k_{(1)}| \geq |k_{(2)}| \geq ... \geq |k_{(m)}|.
\end{align*}Lemma 2.4. Let s > 1. For every 
$\vec{k}=(k_1,...,k_{p+1}) \in \mathbb{Z}^{p+1}$ such that:
\begin{align*}
\sum_{j=1}^{p+1} (-1)^{j-1} k_j = 0,
\end{align*}we have,
\begin{align}
| \psi_{2s}(\vec{k}) | \leq C_s |k_{(1)}| ^{2(s-1)} ( | \Omega(\vec{k}) |+ k_{(3)} ^2),
\end{align} where the constant 
$C_s \gt 0$ only depends on s (and p). As a consequence, we have :
\begin{align}
| \Psi^{(0)}_{2s}(\vec{k}) | \leq C_s |k_{(1)}| ^{2(s-1)} k_{(3)}^2,
\end{align}and,
\begin{align}
| \Psi^{(1)}_{2s}(\vec{k}) | \leq C_s \big( |k_{(1)}| ^{2(s-1)} + |k_{(1)}| ^{2(s-1)} \frac{k_{(3)}^2}{|\Omega(\vec{k})|} \big).
\end{align}These estimates are standard; for proofs, we refer for example to [Reference Sun and Tzvetkov34], Lemma 4.1. We now state a lemma that provides a practical lower bound on the resonant function Ω. This is a very slight adaptation of Lemma 6.1 from [Reference Knezevitch22], where a proof is given for p = 5.
Lemma 2.5. There exists a constant 
$c_p \gt 0$ such that for every 
$ (k_1,...,k_{p+1}) \in \mathbb{Z}^{p+1} \setminus \{0\}$ such that:
\begin{align}
\sum_{j=1}^{p+1} (-1)^{j-1} k_j & = 0 & \textrm{and}& & |k_{(4)}| & \leq \frac{1}{10p} |k_{(3)}|,
\end{align}we have:
\begin{align}
|\Omega(\vec{k})| \geq c_p |k_{(1)}||k_{(3)}|,
\end{align}Remark 2.6. As a consequence, note that if 
$\Omega(\vec{k}) = 0$ and 
$k_1-k_2+...-k_{p+1} = 0$ and 
$k_{(3)} \neq 0$, then we have 
$|k_{(3)}| \lt 10 p |k_{(4)}|$.
2.2. Probabilistic tools
Lemma 2.7. Let 
$s \in \mathbb{R}$ and 
$\sigma \lt s-\frac{1}{2}$. Let Y be the Gaussian random variable defined in (1.4). For 
$N \in \mathbb{N}$, set 
$Y_N:= \pi_N Y$. Then, for every 
$q \in [1,\infty)$, the sequence 
$(Y_N)_N$ converges to Y in 
$L^q(\Omega,W^{\sigma,\infty})$. In particular,
\begin{align*}
\sup_{N \in \mathbb{N}} \mathbb{E}\big[ \| Y_N \|^q_{W^{\sigma,\infty}} \big] \leq C_q \lt \infty.
\end{align*}We refer to [Reference Forlano and Seong17], Lemma 5.3, for a proof of this lemma.
Lemma 2.8. (Wiener Chaos estimate)
Let 
$k \in \mathbb{N}$ and 
$c : \mathbb{Z}^k \rightarrow \mathbb{C}$. Consider the following expression:
\begin{align*}
S(\omega) := \sum_{(n_1,...,n_k)\in \mathbb{Z}^k} c(n_1,...,n_k) g^{\iota_1}_{n_1}(\omega)...g^{\iota_k}_{n_k}(\omega),
\end{align*} where the gn are complex standard i.i.d Gaussians, and 
$\iota_j \in \{-,+\}$ is the complex conjugation or the identity whether 
$\iota_j = -$ or 
$\iota_j = +$ respectively.
Suppose that 
$S\in L^2(\Omega)$, then, there exists C > 0 such that for every 
$q \in [1,\infty)$:
\begin{align*}
\| S \|_{L^q(\Omega)} \leq C q^{\frac{k}{2}} \| S\|_{L^2(\Omega)}.
\end{align*}For a proof of this lemma, we refer to [Reference Simon31] (see also [Reference Thomann and Tzvetkov36] and [Reference Tzvetkov37]).
In this paper, an important element to prove Lq-bounds on the Radon-Nikodym derivatives (1.10) is the Boué-Dupuis variational formula introduced in [Reference Boué and Dupuis5]. See also the related paper by Üstünel [Reference Üstünel1]. This approach was adopted by Barashkov and Gubinelli in [Reference Barashkov and Gubinelli2], and it has since been applied in the context of quasi-invariance in [Reference Coe and Tolomeo11, Reference Forlano16–Reference Forlano and Tolomeo18]. More precisely, we will use the following simplified version of the Boué-Dupuis formula, which has been stated and proven in Lemma 2.6 of [Reference Forlano and Tolomeo18].
Lemma 2.9. (see Lemma 2.6 of [Reference Forlano and Tolomeo18])
Let 
$s\in \mathbb{R}$, 
$N \in \mathbb{N}$, and 
$Y \sim \mu_s$ the Gaussian random variable defined in (1.4). Let 
$\mathcal F:C^\infty(\mathbb{T})\rightarrow\mathbb{R}$ be a measurable function such that 
$\mathbb{E}[|\mathcal F_-(\pi_N Y)|^p]~ \lt \infty$, for some p > 1, where 
$\mathcal F_-$ denotes the negative part of 
$\mathcal F$. Then,
\begin{align}
\log \int e^{\mathcal F(\pi_N u)} d\mu_s \leq \mathbb{E} \big[\sup_{V \in H^s} \{\mathcal F(\pi_N Y + \pi_N V) - \frac{1}{2}\left\lVert V \right\rVert^2_{H^s} \} \big].
\end{align}3. Quantitative quasi-invariance and globalization of the local solutions
In this section, we prove Theorem 1.1. To do so, we use: a local theory that is applicable to (pNLS) and (1.8) for all 
$N \in \mathbb{N}$ (every parameter in Proposition 3.1 will be independent of 
$N \in \mathbb{N}$), a suitable approximation of the original flow Φ by the truncated flow 
$\Phi^N$ (see Proposition 3.2), and a flow tail estimate (see Lemma 3.7). Note that we assume here the crucial bounds (3.3) which are proved later in 
$\S$ 7 (more precisely in paragraph 7.3).
3.1. Local Cauchy theory and approximation
We first recall the local well-posedness of (pNLS) in 
$H^{\sigma}(\mathbb{T})$, 
$\sigma \gt \frac{1}{2}$. Even though we already know that (1.8) is globally well-posed, the fact that it verifies the same local theory than (pNLS) is important. We consider the Duhamel map 
$\Gamma_{u_0}$ in (1.1), and we similarly define 
$\Gamma^N_{u_0}$ replacing 
$|u|^{p-1}u$ by 
$\pi_N (|\pi_Nu|^{p-1}\pi_Nu)$.
Proposition 3.1. Let 
$\sigma \gt \frac{1}{2}$. There exist universal constants 
$C_0 \gt 0$ and 
$c_1 \gt 0$ such that for every 
$R_0 \gt 0$, denoting 
$R := C_0 R_0$ and 
$\delta := c_1 R_0^{1-p}$, we have that for every 
$\| u_0 \|_{H^{\sigma}} \leq R_0$, and every 
$N \in \mathbb{N} $, the maps:
\begin{align*}
\Gamma_{u_0}&: B_R(\delta) \to B_R(\delta) & & \textrm{and,} & \Gamma^{N}_{u_0}: &B_R(\delta) \to B_R(\delta),
\end{align*} are all contractions, with the same universal contraction coefficient 
$\gamma \in (0,1)$, and where 
$B_R(\delta)$ is the closed centered ball of radius R in 
$\mathcal C([-\delta, \delta],H^{\sigma}(\mathbb{T}))$.
Proposition 3.2. Let 
$K \subset H^{\sigma}(\mathbb{T})$ a compact set, and let 
$I \subset \mathbb{R}$ a compact interval containing 0 such that every solution of (pNLS) generated by an initial data in K lives on I. Then,
\begin{align}
\sup_{u_0 \in K} \sup_{t \in I} \| \Phi_t u_0 - \Phi^N_t u_0 \|_{H^{\sigma}} \underset{N \to \infty }{\longrightarrow} 0.
\end{align}Proofs of the two previous propositions can be found in [Reference Knezevitch22, Reference Knezevitch23].
3.2. Transport of cutoff Gaussian measures
We know that the truncated flow 
$\Phi^N$ transports the Gaussian measures as in (1.10); thanks to the fact that the renormalized energy 
$\mathcal E_N$ is preserved by 
$\Phi^N$, we deduce that it transports the cutoff Gaussian measures (1.12) similarly:
Proposition 3.3. Let 
$N \in \mathbb{N}$, 
$t \in \mathbb{R}$, and R > 0. Then,
\begin{align}
(\Phi^N_t)_\# \mu_{s,R,N} = g_{s,N,t} d\mu_{s,R,N}.
\end{align} The proof is standard, but we provide it nonetheless to emphasize the importance of the conservation of the cutoff 
$\chi_R(\mathcal E_N(u))$ by 
$\Phi^N$. First we need the following lemma:
Lemma 3.4. Let X,Y be two topological spaces, and 
$\Upsilon : X \to Y$ an homeomorphism. We consider a measure 
$\mu : \mathcal B(X) \to [0,\infty]$ (on the Borel σ-algebra of X) and a non-negative measurable function 
$f : X \to [0,\infty]$. Then, the density measure 
$f d\mu$ is transported by ϒ as:
\begin{align*}
\Upsilon_\# (f d\mu) = (f \circ \Upsilon^{-1}) \Upsilon_\# \mu
\end{align*}.
Let us briefly recall the proof of this lemma:
Proof. For 
$\psi : Y \to [0,\infty]$ a non-negative measurable function, we have by definition of a push-forward measure:
\begin{align*}
\int_Y \psi(v) d\Upsilon_\# (f d\mu)(v) = \int_X \psi(\Upsilon(u)) f(u) d\mu(u) = \int_Y \psi(v) (f \circ \Upsilon^{-1} )(v) d\Upsilon_\# d\mu(v),
\end{align*} which indeed implies that 
$\Upsilon_\# (f d\mu) =(f \circ \Upsilon^{-1}) \Upsilon_\# \mu $.
Next, we move on to the proof of Proposition 3.3.
Proof. Since 
$\Phi^N_t$ is bijective, with inverse 
$\Phi^N_{-t}$, Lemma 3.4 gives:
\begin{align*}
(\Phi^N_t)_\# \mu_{s,R,N} = (\Phi^N_t)_\# \big( \chi_R(\mathcal E_N(u)) d\mu_s \big) = \chi_R(\mathcal E_N(\Phi^N_{-t}u))(\Phi^N_t)_\# \mu_s.
\end{align*} By (1.10) and the conservation 
$\mathcal E_N(\Phi^N_{-t}u) = \mathcal E_N(u)$, we deduce that:
\begin{align*}
(\Phi^N_t)_\# \mu_{s,R,N} = \chi_R(\mathcal E_N(u))g_{s,N,t} d\mu_s = g_{s,N,t} d\mu_{s,R,N}.
\end{align*}as desired.
Remark 3.5. The fact that the cutoff in (1.12) is conserved by the flow is a crucial element in our analysis. In the context of quasi-invariance, it is also pertinent to consider cutoffs on the 
$H^{\sigma}(\mathbb{T})$-norm (
$\sigma \lt s-\frac{1}{2}$) (see for instance [Reference Coe and Tolomeo11, Reference Knezevitch22, Reference Sun and Tzvetkov34]), considering the measures 
$\gamma_{s,R} := \mathbb{1}_{B_R^{H^{\sigma}}} \mu_s$, which are transported by the truncated flow as:
\begin{align*}
(\Phi^N_t)_\# \gamma_{s,R} = g_{s,t,N} \mathbb{1}_{\Phi^N_t(B_R)} d\mu_s.
\end{align*} We see that the cutoff on BR is transported to the cutoff on 
$\Phi^N_t(B_R)$. This makes difficult any iterative argument as in the next paragraph, especially when we do not have any control on the balls 
$\Phi^N_t(B_R)$.
3.3. The globalization argument
In this paragraph we prove Theorem 1.1. For convenience, we assume here the propositions 5.1 and 5.2 on the cutoff Gaussian measures, see 
$\S$ 5 for the proofs; and the following quantitative bounds on the Radon-Nikodym derivatives, see 
$\S$ 7 for the proof.
Proposition 3.6. Let 
$\frac{3}{2} \geq s \gt s_p$, where sp is defined in (1.6). There exists a constant A > 0, such that for every 
$N \in \mathbb{N}$, R > 0, 
$t \in \mathbb{R}$:
\begin{align}
\| g_{s,N,t}\|_{L^q(d\mu_{s,R,N})} \leq \exp \big( C_{s,R,q}(1+|t|)^A \big),
\end{align} where the constant 
$C_{s,R,q} \gt 0$ is independent on N.
Next, we decompose the proof of Theorem 1.1 in two lemmas; then, the final argument is given at the end of the present paragraph.
Lemma 3.7. (Flow tail estimate)
Let T > 0 and 
$\sigma \lt s - \frac{1}{2}$ close to 
$s - \frac{1}{2}$. Then,
\begin{align}
\mu_{s,R,N}\big(u_0 : \sup_{|t| \leq T} \| \Phi_t^Nu_0 \|_{H^{\sigma}} \gt M \big) \leq e^{C_{s,R} (1+T)^A} e^{-\alpha M^2}.
\end{align}for some constant α > 0.
Proof. Let us consider the universal constants 
$C_0,c_1 \gt 0$ from Proposition 3.1. Introducing 
$\delta = c_1 (\frac{M}{C_0})^{1-p}$, we know that for every initial data 
$v_0 \in H^{\sigma}$,
\begin{align*}
\| v_0 \|_{H^{\sigma}} \leq \frac{M}{C_0} \implies \sup_{-\delta \leq t \leq \delta} \| \Phi^N_t v_0 \|_{H^{\sigma}} \leq M.
\end{align*} Thus, using the additivity of the flow, and denoting 
$m:= \lfloor \frac{T}{\delta}\rfloor$, we have:
\begin{align*}
\bigcap_{k=-m}^{m} \big\{u_0 : \| \Phi^N_{k \delta} u_0\|_{H^{\sigma}} \leq \frac{M}{C_0} \big\} \subset \big\{u_0 : \sup_{|t| \leq T} \| \Phi^N_t \|_{H^{\sigma}} \leq M \big\}.
\end{align*}As a consequence,
\begin{align*}
&\mu_{s,R,N}\big(u_0 : \sup_{|t| \leq T} \| \Phi_t^Nu_0 \|_{H^{\sigma}} \gt M \big) \leq \sum_{k=-m}^m \mu_{s,R,N}\big( u_0 : \| \Phi^N_{k \delta} u_0\|_{H^{\sigma}} \gt \frac{M}{C_0} \big) \\
& = \sum_{k=-m}^m (\Phi^N_{k \delta})_\#\mu_{s,R,N}\big( \| u_0\|_{H^{\sigma}} \gt \frac{M}{C_0} \big) \leq \sum_{k=-m}^m \| g_{s,N,k\delta} \|_{L^2(d\mu_{s,R,N})} \mu_{s,R,N}\big(\| u_0\|_{H^{\sigma}} \gt \frac{M}{C_0} \big)^{\frac{1}{2}}.
\end{align*} Now, on the one hand, 
$\| g_{s,N,k\delta} \|_{L^2(d\mu_{s,R,N})} \leq e^{C_{s,R} (1+T)^A}$ by (3.3); and on the other hand, we have the standard Gaussian tail estimate (see for example Fernique’s theorem in [Reference Bogachev4] or [Reference Kuo24]):
\begin{align*}
\mu_{s,R,N}\big(\| u_0\|_{H^{\sigma}} \gt \frac{M}{C_0} \big) \leq \mu_s\big(\| u_0\|_{H^{\sigma}} \gt \frac{M}{C_0} \big) \lesssim e^{-\alpha M^2},
\end{align*}for some constant α > 0. Using this in the estimate above yields:
\begin{align*}
\mu_{s,R,N}\big(u_0 : \sup_{|t| \leq T} \| \Phi_t^Nu_0 \|_{H^{\sigma}} \gt M \big) &\lesssim (\frac{T}{\delta} + 1) e^{C_{s,R} (1+T)^A} e^{-\alpha M^2} \\
&\lesssim (1+T)e^{C_{s,R} (1+T)^A} (1+M)^{p-1}e^{-\alpha M^2}.
\end{align*} Hence, taking 
$C_{s,R} \gt 0$ slightly larger and α > 0 slightly smaller leads to (3.4).
Lemma 3.8. Let T > 0 and 
$K \subset H^{\sigma}(\mathbb{T})$ a compact set, where 
$\sigma \lt s-\frac{1}{2}$ close to 
$s-\frac{1}{2}$. There exists a Borel subset 
$\Sigma_{K,T}$ of K, with full µs-measure in K, such that every solutions generated by an initial data in 
$\Sigma_{K,T}$ lived on 
$[-T,T]$. In other words,
\begin{align*}
\mu_s( K \setminus \Sigma_{K,T}) &= 0, & & \textrm{and:} & \forall u_0 \in \Sigma_{K,T}&, I_{max}(u_0) \supset [-T,T],
\end{align*} where we recall that 
$I_{max}(u_0)$ is the maximal interval on which the solution of (pNLS) with initial data u 0 exists, see (1.2).
Proof. Without loss of generality, we work on the interval 
$[0,T]$. We introduce an increasing sequence 
$\{R_l\}_{l \in \mathbb{N}}$ of positive numbers tending to infinity. Our strategy consists in constructing first, for every Rl, and all integers M such that 
$K \subset B_M^{H^{\sigma}}$, Borel subsets 
$\Sigma_{K,T,M,R_l}$ of K, such that:
\begin{align*}
\lim_{M \to \infty} \mu_{s,R_l}(K \setminus \Sigma_{K,T,M,R_l}) &= 0 & &\text{and,} & \forall u_0 \in \Sigma_{K,T,M,R_l}&: I_{max}(u_0) \supset [0,T].
\end{align*}Indeed, if we do so, then the following Borel subset of K:
\begin{align*}
\Sigma_{K,T} := \bigcup_{l,M} \Sigma_{K,T,M,R_l},
\end{align*}would satisfy:
\begin{align*}
&\mu_s(K \setminus \Sigma_{K,T}) = \mu_s\big( \bigcap_{l,M} K \setminus \Sigma_{K,T,M,R_l} \big) = \lim_{l' \to \infty} \mu_{s,R_{l'}} \big( \bigcap_{l,M} K \setminus \Sigma_{K,T,M,R_l} \big) \\
&\leq \liminf_{l' \to \infty} \mu_{s,R_{l'}} \big( \bigcap_{M} K \setminus \Sigma_{K,T,M,R_{l'}} \big) \leq \liminf_{l' \to \infty} \big[ \liminf_{M \to \infty } \mu_{s,R_{l'}} (K \setminus \Sigma_{K,T,M,R_{l'}}) \big] =0.
\end{align*}Therefore, this would complete the proof.
In what follows, we construct such sets 
$\Sigma_{K,T,M,R_l}$. Let then 
$l\in \mathbb{N}$, and let M be an integer such that 
$K \subset B^{H^{\sigma}}_M$. We consider 
$\delta \sim (1+M)^{p-1}$ the local existence time from Proposition 3.1 associated to initial data of size smaller than M + 1. We denote 
$m := \lfloor \frac{T}{\delta} \rfloor$. Furthermore, thanks to Proposition 5.2, we introduce an integer N 0 such that:
\begin{align}
N \geq N_0 \implies \forall A \in \mathcal B(H^{\sigma}), |\mu_{s,R_l}(A) - \mu_{s,R_l,N}(A)| \leq \frac{1}{mM}
\end{align} Thanks to the approximation by the truncated flow (see Proposition 3.2), we know that there exists 
$N_1 \geq N_0$ such that:
\begin{align*}
N \geq N_1 \implies \sup_{u_0 \in K} \sup_{0 \leq t \leq \delta} \| \Phi_t u_0 - \Phi^N_t u_0 \|_{H^{\sigma}} \leq 1,
\end{align*}Hence, introducing the set:
\begin{align*}
K_1 := K \cap \big\{ \sup_{0 \leq t \leq \delta} \| \Phi^{N_1}_t u_0 \|_{H^{\sigma}} \leq M \big\},
\end{align*} we deduce that for every 
$u_0 \in K_1$, 
$\| \Phi_\delta u_0 \|_{H^{\sigma}} \leq M + 1$, which ensures by Proposition 3.1 that the solution generated by u 0 lives on 
$[0,2\delta]$, that is 
$I_{max}(u_0) \supset [0,2\delta]$. Next, since K 1 is compact (it is the intersection of the compact K with a closed set), we can apply Proposition 3.2 as above (but now with K 1 and 
$[0,2 \delta]$), and find 
$N_2\geq N_0$ such that every initial data in:
\begin{align*}
K_2 :=K \cap \big\{ \sup_{0 \leq t \leq \delta} \| \Phi^{N_1}_t u_0 \|_{H^{\sigma}} \leq M \big\} \cap \big\{ \sup_{0 \leq t \leq 2\delta} \| \Phi^{N_2}_t u_0 \|_{H^{\sigma}} \leq M \big\},
\end{align*} generates a solution that lives on 
$[0,3\delta]$. Thus, proceeding by induction (recalling that 
$m = \lfloor \frac{T}{\delta} \rfloor$), we are able to construct 
$N_m,...,N_1$, all greater that N 0, such that every initial data in the set:
\begin{align*}
\Sigma_{K,T,M,R_l} := \bigcap_{k=1}^{m} K \cap \big\{\sup_{0 \leq t \leq k\delta} \| \Phi^{N_k}_t u_0 \|_{H^{\sigma}} \leq M \big\},
\end{align*} yields a solution on 
$[0,T]$. Note that this set depends on Rl since we chose every Nk greater than N 0, and that N 0 itself depends on Rl (see (3.5)).
Next, we estimate the 
$\mu_{s,R_l}$-measure of this set. Since every Nk is greater than N 0, we can write (thanks to (3.5)):
\begin{align*}
\mu_{s,R_l}(K \setminus \Sigma_{K,T,M,R_l}) &\leq \sum_{k=1}^m \mu_{s,R_l}\big(\sup_{0 \leq t \leq k\delta} \| \Phi^{N_k}_t u_0 \|_{H^{\sigma}} \gt M\big) \\
& \leq \frac{1}{M} + \sum_{k=1}^m \mu_{s,R_l,N_k}\big(\sup_{0 \leq t \leq k\delta} \| \Phi^{N_k}_t u_0 \|_{H^{\sigma}} \gt M\big).
\end{align*}Applying now (3.4), leads to:
\begin{align*}
\mu_{s,R_l}(K \setminus \Sigma_{K,T,M,R_l}) \leq \frac{1}{M} + C_{s,R_l,T} M^{p-1} e^{-\alpha M^2},
\end{align*} which indeed tends to 0 as 
$M \to \infty$. This completes the proof.
Proof of Theorem 1.1
First, we recall that 
$\sigma \lt s - \frac{1}{2}$ is fixed and close to 
$s-\frac{1}{2}$. Then, we consider 
$\sigma' \in (\sigma ,s-\frac{1}{2})$. For every 
$n \in \mathbb{N}$, we introduce 
$B^{H^{\sigma'}}_n$, the closed centered ball of radius n in 
$H^{\sigma'}(\mathbb{T})$, which is compact in 
$H^{\sigma}(\mathbb{T})$. Moreover, we introduce an increasing sequence 
$\{T_k\}_{k \in \mathbb{N}}$ of positive time tending to infinity. Applying Lemma 3.8, we consider for every 
$n,k \in \mathbb{N}$:
\begin{align*}
\Sigma_{n,k} &:= \Sigma_{B^{H^{\sigma'}}_n, T_k} & &\text{and then we introduce:} & \Sigma &:= \bigcap_{k \in \mathbb{N}} \bigcup_{n \in \mathbb{N}} \Sigma_{n,k}.
\end{align*} Let us now observe that if 
$u_0 \in \Sigma$, then for every 
$k\in\mathbb{N}$, 
$I_{max}(u_0) \subset [-T_k, T_k]$. It means that u 0 generates a global solution of (pNLS). In other words, Σ is a subset of the global well-posedness set 
$\mathcal G$ (defined in (1.5)). Next, we prove that Σ is of full µs-measure. It is sufficient to prove that for every k, 
$\cup_n \Sigma_{n,k}$ is of full µs-measure. Since we know that 
$H^{\sigma'}(\mathbb{T})$ is of full µs-measure, we can write:
\begin{align*}
\mu_s \big( H^{\sigma} \setminus \bigcup_{n \in \mathbb{N}} \Sigma_{n,k}\big) &= \mu_s \big( H^{\sigma'} \setminus \bigcup_{n \in \mathbb{N}} \Sigma_{n,k}\big) = \lim_{n' \to \infty} \mu_s \big( B^{H^{\sigma'}}_{n'} \setminus \bigcup_{n \in \mathbb{N}} \Sigma_{n,k} \big) \\
& \leq \liminf_{n' \to \infty} \mu_s( B^{H^{\sigma'}}_{n'} \setminus \Sigma_{n',k} ) = 0.
\end{align*} As a consequence, 
$\mu_s(\Sigma^c) = 0$, and a fortiori 
$\mu_s(\mathcal G^c) = 0$. This concludes the proof.
4. Quantitative quasi-invariance for the original flow and application
This section is dedicated to the proof of Theorem 1.2 and its application to the growth of Sobolev norms in Corollary 1.4. Again we assume (3.3) and Proposition 5.1, which are proven in 
$\S$ 7 and 
$\S$ 5, respectively.
4.1. Proof of the quasi-invariance
In the previous section, we have proven that the flow was global for µs-every initial data. Now, we prove that µs is quasi-invariant along this flow, with quantitative bounds on the Radon-Nikodym derivative. Moreover, we prove that the renormalized energy (defined as a limit in 
$L^q(d\mu_s)$ of truncated renoramlized energies) is indeed preserved by the flow. We rely on the following classical lemma from functional analysis.
Lemma 4.1. Let E be a Banach space. Let 
$(L_n)_{n \in \mathbb{N}}$ be a sequence of linear forms that are uniformly bounded, in the sense that:
\begin{align*}
\sup_{n\in \mathbb{N}} \| L_n \|_{E \to \mathbb{R}} \lt +\infty,
\end{align*}Suppose that for a dense linear subset D of E, we have:
\begin{align*}
\forall x \in D, \lim_{n \to \infty} L_n(x) =: L(x) \in \mathbb{R} \text{exists}.
\end{align*} Then, the map 
$L : D \to \mathbb{R}$ is linear and extends continuously, and uniquely, in a linear form (still denoted L):
\begin{align*}
L &: E \to \mathbb{R}, & & \text{such that:} & \| L \|_{E \to \mathbb{R}} \leq \sup_{n \in \mathbb{N}} \| L_n \|_{E \to \mathbb{R}}.
\end{align*} Moreover, for every 
$x\in E$, 
$L_n(x) \to L(x)$.
Proof of Theorem 1.2
Let 
$q \in (1,\infty)$ and 
$q' \in (1,\infty)$ its conjugate exponent. We want to apply Lemma 4.1 with E the Banach space 
$L^{q'}(d\mu_s)$, whose dual is identified to 
$L^q(d\mu_s)$, and D the linear subspace of bounded continuous function on 
$H^{\sigma}(\mathbb{T})$ (which is indeed dense in 
$L^{q'}(d\mu_s)$).
Let R > 0. Let 
$\psi : H^{\sigma} \to \mathbb{R}$ a bounded continuous function. Then, for 
$N \in \mathbb{N}$, we have from the definition of the Radon-Nikodym derivative (1.10) and the invariance of 
$\mathcal E_N$ by 
$\Phi^N_t$,
\begin{align}
\int \psi(u) \chi_R(\mathcal E_N(u)) g_{s,N,t} d\mu_s = \int \psi(\Phi^N_t u) \chi_R(\mathcal E_N(u)) d\mu_s.
\end{align} Thanks to both (3.1) and (5.1), which in particular ensure that 
$\psi(\Phi^N_t u)$ and 
$\chi_R(\mathcal E_N(u))$ converge in 
$L^2(d\mu_s)$ to 
$\psi(\Phi_t(u))$ and 
$\chi_R(\mathcal E(u))$ respectively, passing to the limit 
$N \to \infty$ in the equality above leads to:
\begin{align}
\lim_{N \to \infty} \int \psi(u) \chi_R(\mathcal E_N(u)) g_{s,N,t} d\mu_s = \int \psi(\Phi_t u) \chi_R(\mathcal E(u)) d\mu_s = \int \psi(u) d(\Phi_t)_\# d\mu_{s,R}.
\end{align} Moreover, thanks to (3.3), we know that that the sequence 
$\chi_R(\mathcal E_N(u)) g_{s,N,t}$ is uniformly bounded in 
$L^q(d\mu_s)$. Hence, with (4.1) and (4.2), we can appeal to Lemma 4.1, and use the standard isomorphism 
$L^q(d\mu_s) \simeq (L^{q'}(d\mu_s))'$, to conclude that there exists 
$g_{s,t,R} \in L^q(d\mu_s)$ such that:
\begin{align}
(\Phi_t)_\#\mu_{s,R} &= g_{s,t,R} d\mu_s & &\text{with:} & \| g_{s,t,R}\|_{L^q(d\mu_s)} \leq \exp \big( C_{s,R,q}(1+|t|)^{A} \big),
\end{align}and,
\begin{align}
\chi_R(\mathcal E_N(u)) g_{s,N,t} \longrightarrow g_{s,t,R}, \text{weakly in}\ L^q(d\mu_s).
\end{align} Since R > 0 is arbitrary, we deduce that 
$(\Phi_t)_\# \mu_s$ is absolutely continuous with respect to µs; indeed, if a Borel set A is such that 
$\mu_s(A)=0$, then:
\begin{align*}
(\Phi_t)_\# \mu_s (A) = \lim_{R \to \infty}(\Phi_t)_\# \mu_{s,R} (A) \leq \liminf_{R \to \infty} \| g_{s,t,R}\|_{L^q(d\mu_s)} \mu_s(A)^{\frac{1}{q'}} = 0.
\end{align*} Invoking now the Radon-Nikodym theorem, there exists a non-negative measurable function 
$g_{s,t}$ such that:
\begin{align}
(\Phi_t)_\# \mu_s = g_{s,t} \mu_s.
\end{align} Let us now prove that 
$\mathcal E(u) = \mathcal E(\Phi_tu)$ µs-almost everywhere. Again, we consider 
$\psi : H^{\sigma} \to \mathbb{R}$ a bounded continuous function, and we write:
\begin{align*}
&\int \psi(u) \mathcal E(\Phi_t(u)) d\mu_{s,R} = \int \psi(\Phi_{-t}(u)) \mathcal E(u) d (\Phi_t)_\# \mu_{s,R} \\
& = \int \psi(\Phi_{-t}(u)) \mathcal E(u) g_{t,s,R} d\mu_s
= \lim_{N \to \infty} \int \psi(\Phi^N_{-t}(u)) \mathcal E_N(u) \chi_R(\mathcal E_N(u)) g_{s,N,t} d\mu_s.
\end{align*} This limit is justified thanks to the weak convergence in 
$L^q(d\mu_s)$ in (4.4) and the convergence in 
$L^q(d\mu_s)$ of 
$\psi(\Phi^N_{-t}(u)) \mathcal E_N(u) $ to 
$\psi(\Phi_{-t}(u)) \mathcal E(u)$ (see the propositions 3.2 and 5.1). Using now the invariance of 
$\mathcal E_N$ by the truncated flow, and again Proposition 5.1, we can continue the equality above as:
\begin{align*}
\int \psi&(u) \mathcal E(\Phi_t(u)) d\mu_{s,R} = \lim_{N \to \infty} \int \psi(u) \mathcal E_N(u) \chi_R(\mathcal E_N(u)) d\mu_s \\
& = \int \psi(u) \mathcal E(u) \chi_R(\mathcal E(u)) d\mu_s = \int \psi(u) \mathcal E(u)d\mu_{s,R}.
\end{align*} Since ψ is arbitrary, we can deduce that 
$\mathcal E(\Phi_t(u)) = \mathcal E(u)$ 
$\mu_{s,R}$-almost everywhere; and then, since R is also arbitrary, we can conclude that this equality holds µs-almost everywhere.
Finally, thanks to (4.5), the invariance of 
$\mathcal E$ by the flow, and Lemma 3.4, we have for every R > 0Footnote 1:
\begin{align*}
(\Phi_t)_\# \mu_{s,R} = (\Phi_t)_\# \big( \chi_R(\mathcal E(u)) d\mu_s\big) = \chi_R(\mathcal E(\Phi_{-t}u)) g_{s,t} d\mu_s = \chi_R(\mathcal E(u)) g_{s,t} d\mu_s = g_{s,t} d\mu_{s,R}.
\end{align*}Using (4.3) and (4.4), we also obtain:
\begin{align*}
\| \chi_R(\mathcal E(u)) g_{s,t}\|_{L^q(d\mu_s)} \leq \exp \big( C_{s,R,q}(1+|t|)^{M_0} \big),
\end{align*}and,
\begin{align*}
\chi_R(\mathcal E_N(u)) g_{s,N,t} \longrightarrow \chi_R(\mathcal E(u)) g_{s,t} \text{weakly in} L^q(d\mu_s).
\end{align*}This completes the proof.
Remark 4.2. The renormalized energy 
$\mathcal E$ being by nature an element of 
$L^0(\mu_s)$, the space of measurable functions modulo µs-almost everywhere equivalence, the definition of the composition 
$\mathcal E \circ \Phi_t$ (for a fixed time 
$t \in \mathbb{R}$) is subtle. In fact, it is a well-defined element in 
$L^0(\mu_s)$ precisely thanks to the quasi-invariance. Indeed, for any other measurable function 
$\mathcal E'$, if 
$\mathcal E'$ and 
$\mathcal E$ coincide µs-almost everywhere then so do 
$\mathcal E \circ \Phi_t$ and 
$\mathcal E' \circ \Phi_t$, since:
\begin{align*}
\mu_s \big( \{\mathcal E \circ \Phi_t \neq \mathcal E' \circ \Phi_t \}\big) = (\Phi_t)_\# \mu_s \big( \{\mathcal E \neq \mathcal E' \}\big).
\end{align*} Similarly, considering now Φ as a function of both variables t and u, it is also possible to make sense of the composition 
$\mathcal E \circ \Phi$ as a function in 
$L^0(\mathcal L^1\otimes \mu_s)$, where 
$\mathcal L^1$ is the Lebesgue measure on 
$\mathbb{R}$, because the measure 
$\Phi_\# (\mathcal L^1 \otimes \mu_s)$ is absolutely continuous with respect to µs due to Fubini–Tonelli theorem; more precisely, if A is a Borel set such that 
$\mu_s(A) = 0$, then:
\begin{align*}
&\Phi_\# (\mathcal L^1 \otimes \mu_s)(A) = \int_{\mathbb{R} \times H^{\sigma}} \mathbb{1}_A(\Phi_t(u)) \mathcal L^1 \otimes \mu_s(dt,du) \\ &
= \int_{\mathbb{R}} \big( \int_{H^{\sigma}} \mathbb{1}_A(\Phi_t(u)) \mu_s (du) \big) \mathcal L^1(dt) = \int_{\mathbb{R}} (\Phi_t)_\# \mu_s (A) \mathcal L^1(dt) = 0.
\end{align*} Finally, thanks to (1.13) and the definition of the product measure, we have 
$\mathcal E \circ \Phi = \mathcal E$ in 
$L^0(\mathcal L^1 \otimes \mu_s)$ since:
\begin{align*}
\mathcal L^1 \otimes \mu_s (\mathcal E \circ \Phi \neq \mathcal E) = \int_{\mathbb{R}} \mu_s ( u : \mathcal E(\Phi_t(u)) \neq \mathcal E(u) ) \mathcal L^1(dt) = 0.
\end{align*}This proves (1.15) as stated in the introduction.
4.2. Growth of Sobolev norms
Here, we prove Corollary 1.4 by adapting Bourgain’s invariant argument. In the context of invariant Gibbs measures, one may expect logarithmic bounds on the Sobolev norms of the solutions. However, for quasi-invariant Gaussian measures, the Radon-Nikodym derivatives depend on time, and we need to control at every time its 
$L^q(d\mu_{s,R})$-norm. Here, the exponential control in (1.14) yields polynomial bounds on the Sobolev norms of the solutions. The strategy of the following proof is not new (see for example [Reference Höfer and Nikov21]).
Proof of Corollary 1.4
For T > 0, proceeding as in the proof of (3.4), we have the flow tail estimate:
\begin{align*}
\mu_{s,R}\big(u_0 : \sup_{|t| \leq T} \| \Phi_tu_0 \|_{H^{\sigma}} \gt M \big) \leq e^{C_{s,R} (1+T)^A} e^{-\alpha M^2},
\end{align*} for any M > 0. As a consequence, for 
$B \gt \frac{A}{2}$:
\begin{align*}
\sum_{T \in \mathbb{N}} \mu_{s,R}\big(u_0 : \sup_{|t| \leq T} \| \Phi_t u_0 \|_{H^{\sigma}} \gt T^{B} \big) \lt +\infty.
\end{align*} By the Borel-Cantelli lemma, it implies that the set formed by the elements u 0 such that 
$ \sup_{|t| \leq T} \| \Phi_t u_0 \|_{H^{\sigma}} \gt T^{B}$ for infinitely many 
$T \in \mathbb{N}$ is of zero 
$\mu_{s,R}$-measure. It means that for 
$\mu_{s,R}$-almost every u 0, there exists 
$T_0= T_0(u_0)\in \mathbb{N}$ such that:
\begin{align*}
T \geq T_0 \implies \sup_{|t| \leq T} \| \Phi_t u_0 \|_{H^{\sigma}} \leq T^{B}.
\end{align*} This implies (1.16) for 
$\mu_{s,R}$-almost every u 0, and since R > 0 is arbitrary, it also holds for µs-almost every u 0.
5. Renormalized energy and associated cutoff Gaussian measures
In this section, we consider 
$s \in (\frac{5}{4}, \frac{3}{2}]$ and 
$\sigma \lt s - \frac{1}{2}$ (close to 
$s - \frac{1}{2}$). The restriction 
$s \leq \frac{3}{2}$ indicates that we are interested in Gaussian measures µs supported in Sobolev spaces of regularity (strictly) smaller than 1 (that is below the energy level). The condition 
$s \gt \frac{5}{4}$ corresponds to a regime where we are able to make sense of the renormalized energy as limit the of truncated renormalized energies (1.11).
Proposition 5.1. Let 
$\frac{5}{4} \lt s \leq \frac{3}{2}$. Then, for every 
$q \in [1, \infty)$, the sequence 
$(\mathcal E_N)_N$ is Cauchy in 
$L^q(d\mu_s)$, and therefore converges to a limit function 
$\mathcal E : H^{\sigma}(\mathbb{T}) \rightarrow \mathbb{R}^{+}$.
As a consequence, introducing a smooth-cutoff function 
$\chi : \mathbb{R} \rightarrow [0,1]$ such that 
$\chi \equiv 1$ on 
$[-\frac{1}{2},\frac{1}{2}]$ and 
$\chi \equiv 0$ on 
$[-1,1]^c$, we obtain that for every R > 0:
\begin{align}
\chi_R \circ \mathcal E_N \underset{N \rightarrow \infty}{\longrightarrow} \chi_R \circ \mathcal E, \text{in}\ L^q(d\mu_s),
\end{align} where 
$\chi_R = \chi(\cdot/R)$.
Proof. Note first that, thanks to Lemma 2.7, 
$u \mapsto \| u \|_{L^2(\mathbb{T})}^2$ already belongs to 
$L^q(d\mu_s)$, and that 
$u \mapsto \| \pi_N u \|_{L^{p+1}(\mathbb{T})}^{p+1}$ converges to 
$u \mapsto \| u \|_{L^{p+1}(\mathbb{T})}^{p+1}$ in 
$L^q(d\mu_s)$, 
$q \in [1,\infty)$. Thus, we are reduced to prove that:
\begin{align*}
u \longmapsto \| \partial_x \pi_N u \|_{L^2(\mathbb{T})}^2 - \sigma_N,
\end{align*} is a Cauchy sequence in 
$L^q(d\mu_s)$. Let then M > N integers. From Wiener chaos (see Lemma 2.8):
\begin{align}
& \big\| \big( \frac{1}{2} \| \partial_x \pi_M u \|_{L^2(\mathbb{T})}^2 - \sigma_M\big) - \big( \frac{1}{2}\| \partial_x \pi_N u \|_{L^2(\mathbb{T})}^2 - \sigma_N\big) \big\|_{L^q(d\mu_s)} \nonumber\\
&\quad \quad = \frac{1}{2} \mathbb{E} \big[ \big| \sum_{N \lt |n| \leq M} \frac{|n|^2}{\langle n \rangle^{2s}} (|g_n|^2-1) \big|^q \big]^{\frac{1}{q}} \lesssim q \mathbb{E} \big[ \big| \sum_{N \lt |n| \leq M} \frac{|n|^2}{\langle n \rangle^{2s}} (|g_n|^2-1) \big|^2 \big]^{\frac{1}{2}} \nonumber\\
&\quad \quad \quad \quad \quad \quad \lesssim \big( \sum_{N \lt |n| \leq M} \langle n \rangle^{4(1-s)} \big)^{\frac{1}{2}}.
\end{align} This concludes the proof of the first point in Proposition 5.1, since the right-hand side tends to 0 as 
$N,M \rightarrow \infty$ when 
$s \gt \frac{5}{4}$.
Let us now prove the second point. From the convergence in 
$L^q(d\mu_s)$, we deduce that 
$\mathcal E_N$ converges to 
$\mathcal E$ in µs-measure; and, χR being uniformly continuous, we obtain that 
$\chi_R \circ \mathcal E_N$ converges to 
$\chi_R \circ \mathcal E$ in µs-measure. Besides, 
$\chi_R \circ \mathcal E_N$ is uniformly bounded in 
$L^q(d\mu_s)$, 
$q \in (1,\infty)$. Thus, from the two last points, we deduce that 
$\chi_R \circ \mathcal E_N$ converges to 
$\chi_R \circ \mathcal E$ in 
$L^{q'}(d\mu_s)$ for every 
$q' \in [1,q)$; and as 
$q \in (1,\infty)$ is arbitrary, the convergence holds for any 
$q' \in [1,\infty)$.
Proposition 5.2. Let R > 0. For any ɛ > 0, there exists 
$N_\varepsilon \in \mathbb{N}$ such for every 
$N \geq N_\varepsilon$, and every Borel set 
$A \subset H^{\sigma}$,
\begin{align*}
|\mu_{s,R}(A) - \mu_{s,R,N}(A)| \leq \varepsilon.
\end{align*}Proof. This follows from the convergence of 
$\chi_R \circ \mathcal E_N$ to 
$\chi_R \circ \mathcal E$ in 
$L^1(d\mu_s)$ (see Proposition 5.1).
5.1. Estimates with bounded renormalized energy
This paragraph is devoted to a proposition that highlights the benefits of working with bounded renormalized energies. We use the following notation:
Notation 5.3. For a real number γ, the symbol 
$\gamma+$ refers to a real number greater than γ that can be chosen arbitrarily close to it.
Proposition 5.4. Let 
$s \in (\frac{5}{4}, \frac{3}{2}]$. Let 
$N \in \mathbb{N}$, R > 0, and 
$V \in H^s$. Denote 
$U:=\pi_N(Y+V)$, where 
$Y \sim \mu_s$ is the Gaussian random variable defined in (1.4). Suppose that:
\begin{align*}
\mathcal E_N(U) \leq R.
\end{align*} Then, for every 
$\alpha \lt s -\frac{1}{2}$, there exists a positive random variable 
$W_{s,R,N}$ such that:
\begin{align}
\| U \|_{H^\alpha} \leq W_{s,R,N} (1+ \| V \|_{H^s})^{\alpha(3-2s)+},
\end{align}which satisfies, for every m > 0,
\begin{align}
\sup_{N \in \mathbb{N}} \mathbb{E}[W_{s,R,N}^m] \leq C_{s,R} \lt \infty.
\end{align} As a consequence, for every ɛ > 0, there exists a constant 
$C_{\varepsilon} \gt 0$ such that:
\begin{align}
\|\pi_N(|\pi_N U|^{p-1}\pi_N U)\|_{H^{\frac{1}{2} + \varepsilon }(\mathbb{T})} \leq C_{\varepsilon} W'_{s,R,N} (1+ \| V \|_{H^s})^{(3-2s)[\frac{p}{2} + \varepsilon]+},
\end{align} for a positive random variable 
$W'_{s,R,N}$, with all moment finite (uniformly in 
$N \in \mathbb{N}$), as in (5.4).
Proof. For commodity, we denote 
$V_N := \pi_N V$ and 
$Y_N := \pi_N Y$. First, by assumption, we have:
\begin{align*}
\mathcal E_N(U) = \| U \|_{L^2(\mathbb{T})}^2 + \Big| \frac{1}{p+1} \| U \|_{L^{p+1}(\mathbb{T})}^{p+1} + \big( \frac{1}{2} \| \partial_x U \|^2_{L^2(\mathbb{T})} - \sigma_N \big) \Big| \leq R.
\end{align*}So, as an immediate consequence,
\begin{align}
\| U \|_{L^2}^2 \leq R
\end{align} Next, we expand the term 
$\frac{1}{2} \| \partial_x U \|^2_{L^2} - \sigma_N$ as:
\begin{align*}
\frac{1}{2} \| \partial_x U \|^2_{L^2} - \sigma_N = \frac{1}{2} \| \partial_x V_N \|^2_{L^2} + \text{Re} \int_{\mathbb{T}} \partial_x V_N \overline{\partial_x Y_N}dx + \frac{1}{2} \| \partial_x Y_N \|^2_{L^2} - \sigma_N,
\end{align*}and we deduce that:
\begin{multline*}
\frac{1}{2} \| \partial_x V_N\|^2_{L^2} \leq \mathcal E_N(U) + \big| \int_{\mathbb{T}} \partial_x V_N \overline{\partial_x Y_N}dx \big| + \big| \frac{1}{2} \| \partial_x Y_N \|^2_{L^2} - \sigma_N \big| \\
\leq R + \| Y_N\|_{H^{s-\frac{1}{2}-}} \|V_N \|_{H^{\frac{5}{2}-s +}} + \big| \frac{1}{2} \| \partial_x Y_N \|^2_{L^2} - \sigma_N \big|.
\end{multline*}Using (5.6), it follows that
\begin{align}
\frac{1}{2} \| V_N\|_{H^1}^2 \leq 3 R + \| Y_N\|_{H^{s-\frac{1}{2}-}} \|V_N \|_{H^{\frac{5}{2}-s +}} + 2 \|Y_N\|_{L^2}^2 + \big| \frac{1}{2} \| \partial_x Y_N \|^2_{L^2} - \sigma_N \big|.
\end{align} We interpolate now 
$H^{\frac{5}{2}-s+}$ between H 1 and Hs (note that 
$ 1 \leq \frac{5}{2}-s \lt \frac{5}{4} \lt s$) and then apply Young’s inequality:
\begin{align}
\| Y_N\|_{H^{s-\frac{1}{2}-}} \|V_N \|_{H^{\frac{5}{2}-s +}} \leq & \| Y_N\|_{H^{s-\frac{1}{2}-}} \|V_N \|^{2 {s-\frac{5}{4}}{s-1} - }_{H^1} \|V_N \|_{H^s}^{\frac{\frac{3}{2}-s}{s-1}+}\nonumber\\
& \quad \leq C \| Y_N\|^M_{H^{s-\frac{1}{2}-}} + \frac{1}{4} \|V_N \|^2_{H^1} + \|V_N \|^{2(3-2s)+}_{H^s},
\end{align} where 
$C,M \gt 0$ are large constants. Here we used the following identities:
\begin{align*}
\begin{cases}
\frac{5}{2}-s+ = \big(1- (\frac{\frac{3}{2}-s}{s-1} +)\big) \cdot 1 + \big( \frac{\frac{3}{2}-s}{s-1} + \big) \cdot s \text{(for the interpolation)}\\
1 = ( \frac{s-\frac{5}{4}}{s-1}+) + (1 - \frac{s-\frac{5}{4}}{s-1} -) + \frac{1}{M} \text{(for Young's inequality)}.
\end{cases}
\end{align*}We emphasize that the second identity is valid since M can be chosen arbitrary large.
Finally, plugging (5.8) into (5.7) yields:
\begin{align*}
\frac{1}{4} \| V_N\|_{H^1}^2 \leq 3 R + \|V_N \|^{2(3-2s)+}_{H^s} + C \| Y_N\|^M_{H^{s-\frac{1}{2}-}} + 2 \|Y_N\|_{L^2}^2 + \big| \frac{1}{2} \| \partial_x Y_N \|^2_{L^2} - \sigma_N \big|.
\end{align*}Hence,
\begin{align}
\| V_N\|_{H^1} \leq W_{s,R,N}(1 + \| V_N \|_{H^s})^{3-2s+},
\end{align}where,
\begin{align}
W_{s,R,N} \lesssim \big( R + \| Y_N\|^M_{H^{s-\frac{1}{2}-}} + \|Y_N\|_{L^2}^2 + \big| \frac{1}{2} \| \partial_x Y_N \|^2_{L^2} - \sigma_N \big| \big)^{\frac{1}{2}}.
\end{align} Let us now conclude the proof. Fix 
$\alpha \lt s- \frac{1}{2}$. Then by interpolation, by (5.6), and by (5.9), we can write:
\begin{multline*}
\| U \|_{H^\alpha} \leq \| Y_N \|_{H^\alpha} + \| V_N \|_{H^\alpha} \leq \| Y_N \|_{H^\alpha} + \| V_N \|_{L^2}^{1-\alpha} \| V_N \|_{H^1}^\alpha \\
\leq \| Y_N \|_{H^\alpha} + (\| U \|_{L^2} + \| Y_N \|_{L^2})^{1-\alpha} W_{s,R,N}^\alpha (1 + \| V_N \|_{H^s})^{\alpha(3-2s)+} \\
\leq W_{s,R,N} (1 + \| V_N \|_{H^s})^{\alpha(3-2s)+},
\end{multline*}where,
\begin{align*}
W_{s,R,N} = \| Y_N \|_{H^\alpha} + ( \sqrt{R} + \| Y_N \|_{L^2})^{1-\alpha} W_{s,R,N}^\alpha.
\end{align*} Thanks to Lemma 2.7, (5.2), and the expression of 
$W_{s,R,N}$ in (5.10), we indeed obtain that the 
$W_{s,R,N}$ satisfy the condition (5.4).
To obtain (5.5), we use the standard estimate:
\begin{align*}
\|\pi_N(|\pi_N U|^{p-1}\pi_N U)\|_{H^{\frac{1}{2} + \varepsilon }(\mathbb{T})} \leq C_{\varepsilon} \| U \|_{H^{\frac{1}{2} + \varepsilon }(\mathbb{T})} \| U \|^{p-1}_{H^{\frac{1}{2}}(\mathbb{T})},
\end{align*}and then apply (5.3). This completes the proof.
6. Poincaré-Dulac normal form reduction and deterministic estimate
This section is organized in two paragraphs; in the first one, we introduce a “correction to the energy” thanks to a Poincaré-normal form reduction, and in the second one we prove a related deterministic estimate. In our analysis, this normal form is essentially an integration by part in time in the difference 
$ \| \Phi^N_{-t}u\|_{H^s(\mathbb{T})}^2 - \| \pi_N u\|_{H^s(\mathbb{T})}^2$ appearing in the formula of the Radon-Nikodym derivatives (1.10). In particular, we gain an inverse factor of the resonant function Ω (see Definition (2.2)). It has already been performed in [Reference Sun and Tzvetkov34] (on the three dimensional torus) and [Reference Knezevitch23], for p = 5. For a general odd power 
$p \geq 5$, the computations are identical and are thus not provided here. Also, note that normal form reductions has already been used in the context of quasi-invariance, see for example [Reference Forlano and Trenberth19, Reference Oh and Seong26–Reference Oh and Tzvetkov28].
6.1. Poincaré-Dulac normal form reduction
Definition 6.1. Firstly, we define 
$\mathcal M_s : C^\infty(\mathbb{T})^{p+1} \rightarrow \mathbb{R}$ and 
$\mathcal T_s : C^\infty(\mathbb{T})^{p+1} \rightarrow \mathbb{R}$ the two following multi-linear maps:
\begin{align*}
\mathcal M_s(u_1,...,u_{p+1}):= \sum_{k_1-k_2+...-k_{p+1}=0} \Psi^{(0)}_{2s}(\vec{k}) \prod_{\substack{j=1 \\ odd}}^{p}\widehat{u}_j(k_j) \prod_{\substack{j=2 \\ even}}^{p+1} \overline{\widehat{u}_j}(k_j),
\end{align*}and,
\begin{align*}
\mathcal T_s(u_1,...,u_{p+1}):= \sum_{k_1-k_2+...-k_{p+1}=0} \Psi^{(1)}_{2s}(\vec{k})\prod_{\substack{j=1 \\ odd}}^{p}\widehat{u}_j(k_j) \prod_{\substack{j=2 \\ even}}^{p+1} \overline{\widehat{u}_j}(k_j),
\end{align*} where we recall that 
$\Psi^{(0)}_{2s}$ and 
$\Psi^{(1)}_{2s}$ are defined in Definition 2.2. Then, for 
$N\in \mathbb{N}$, we define the truncated version of these maps as :
\begin{align*}
\mathcal T_{s,N}(u_1,...,u_{p+1})& := \mathcal T_s(\pi_N u_1,...,\pi_N u_{p+1}), \quad \\ \mathcal M_{s,N}(u_1,...,u_{p+1}) &:= \mathcal M_s(\pi_N u_1,...,\pi_N u_{p+1}).
\end{align*}Finally, we consider:
\begin{align*}
\mathcal N_{s,N}(u_1,...,u_{p+1}) := \mathcal T_s(\pi_N(|\pi_N u_1|^{p-1} \pi_N u_1),\pi_Nu_2,...,\pi_Nu_{p+1}).
\end{align*}Definition 6.2. (Energy correction and modified energy)
For 
$N \in \mathbb{N}$, we define the energy correction 
$R_{s,N}$ as the function :
\begin{align}
R_{s,N}(u):= \frac{1}{p+1} \text{Re} \mathcal T_{s,N}(u),
\end{align} Moreover, we introduce the modified energy 
$E_{s,N}$ as the function:
\begin{align}
E_{s,N}(u) := \frac{1}{2} \left\lVert \pi_N u \right\rVert_{H^s(\mathbb{T})}^2 + R_{s,N}(u),
\end{align}and we also consider the modified energy derivative as the function :
\begin{align}
Q_{s,N}(u):= -\frac{1}{p+1} \text{Im} \mathcal M_{s,N}(u) + \text{Im} \mathcal N_{s,N}(u).
\end{align}Proposition 6.3. (Poincaré-Dulac normal form reduction)
For all 
$N\in \mathbb{N}$, and every 
$u :\mathbb{T} \rightarrow \mathbb{C}$, we have:
\begin{align}
\frac{d}{dt}E_{s,N}(\Phi^N_t u) = Q_{s,N}(\Phi^N_t u).
\end{align} For a proof of this proposition, we refer to the computations performed in [Reference Sun and Tzvetkov34], 
$\S$ 2 (see also [Reference Knezevitch23] 
$\S$ 2).
6.2. A deterministic estimate
Lemma 6.4. (Deterministic dyadic estimates)
Let s > 1. Let 
$\Psi_{2s} \in \{\Psi_{2s}^{(0)}, \Psi_{2s}^{(1)}\}$. Let 
$( f^{(1)}_{k_1} )_{k_1 \in \mathbb{Z}}$,...,
$( f_{k_{p+1}}^{(p+1)} )_{k_{p+1} \in \mathbb{Z}}$ be sequences of complex numbers that satisfy 
$f^{(j)}_{k_j}~=~\mathbb{1}_{|k_j| \sim N_j} f^{(j)}_{k_j} $, where 
$N_1, ..., N_{p+1}$ are dyadic integers. Then, there exists a constant 
$C_s \gt 0$, only depending on s, such that:
\begin{align}
\sum_{k_1-k_2+...-k_{p+1}=0 } | \Psi_{2s}(\vec{k}) | \prod_{j=1}^{p+1} | f_{k_j}^{(j)} | \leq C_s N_{(1)}^{2(s-1)+} N_{(3)}^{2} (N_{(7)}...N_{(p+1)})^{\frac{1}{2}} \prod_{j=1}^{p+1} \| f^{(j)}_{k_j}\|_{l^2}.
\end{align} Furthermore, in the regime 
$N_{(4)} \ll N_{(3)}$, we have the refined estimate:
\begin{align}
\sum_{k_1-k_2+...-k_{p+1}=0 } | \Psi_{2s}(\vec{k}) | \prod_{j=1}^{p+1} | f_{k_j}^{(j)} | \leq C_s N_{(1)}^{2(s-1)} (N_{(3)}...N_{(6)}N_{(7)}...N_{(p+1)})^{\frac{1}{2}} \prod_{j=1}^{p+1} \| f^{(j)}_{k_j}\|_{l^2}.
\end{align} Here, 
$N_{(1)} \geq ... \geq N_{(p+1)}$ refers to a non-increasing rearrangement of 
$N_1,...,N_{p+1}$.
This lemma has already been proven in the case p = 5 in [Reference Knezevitch22], Lemma 7.5 and Lemma 7.6, see also [Reference Knezevitch23] Lemma 9.7. Note that the proof incorporated implicitly dispersive effects through the L 6-Strichartz estimate due to Bourgain in [Reference Bourgain6]:
\begin{align*}
\| g \|_{L^6(\mathbb{T}_t \times \mathbb{T}_x)} \lesssim \| g \|_{H^{0+}(\mathbb{T})}.
\end{align*}It would have been natural to try to mimic the proof for a general p (odd), using the Strichartz estimate:
\begin{align}
\| g \|_{L^{p+1}(\mathbb{T}_t \times \mathbb{T}_x)} \lesssim \| g \|_{H^{\frac{1}{2}- \frac{3}{p+1}+}(\mathbb{T})},
\end{align} due to Bourgain and Demeter in [Reference Bourgain and Demeter10]. However, the loss of derivatives in (6.7) will lead to a worse estimate than the one stated above. Here, we prove Lemma 6.4 for 
$p\geq 7$, it will follow as a consequence of the known result for p = 5.
Proof. Let 
$p \geq 7$ (odd). For commodity we assume that 
$N_1 \geq ... \geq N_{p+1}$, that is 
$N_j = N_{(j)}$.
• First, we prove (6.5). We rearrange the sum as:
\begin{align*}
\sum_{k_1-k_2+...-k_{p+1}=0 } | \Psi_{2s}(\vec{k}) | \prod_{j=1}^{p+1} | f_{k_j}^{(j)} | = \sum_{k_7,...,k_{p+1}} \prod_{j=7}^{p+1} | f_{k_j}^{(j)} | \sum_{\substack{k_1-k_2+...-k_6 \\ = -k_7 +...+k_{p+1}}} | \Psi_{2s}(\vec{k}) | \prod_{j=1}^{6} | f_{k_j}^{(j)} |
\end{align*}and we know that (Lemma 7.5 in [Reference Knezevitch22]):
\begin{align*}
\sum_{\substack{k_1-k_2+...-k_6 \\ = -k_7 +...+k_{p+1}}} | \Psi_{2s}(\vec{k}) | \prod_{j=1}^{6} | f_{k_j}^{(j)} | \leq C_s N_{(1)}^{2(s-1)+} N_{(3)}^{2} \prod_{j=1}^{6} \| f^{(j)}_{k_j}\|_{l^2}.
\end{align*}Thus,
\begin{align*}
\sum_{k_1-k_2+...-k_{p+1}=0 } | \Psi_{2s}(\vec{k}) | \prod_{j=1}^{p+1} | f_{k_j}^{(j)} | \leq C_s N_{(1)}^{2(s-1)+} N_{(3)}^{2} \prod_{j=1}^{6} \| f^{(j)}_{k_j}\|_{l^2} \sum_{k_7,...,k_{p+1}} \prod_{j=7}^{p+1} | f_{k_j}^{(j)} |.
\end{align*} Using now the Cauchy-Schwarz inequality, and the localization of the 
$f^{(j)}_{k_j}$, leads to (6.5).
• Second, we prove (6.6). Here, we are in the regime 
$N_{(4)} \ll N_{(3)}$, so we are in a position to use the lower bound on Ω from Lemma 2.5. Besides, thanks to Remark 2.6, we see that the case 
$\Psi_{2s}(\vec{k})=\Psi^{(0)}_{2s}(\vec{k})$ leads to a zero-contribution, so we only need to treat the case 
$\Psi_{2s}(\vec{k})=\Psi^{(1)}_{2s}(\vec{k})$. Now, combining (2.4) and (2.6), we can write:
\begin{align*}
\sum_{k_1-k_2+...-k_{p+1}=0 } | \Psi_{2s}(\vec{k}) | \prod_{j=1}^{p+1} | f_{k_j}^{(j)} | \leq C_s N_{(1)}^{2(s-1)} \sum_{k_3,...,k_{p+1}} \prod_{j=3}^{p+1} | f_{k_j}^{(j)} | \sum_{k_2} |f^{(2)}_{k_2}f^{(1)}_{k_2 - k_3 + ... +k_{p+1}} |.
\end{align*} Applying the Cauchy-Schwarz inequality first in the k 2 summation and then in the 
$k_3,....,k_{p+1}$ summation leads to (6.6).
7. Uniform Lq integrability of the Radon-Nikodym derivatives
This section is central; it is dedicated to the proof of Proposition 3.6, which provides a (uniform in N) 
$L^q(d\mu_{s,R,N})$-bound on the truncated Radon-Nikodym derivatives 
$g_{s,N,t}$. To do so, we combine the Boué-Dupuis variational formula (see Lemma 2.9), the Poincaré-Dulac normal form reduction from Proposition 6.3, introducing weighted Gaussian measures, and finally the renormalized-energy cutoff through Proposition 5.4.
7.1. The weighted Gaussian measures
The idea is to replace the energy 
$\frac{1}{2} \| . \|^2_{H^s}$ in the formula of the truncated Radon-Nikodym derivatives in (1.10) by the modified energy 
$E_{s,N}$ in (6.2) produced by the Poincaré-Dulac normal form reduction.
Definition 7.1. For 
$N \in \mathbb{N}$ and R > 0, we define the weighted Gaussian measure:
\begin{align}
\rho_{s,R,N} := e ^{-R_{s,N}(u)} \mu_{s,R,N},
\end{align} where 
$R_{s,N}$ is the energy correction in (6.1), and 
$\mu_{s,R,N}$ is the cutoff Gaussian measure in (1.12).
From the forthcoming Lemma 7.5, we have, for every 
$q \in [1,\infty)$:
\begin{align}
\| e^{- R_{s,N}} \|_{L^q(d\mu_{s,R,N})} \leq C_{s,R,q} \lt +\infty,
\end{align} which ensures that 
$\rho_{s,R,N}$ has a finite mass. As expected, the weighted Gaussian measures are transported by the truncated flow as follows:
Proposition 7.2. Let 
$N \in \mathbb{N}$, 
$t \in \mathbb{R}$, and R > 0. Then,
\begin{align}
(\Phi^N_t)_\# \rho_{s,R,N} &= f_{s,N,t} d\rho_{s,R,N}, & f_{s,N,t} &:= \exp \big( -(E_{s,N}(\Phi^N_{-t}(u)) - E_{s,N}(u)) \big),
\end{align} where 
$\rho_{s,R,N}$ is the weighted Gaussian measure defined in (7.1)
Proof. Lemma 3.4 combined with (3.2) gives:
\begin{align*}
(\Phi^N_t)_\# \rho_{s,R,N} &= (\Phi^N_t)_\# \big( e^{-R_{s,N}(u)} d\mu_{s,R,N}\big) = e^{-R_{s,N}(\Phi^N_{-t}u)} g_{s,N,t} d\mu_{s,R,N} \\
&= e^{-(R_{s,N}(\Phi^N_{-t}u) - R_{s,N}(u))} g_{s,N,t} d\rho_{s,R,N},
\end{align*}which is the desired equality recalling the definitions (1.10) and (6.2).
7.2. Two lemmas
In this paragraph we establish two lemmas that, when combined, imply the desired Proposition 3.6; for convenience, we devote the next paragraph 7.3 to the proof of this implication.
The first lemma is inspired by [Reference Coe and Tolomeo11, Reference Forlano16, Reference Forlano and Tolomeo18]. Here, we show that it is also compatible with weighted Gaussian measures. Again, we will benefit from the conservation of the cutoff; in [Reference Coe and Tolomeo11] the authors were confronted to a situation were this property was not satisfied.
Lemma 7.3. Let 
$t \in \mathbb{R}$ and 
$N \in \mathbb{N}$. Then, for every 
$q \in (1,\infty)$,
\begin{align}
\sup_{\tau \in [0,t]} \| f_{s,N,\tau}\|^q_{L^q(\rho_{s,R,N})} \leq \int \exp \big( q |t Q_{s,N}(u)| \big) d\rho_{s,R,N},
\end{align}Remark 7.4. In [Reference Forlano and Tolomeo18], the (analogous) term 
$Q_{s,N}$ could not be considered directly; it was essential to use time oscillations considering instead the term 
$ \delta^{-1} \int_0^\delta Q_{s,N}(\Phi^N_{t'}u) dt'$, with δ > 0 small. Since this issue does not arise in our situation, we can let δ → 0, yielding the term 
$Q_{s,N}(u)$.
Proof. First, using the definition of the Radon-Nikodym derivative, we have for 
$\tau \in [0,t]$:
\begin{align*}
\begin{split}
\| f_{s,N,\tau} & \|^q_{L^q(\rho_{s,R,N})} = \int f_{s,N,\tau}^{q-1} f_{s,N,\tau} d\rho_{s,R,N} \\
& = \int \exp\big(-(q-1)(E_{s,N}(\Phi^N_{-\tau}u) - E_{s,N}(u)) \big) (\Phi^N_\tau)_\# d\rho_{s,R,N} \\
& =\int \exp\big((q-1)(E_{s,N}(\Phi^N_{\tau}u) - E_{s,N}(u)) \big) d\rho_{s,R,N}.
\end{split}
\end{align*} Then, we introduce a parameter 
$n \in \mathbb{N}$, which is intended to tend to infinity. By subdividing 
$[0,\tau]$ into intervals of length 
$\frac{\tau}{n}$, applying Jensen’s inequality, using again the definition of the Radon-Nikodym derivative, and then Hölder’s inequality (considering qʹ the conjugate exponent of q), we obtain:
\begin{align*}
\begin{split}
\| &f_{s,N,\tau} \|^q_{L^q(\rho_{s,R,N})} = \int \exp\big( (q-1)n \sum_{k=0}^{n-1} \frac{1}{n}( E_{s,N}(\Phi^N_{(k+1)\frac{\tau}{n}}u) - E_{s,N}(\Phi^N_{k\frac{\tau}{n}}u))\big) d\rho_{s,R,N} \\
& \leq \sum_{k=0}^{n-1} \frac{1}{n} \int \exp \big( (q-1)n ( E_{s,N}(\Phi^N_{\frac{\tau}{n}} \Phi^N_{k\frac{\tau}{n}}u) - E_{s,N}(\Phi^N_{k\frac{\tau}{n}}u)) \big) d\rho_{s,R,N} \\
& = \sum_{k=0}^{n-1} \frac{1}{n} \int \exp \big( (q-1)n ( E_{s,N}( \Phi^N_{\frac{\tau}{n}}u) - E_{s,N}(u)) \big) f_{s,N,\frac{k \tau}{n}} d\rho_{s,R,N} \\
& \leq \sum_{k=0}^{n-1} \frac{1}{n} \big[ \int \exp \big( q n ( E_{s,N}( \Phi^N_{\frac{\tau}{n}}u) - E_{s,N}(u)) \big) d\rho_{s,R,N} \big]^{\frac{1}{q'}} \| f_{s,N,\frac{k \tau}{n}} \|_{L^q(d\rho_{s,R,N})} \\
& \leq \sup_{t' \in [0,t]}\| f_{s,N,t'} \|_{L^q(d\rho_{s,R,N})} \big[ \int \exp \big( q n ( E_{s,N}( \Phi^N_{\frac{\tau}{n}}u) - E_{s,N}(u)) \big) d\rho_{s,R,N} \big]^{\frac{1}{q'}}.
\end{split}
\end{align*} Thanks to (6.4), we obtain by passing to the limit 
$n \to \infty$:
\begin{align}
\| f_{s,N,\tau} \|^q_{L^q(\rho_{s,R,N})} \leq \sup_{t' \in [0,t]}\| f_{s,N,t'} \|_{L^q(d\rho_{s,R,N})} \big[ \int \exp \big( q \tau Q_{s,N}(u) \big) d\rho_{s,R,N} \big]^{\frac{1}{q'}},
\end{align} and then, passing to the sup over 
$\tau \in [0,t]$ yields:
\begin{align*}
\sup_{\tau \in [0,t]} \| f_{s,N,\tau} \|^q_{L^q(\rho_{s,R,N})} \leq \sup_{\tau \in [0,t]} \int \exp \big( q \tau Q_{s,N}(u) \big) d\rho_{s,R,N} \leq \int \exp \big( q |t Q_{s,N}(u)| \big) d\rho_{s,R,N},
\end{align*}which is the desired inequality. To conclude, let us justify the passage to the limit in (7.5). Here, we can apply the dominated convergence theorem; indeed:
\begin{align*}
n | E_{s,N}( \Phi^N_{\frac{\tau}{n}}u) - E_{s,N}(u)| \leq | \tau | \sup_{t' \in [0,\frac{\tau}{n}]} |\frac{d}{dt'} E_{s,N}(\Phi^N_{t'}u)| = | \tau | \sup_{t' \in [0,\frac{\tau}{n}]} |Q_{s,N}(\Phi^N_{t'}u)|,
\end{align*} and thanks to the 
$L^2(\mathbb{T})$-norm conservation, the fact that the 
$L^2(\mathbb{T})$-norm is bounded in the integral due to the cutoff 
$\chi_R \circ \mathcal E_N$, and the expression of 
$Q_{s,N}$ in (6.3), we can then write:
\begin{align*}
n | E_{s,N}( \Phi^N_{\frac{\tau}{n}}u) - E_{s,N}(u)| \leq |\tau |C_{R,N} \lt + \infty,
\end{align*} where 
$C_{R,N} \gt 0$ is independent of n.
Lemma 7.5. Let 
$s \gt s_p$, where sp is defined in (1.6). Then, for A > 0 large enough, we have that for every R > 0, there exists a constant 
$C_{s,R} \gt 0$, such that for every λ > 0, 
$N\in\mathbb{N}$, and 
$\mathcal F_{s,N} \in \{\mathcal M_{s,N}, \mathcal T_{s,N}, \mathcal N_{s,N}\}$,
\begin{align}
\log \int e^{\lambda | \mathcal F_{s,N}(u)|}d\mu_{s,R,N} \leq C_{s,R} \lambda^A.
\end{align}As a consequence, we have:
\begin{align}
\log \int e^{\lambda | R_{s,N}(u)|}d\mu_{s,R,N} + \log \int e^{\lambda | Q_{s,N}(u)|}d\mu_{s,R,N} \leq C_{s,R} \lambda^A.
\end{align}Notations 7.6. (Preparation for the proof of Lemma 7.5)
Let 
$N\in \mathbb{N}$. Firstly, for 
$V \in H^s(\mathbb{T})$, we denote:
\begin{align*}
U := \pi_N (V + Y),\ \text{where}\ Y \sim \mu_s\ \text{is the Gaussian random variable defined in (1.4)}.
\end{align*} Secondly, for 
$\mathcal F_{s,N} \in \{\mathcal M_{s,N}, \mathcal T_{s,N}, \mathcal N_{s,N} \}$, and considering 
$F_N : U \mapsto \pi_N(|\pi_N U|^{p-1}\pi_N U)$, we observe that there exists a unique
\begin{align*}
(\Psi_{2s},G) \in \{(\Psi^{(0)}_{2s},id) \} \cup \{\Psi^{(1)}_{2s} \} \times \{id, F_N\}
\end{align*}such that:
\begin{align*}
\mathcal F_{s,N}(U)= \sum_{k_1-k_2+...-k_{p+1}=0} \Psi_{2s}(\vec{k}) G_{k_1}\overline{U_{k_2}}...\overline{U_{k_{p+1}}},
\end{align*} with the abuse of notation 
$G = G(U)$, that we use for commodity (and where 
$G_k, U_k$ correspond to the k-th Fourier coefficient). This concise formulation for 
$\mathcal F_{s,N}$ will allow us to avoid treating the terms 
$\mathcal M_{s,N}, \mathcal T_{s,N}$, and 
$\mathcal N_{s,N}$ separately.
Thirdly, for a set of dyadic integers 
$(N_1,...,N_{p+1}) \in (2^{\mathbb{N}})^{p+1}$, we denote 
$\vec{N} = (N_1,...,N_{p+1})$, and we consider the dyadic block for 
$\mathcal F_{s,N}$:
\begin{align}
\mathcal F_{s,\vec{N}}(U) := \sum_{k_1-k_2+...-k_{p+1}=0} \Psi_{2s}(\vec{k}) G^{N_1}_{k_1}\overline{U^{N_2}_{k_2}}...\overline{U^{N_{p+1}}_{k_{p+1}}},
\end{align} where, for 
$W: \mathbb{T} \to \mathbb{C}$ and 
$L \in 2^{\mathbb{N}}$, we adopt the notation 
$W^L = P_{L} \pi_N W$, with PL the projector onto frequencies of size ∼ L.
Finally, we denote 
$N_{(1)} \geq ... \geq N_{(p+1)}$ a non-increasing rearrangement of the dyadic integers 
$N_1,...,N_{p+1}$.
Proof of Lemma 7.5
Let 
$N \in \mathbb{N}$ and 
$\mathcal F_{s,N} \in \{\mathcal M_{s,N}, \mathcal T_{s,N}, \mathcal N_{s,N} \}$. We adopt the Notations 7.6. We start by applying the simplified Boué-Dupuis formula (2.7) to the function 
$\mathcal F = \lambda \mathbb{1}_{\mathcal E_N \leq R}|\mathcal F_{s,N}|$:
\begin{align*}
\log \int e^{\lambda | \mathcal F_{s,N}(u)|}d\mu_{s,R,N} &\leq \log \int e^{\mathbb{1}_{\mathcal E_N(u) \leq R} \lambda | \mathcal F_{s,N}(u)|} d\mu_s \\
& \leq \mathbb{E} \big[\sup_{V \in H^s} \{\lambda \mathbb{1}_{\mathcal E_N(U) \leq R}|\mathcal F_{s,N}(U)| - \frac{1}{2}\left\lVert V \right\rVert^2_{H^s} \} \big].
\end{align*} Then, decomposing 
$\mathcal F_{s,N}$ into dyadic blocks, we obtain:
\begin{align}
\log \int e^{\lambda | \mathcal F_{s,N}(u)|}d\mu_{s,R,N} \leq \mathbb{E} \big[\sup_{V \in H^s} \{\lambda \mathbb{1}_{\mathcal E_N(U) \leq R}\sum_{\vec{N}} | \mathcal F_{s,\vec{N}}(U)| - \frac{1}{2}\left\lVert V \right\rVert^2_{H^s} \} \big].
\end{align}Now, our goal is to prove that:
\begin{align}
\mathbb{1}_{\mathcal E_N(U) \leq R}|\mathcal F_{s,\vec{N}}(U)| \leq N_{(1)}^{0-} X_{s,R,N} (1 + \| V \|_{H^s})^{2-},
\end{align} with 
$X_{s,R,N}$ a non-negative random variable satisfying:
\begin{align}
\forall m \gt 0, \mathbb{E} [ X_{s,R,N}^m] \leq C_{s,R,m} \lt +\infty.
\end{align}Indeed, if we do so, then we will be able to continue (7.9) as follows:
\begin{align*}
\log \int e^{\lambda | \mathcal F_{s,N}(u)|}d\mu_{s,R,N} \lesssim \mathbb{E}\big[ \sup_{V \in H^s} \lambda X_{s,R,N}(1 + \| V \|_{H^s})^{2-} - \frac{1}{2}\left\lVert V \right\rVert^2_{H^s} \big] \lesssim \lambda^A \mathbb{E}[X_{s,R,N}^A],
\end{align*}for A > 0 large; and finally using the property (7.11) in the estimate above will ensure (7.6).
Hence, let us turn to the proof of (7.10). In 
$\mathcal F_{s,\vec{N}}(U)$ (see (7.8)), only N 1 has a special position, due to the presence of G. In our analysis, the size of N 1 with respect to the other frequencies does not play a crucial role, and in all cases we estimate G N 1 applying Proposition 5.4. Hence, for convenience, we assume in what follows that 
$N_j = N_{(j)}$ (for 
$j=1,...,p+1$), as the other cases are similar. Finally, throughout the proof we use the following fact:
\begin{align*}
\big( k_1-k_2+...-k_{p+1}=0 \text{ and } \forall j, |k_j| \sim N_j \big) \implies N_{(1)} \sim N_{(2)},\ \text{that is } N_1 \sim N_2,
\end{align*}without explicitly referring to it.
Now, we consider the two following frequency regime:
\begin{align*}
\mathcal F_{s,\vec{N}}(U) = \mathbb{1}_{N_4 \ll N_3} \mathcal F_{s,\vec{N}}(U) + \mathbb{1}_{N_4 \sim N_3} \mathcal F_{s,\vec{N}}(U),
\end{align*}and we treat separately each of these terms.
The first regime: When 
$N_4 \ll N_3$. We are here in a position to use the refined estimate (6.6). Doing so leads to:
\begin{align*}
|\mathcal F_{s,\vec{N}}(U)| \lesssim N_{(1)}^{2(s-1)} \| G^{N_1} \|_{L^2} \| U^{N_2} \|_{L^2} \prod_{j=3}^{p+1} \| U^{N_j} \|_{H^{\frac{1}{2}}}.
\end{align*} Applying now (5.5) for G N 1 (true for 
$G=F_N$, and also clearly for G = id), (5.3) with 
$\alpha = 2(s-\frac{5}{4})$ to U N 2, and (5.3) with 
$\alpha = \frac{1}{2}$ to 
$U^{N_j}$ (
$j \geq 3$), yields:
\begin{align*}
\begin{split}
\mathbb{1}_{\mathcal E_N(U) \leq R} |\mathcal F_{s,\vec{N}}(U)| &\lesssim N_1^{-\varepsilon} X_{s,R,N} (1+ \| V \|_{H^s})^{(3-2s) [\frac{p}{2} +\varepsilon + 2(s-\frac{5}{4}) + \frac{p-1}{2}] + } \\
& = N_1^{-\varepsilon} X_{s,R,N} (1+ \| V \|_{H^s})^{(3-2s) [2s +p -3 +\varepsilon] + },
\end{split}
\end{align*} where we gathered in 
$X_{s,R,N}$ all the positive random variables appearing when using Proposition 5.4. Thanks to this proposition, we indeed see that 
$X_{s,R,N}$ satisfies (7.11); and then, thanks to Lemma 2.1, the inequality above implies the desired estimate (7.10) for ɛ > 0 small enough.
The second regime: When 
$N_4 \sim N_3$. Here, we apply (6.5) and obtain:
\begin{align*}
\mathbb{1}_{\mathcal E_N(U) \leq R} |\mathcal F_{s,\vec{N}}(U)| \lesssim \mathbb{1}_{\mathcal E_N(U) \leq R} N_1^{2(s-1) +} N_3^2 \| G^{N_1} \|_{L^2} \prod_{j=2}^{6} \| U^{N_j} \|_{L^2} \prod_{j=7}^{p+1} \| U^{N_j} \|_{H^{\frac{1}{2}}}.
\end{align*} In this inequality, we simply bound 
$\| U^{N_5} \|_{L^2}$ and 
$\| U^{N_6} \|_{L^2}$ by R (see (5.6)). Then, we use again (5.5) for G N 1; and, we use (5.3) with 
$\alpha= \sigma = s -\frac{1}{2}-$ for U N 2 and U N 3, and with 
$\alpha = \frac{1}{2}$ for 
$U^{N_j}$ (j = 4 and 
$j \geq 7$). In doing so we obtain:
\begin{align*}
\begin{split}
\mathbb{1}_{\mathcal E_N(U) \leq R} |\mathcal F_{s,\vec{N}}(U)| & \lesssim N_1^{s - 2 -\varepsilon + } N_3^{2 - s +} X_{s,R,N} (1+ \| V \|_{H^s})^{(3-2s) [ \frac{p}{2} + \varepsilon + 2(s - \frac{1}{2}) + \frac{p-4}{2}] + } \\
& \lesssim N_1^{-\varepsilon +} X_{s,R,N} (1+ \| V \|_{H^s})^{(3-2s) [2s + p - 3 +\varepsilon] + },
\end{split}
\end{align*} with again 
$X_{s,R,N}$ satisfying (7.11) (thanks to (5.4) in Proposition 5.4). Since we can replace in the inequality above 
$-\varepsilon +$ by 
$-\varepsilon + \delta_0$, where ɛ and δ 0 can be chosen independently, we indeed obtain (thanks to Lemma 2.1) the desired estimate (7.10) (taking 
$\delta_0 \leq \frac{\varepsilon}{2}$ and ɛ > 0 small enough). The proof is now complete.
7.3. Proof of the Lq integrability
Here, we prove Prpoposition 3.6 combining the two previous lemmas 7.3 and 7.5. First, they imply that:
\begin{align}
\| f_{s,N,t}\|^q_{L^q(\rho_{s,R,N})} \leq \exp \big( C_{s,R,q} |t|^A \big),
\end{align} for every 
$ t \in \mathbb{R}$, R > 0 and 
$q \in (1,\infty)$. Next, recalling the definitions of the Radon-Nikodym derivatives 
$g_{s,N,t}$ and 
$f_{s,N,t}$ in (1.10) and (7.3) respectively, we write:
\begin{align*}
&\| g_{s,N,t} \|^q_{L^q(d\mu_{s,R,N})} = \int e^{qR_{s,N}(\Phi^N_{-t}u) - qR_{s,N}(u)} f_{s,N,t}^q d\mu_{s,R,N} \\
&= \int [e^{qR_{s,N}(\Phi^N_{-t}u)} f_{s,N,t}^{\frac{1}{3}} e^{-\frac{1}{3} R_{s,N}(u)}] [e^{-(q-\frac{2}{3})R_{s,N}(u)}] [f_{s,N,t}^{q-\frac{1}{3}} e^{-\frac{1}{3}R_{s,N}(u)}] d\mu_{s,R,N} \\
& \leq \big( \int e^{3qR_{s,N}(\Phi^N_{-t}u)} f_{s,N,t} d\rho_{s,R,N} \big)^{\frac{1}{3}} \big( \int e^{(-3q + 2)R_{s,N}(u)}d\mu_{s,R,N} \big)^{\frac{1}{3}} \big( \int f_{s,N,t}^{3q-1} d\rho_{s,R,N} \big)^{\frac{1}{3}} \\
& = \big( \int e^{3qR_{s,N}(u)} d\rho_{s,R,N} \big)^{\frac{1}{3}} \big( \int e^{(-3q + 2)R_{s,N}(u)}d\mu_{s,R,N} \big)^{\frac{1}{3}} \big( \int f_{s,N,t}^{3q-1} d\rho_{s,R,N} \big)^{\frac{1}{3}}.
\end{align*}Then, by (7.6) and (7.12), we deduce that:
\begin{align*}
\| g_{s,N,t} \|^q_{L^q(d\mu_{s,R,N})} \leq \exp \big( C_{s,R,q}|t|^A \big),
\end{align*}as promised.
Acknowledgements
The author would like to thank Chenmin Sun, Leonardo Tolomeo, and Nikolay Tzvetkov for suggesting this problem. The author is also grateful to Leonardo Tolomeo for an interesting discussion, and to Chenmin Sun and Nikolay Tzvetkov for their support and advice during the writing of this manuscript. Finally, the author thanks the anonymous referee for valuable comments and suggestions. This work is partially supported by the ANR project Smooth ANR-22-CE40-0017.
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