Normal form approach to the one-dimensional periodic cubic nonlinear Schr\"odinger equation in almost critical Fourier-Lebesgue spaces

In this paper, we study the one-dimensional cubic nonlinear Schr\"odinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces $\mathcal{F} L^p(\mathbb{T})$, $1 \leq p<\infty$. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in $\mathcal{F} L^p(\mathbb{T})$ for $1\leq p \leq \frac32$.

x u ± |u| 2 u = 0 u| t=0 = u 0 , (x, t) ∈ T × R. (1.1) The equation (1.1) arises from various physical settings such as nonlinear optics and quantum physics. See [37] for the references therein. It is also known to be one of the simplest completely integrable PDEs [38,1,2,17,27]. The Cauchy problem (1.1) has been studied extensively both on the real line and on the circle. See [33,21] for the references therein. In this paper, we study the periodic cubic NLS (1.1) in the Fourier-Lebesgue spaces FL p (T) defined via the norm: with a usual modification when p = ∞. For any 2 ≤ p ≤ q ≤ ∞, we have the following continuous embeddings: The space FL 1 (T) is the Wiener algebra. The space FL ∞ (T) is the space of pseudomeasures, which contains all finite Borel measures on T but also more singular distributions. See [25]. Our main interest is to study (1.1) in FL p (T) for p ≫ 1.
On the one hand, the cubic NLS (1.1) is known to be globally well-posed in FL 2 (T) = L 2 (T) [6]. On the other hand, combining the known results [19,21,34], we can easily show that it is ill-posed in the Fourier-Lebesgue space FL p (T) for p > 2 in a very strong sense. See Proposition 1.1 below. This necessitates us to renormalize the nonlinearity and consider the following renormalized cubic NLS: (1.2) Note that the renormalized cubic NLS (1.2) is "equivalent" to the original cubic NLS (1.1) for smooth solutions in the following sense. For u ∈ C(R; L 2 (T)), we define the following invertible gauge transformation G by Then, thanks to the L 2 -conservation, it is easy to see that u ∈ C(R; L 2 (T)) is a solution to (1.1) if and only if G(u) is a solution to the renormalized cubic NLS (1.2). This renormalization removes a certain singular component from the nonlinearity and, as a result, the renormalized cubic NLS (1.2) behaves better than the cubic NLS (1.1) outside L 2 (T). The study of (1.2) outside L 2 (T) has attracted much attention in recent years [8,9,19,33,12,21,34,31,36].
In [19], Grünrock-Herr adapted the Fourier restriction norm method to the Fourier-Lebesgue space setting and proved local well-posedness of the renormalized cubic NLS (1.2) in FL p (T) for 1 ≤ p < ∞ by a standard contraction argument. See also the work by Christ [9]. In [36], by using the completely integrable structure of the equation, we established the following global-in-time a priori bound: sup t∈R u(t) F L p ≤ C( u 0 F L p ) (1.4) for any smooth solution u to the renormalized cubic NLS (1.2) and 2 ≤ p < ∞, which implied global well-posedness of (1.2) in FL p (T) for 1 ≤ p < ∞. 1 As a corollary to the local well-posedness of the renormalized cubic NLS in [19], one easily obtains the following non-existence result for the original cubic NLS (1.1) outside L 2 (T). In [21], the first author (with Z. Guo) proved an analogous non-existence result for (1.1) in negative Sobolev spaces. The argument was based on an a priori bound for smooth solutions to the renormalized cubic NLS (1.2) in negative Sobolev spaces and exploiting a fast oscillation in (1.3). The proof of the local well-posedness in [19] yields an a priori bound for smooth solutions to the renormalized cubic NLS (1.2) in FL p (T). Then, we can prove Proposition 1.1 by proceeding as in [21,35]. We omit details.
In the following, we only consider the focusing case (i.e. with the + sign in (1.1) and (1.2)) for simplicity. Our main results equally apply to the defocusing case. • the nonlinearity N (u) exists in the sense of (ii) above, • u satisfies (1.2) in the distributional sense on T × (−T, T ), where the nonlinearity N (u) is interpreted as above.
In [8,9], Christ introduced this notion in studying the renormalized cubic NLS (1.2) in the low regularity setting. See also [20] for a similar notion of weak solutions, where the nonlinearity is defined as a distributional limit of smoothed nonlinearities.
Next, we introduce the following notion of sensible weak solutions. See also [36,14]. in FL p (T), the corresponding (classical) solutions u m with u m | t=0 = u 0,m converge to u in C([−T, T ]; FL p (T)). Moreover, we impose that there exists a distribution v such that N (u m ) converges to v in the space-time distributional sense, independent of the choice of the approximating sequence.
Note that, by using the equation, the convergence of u m to u in C([−T, T ]; FL p (T)) implies that N (u m ) converges to some v in the space-time distributional sense; see (2.9) below. Hence, the last part of Definition 1.3 is not quite necessary. We, however, keep it for clarity.
We point out that these notions of weak solutions in Definitions 1.2 and 1.3 are rather weak. The cubic nonlinearity N (u) for a weak solution u in the sense of Definitions 1.2 or 1.3 does not directly make sense as a distribution in general and we need to interpret it as a (unique) limit of smoothed nonlinearities N (T N u) or the nonlinearities N (u m ) of smooth approximating solutions u m . This in particular implies that weak solutions in the sense of Definitions 1.2 or 1.3 do not have to satisfy the equation even in the distributional sense.
On the one hand, sensible weak solutions are unique by definition. On the other hand, weak solutions in the extended sense are not unique in general. In fact, Christ [8] proved non-uniqueness of weak solutions in the extended sense for the renormalized cubic NLS (1.2) in negative Sobolev spaces.
Our main goal in this paper is (i) to develop further the normal form approach to study the (renormalized) cubic NLS, introduced in [22], and provide the solution theory for (1.2) in almost critical Fourier-Lebesgue spaces (Theorem 1.4) in the sense of Definitions 1.2 and 1.3 without using any auxiliary function spaces, in particular, without using the Fourier restriction norm method as in [6,19] and (ii) to prove unconditional uniqueness of the (renormalized) cubic NLS in FL p (T) for 1 ≤ p < 3 2 (Theorem 1.5). In proving these results, we apply an infinite iteration of normal form reductions and transform the (renormalized) cubic NLS into the so-called normal form equation. We then prove unconditional wellposedness of the normal form equation in FL p (T) for any 1 ≤ p < ∞; see Theorem 1.9 below.
We now state our main results. This theorem follows from the local well-posedness by Grünrock-Herr [19], combined with the a priori bound (1.4) from [36]. As pointed out above, however, our main goal is to present an argument independent of the Fourier restriction norm method. We instead employ the normal form approach developed in [22]. Our approach does not involve any auxiliary function spaces and consequently allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in FL In [22], the first author (with Z. Guo and S. Kwon) proved an analogous result in L 2 (T) by implementing an infinite iteration of normal form reductions, 3 yielding unconditional uniqueness of the cubic NLS (1.1) in H 1 6 (T). The proof of Theorem 1.4 is also based on the same normal form approach. See the next subsection. Note that when p is very large, Theorem 1.4 is significantly harder to prove than the L 2 -result in [22] due to a much weaker FL p -topology.
Given u 0 ∈ FL p (T), let u be the global solution to (1.2) with u| t=0 = u 0 constructed in Theorem 1.4. Then, by the uniqueness of sensible solutions mentioned above, u must coincide with the global solution constructed in [6,19,36]. In particular, the solution u belongs to the class (T) ⊂ L 3 (T) allows us to make sense of the cubic nonlinearity in a direct manner. In this case, we have the following uniqueness statement.
Then, given any u 0 ∈ FL p (T), the solution u to (1.1) or (1.2) with u| t=0 = u 0 constructed in Theorem 1.4 is unique in C(R; FL p (T)).
Namely, unconditional uniqueness holds for both the cubic NLS (1.1) and the renormalized cubic NLS (1.2) in FL p (T), provided that 1 ≤ p ≤ 3 2 . In [22], the first author (with Z. Guo and S. Kwon) proved unconditional uniqueness in H 1 6 (T) and Theorem 1.5 extends this result to the Fourier-Lebesgue setting. We also mention a recent work by Herr-Sohinger [24] where they proved unconditional uniqueness of the cubic NLS (1.1) in L p ([−T, T ] × T) for p > 3. The main difference between unconditional uniqueness and uniqueness for sensible weak solutions is that the former does not assume that a solution comes with a sequence of smooth approximating solutions, while, by definition, sensible weak solutions are equipped with smooth approximating solutions. Remark 1.6. When p = ∞, the Fourier-Lebesgue space FL ∞ (T) does not admit smooth approximations and hence is not suitable for well-posedness study. Given s ∈ R and 1 ≤ p ≤ ∞, define FL s,p (T) by the norm: Note that FL p (T) = FL 0,p (T). For s < − 1 p , we have FL ∞ (T) ⊂ FL s,p (T) and thus we may wish to study well-posedness in FL s,p (T) for finite p with s < − 1 p since this space admits smooth approximations. On the other hand, the scaling critical regularity 4 That is, unless we use the uniqueness property of sensible solutions and conclude that they belong to the class (1.6) by comparing with the solutions constructed in [6,19,36].
for the cubic NLS (1.1) with respect to the Fourier-Lebesgue spaces FL s,p (T) is given by s crit = − 1 p . In particular, the cubic NLS (1.1) and the renormalized cubic NLS (1.2) are known to be ill-posed in the (super)critical regime. 5 When s < 0, it is easy to modify the argument in [7,10,12] and show that the solution map is not locally uniformly continuous in FL s,p (T). Furthermore, when s ≤ s crit = − 1 p , the cubic NLS (1.1) and the renormalized cubic NLS (1.2) admit norm inflation; given any ε > 0, there exist a solution u to (1.1) or (1.2) and t ∈ (0, ε) such that See [28]. The norm inflation in particular implies discontinuity of the solution map at the trivial function 6 u ≡ 0. Lastly, a typical function in FL ∞ (T) is the Dirac delta function and (1.1) and (1.2) on T are known to be ill-posed with the Dirac delta function as initial data; see [14]. See also Kenig-Ponce-Vega [26] and Banica-Vega [4,5] for the works on the cubic NLS (1.1) on the real line with the Dirac delta function as initial data.
Remark 1.7. Following the argument in [22], we can easily extend Theorem 1.4 to FL s,p (T) for s > 0 and 1 ≤ p < ∞. Similarly, the unconditional uniqueness result in Theorem 1.5 can be extended to FL s,p (T) for (i) s > 0 and 1 ≤ p ≤ 3 2 and (ii) p > 3 2 and s > 2p−3 3p . Note that in these ranges of (s, p), we have FL s,p (T) ֒→ FL Let S(t) = e it∂ 2 x denote the linear Schrödinger propagator. We introduce the interaction representation: x u(t). (1.8) On the Fourier side, we have u(n, t) = e in 2 t u(n, t). Then, (1.2) can be written as 8 ∂ t u n = i n=n 1 −n 2 +n 3 n 2 =n 1 ,n 3 e iΦ(n)t u(n 1 ) u(n 2 ) u(n 3 ) − i| u(n)| 2 u(n) =: N 1 (u)(n) + R(u)(n). (1.9) Here, the phase function Φ(n) is defined by Φ(n) : = Φ(n, n 1 , n 2 , n 3 ) = n 2 − n 2 1 + n 2 2 − n 2 3 = 2(n 2 − n 1 )(n 2 − n 3 ) = 2(n − n 1 )(n − n 3 ), (1.10) 5 In fact, it is shown in [28] that the cubic NLS (1.1) and the renormalized cubic NLS (1.2) are ill-posed even in the logarithmically subcritical regime. 6 One can easily combine the argument in [28,31] to prove norm inflation at general initial data, concluding discontinuity of the solution map at every function FL s,p (T), provided that s ≤ scrit = − 1 p . 7 In the following, we restrict our attention to the renormalized cubic NLS (1.2). See Subsection 2.4 for required modifications to handle the cubic NLS (1.1) in Theorem 1.5. 8 Due to the presence of the time-dependent phase factor e iΦ(n)t , the non-resonant part N1(u), viewed as a trilinear operator is non-autonomous. For notational simplicity, however, we suppress such t-dependence when there is no confusion. We apply this convention to all the multilinear operators appearing in this paper.
where the last two equalities hold under n = n 1 − n 2 + n 3 . From (1.10), we see that N 1 corresponds to the non-resonant part (i.e. Φ(n) = 0) of the nonlinearity and R corresponds to the resonant part. Note that the Duhamel formulation for (1.2): is now expressed as a system of integral equations: for n ∈ Z. In the following, the space FL The main idea is to apply a normal form reduction to (1.9), namely integration by parts in (1.11), to exploit the oscillatory nature of the non-resonant contribution. As in [22,29], we implement an infinite iteration of normal form reductions and derive the following normal form equation: are time-dependent (2j + 1)-linear operators. As we see in Section 3, multilinear dispersion effects are already embedded in these multilinear terms, which allows us to prove that these multilinear operators are bounded in C([−T, T ]; FL p (T)) for any 1 ≤ p < ∞. Moreover, we show that the normal form equation (1.12) is equivalent to (1.9) and the renormalized cubic NLS (1.2) in C(R; FL 3 2 (T)). See Proposition 2.1. As a consequence, we can easily prove local well-posedness of the normal form equation (1.12) in FL p (T) by a simple contraction argument without any auxiliary function spaces.
In [22], an analogous result was shown in L 2 (T). When p > 2, the FL p -norm is weaker than the L 2 -norm. In particular, when p ≫ 1, this fact makes it much harder to show convergence of the series in the normal form equation (1.12) with respect to the FL ptopology.
Once we establish the relevant multilinear estimates (Proposition 2.1), the proof of unconditional local well-posedness for the normal form equation (1.12) follows from a simple contraction argument. Moreover, we show that the local existence time T depends only on the size of the initial data u 0 F L p and consequently, we conclude that solutions exist globally in time in view of the global-in-time bound (1.4) from [36]. See also Appendix A.
Finally, note that Theorem 1.5 follows easily thanks to the equivalence of (1.2) and the normal form equation (1.12) for regular solutions belonging to C(R; FL 3 2 (T)). The contraction argument in proving Theorem 1.9 yields the following Lipschitz bound: (1.14) Then, (1.13) and (1.14) together with the multilinearity of the summands in (1.14) and the unitarity of the linear operator S(t) in FL p (T) allow us to conclude convergence of smoothed nonlinearities N (T N u) or the nonlinearities N (u m ) of smooth approximating solutions u m required in Definitions 1.2 and 1.3. This is a sketch of the proof of Theorem 1.4.
In Section 2, we present the proofs of the main results, assuming the bounds on the multilinear operators {N In Section 3, we implement an infinite iteration of normal form reductions as in [22] and prove Proposition 2.1.
Given u ∈ C([−T, T ]; FL p (T)), we can not, in general, make sense of the cubic nonlinearity N (u) as a distribution since FL p (T) ⊂ L 3 (T). In other words, we can not estimate the cubic nonlinearity without relying on some auxiliary function space. In (1.14), we re-expressed the cubic nonlinearity into series of the multilinear terms of increasing degrees. On the one hand, this transformation brings algebraic complexity. On the other hand, the right-hand side of (1.14) is convergent for u ∈ C([−T, T ]; FL p (T)), allowing us to make sense of the right-hand side of (1.14) as a distribution. Namely, while the left-hand side of (1.14) and the right-hand side of (1.14) coincide for regular solutions u ∈ C([−T, T ]; FL 3 2 (T)), the right-hand side of (1.14) provides a better formulation of the nonlinearity for rougher functions u ∈ C([−T, T ]; FL p (T)), 3 2 < p < ∞. In this sense, we can view the right-hand side of (1.14) as a further renormalization of the renormalized nonlinearity N (u) in (1.5).
By expressing the normal form equation (1.12) in terms of the original function u(t) = S(t)u(t), we obtain (1. 16) As we see in Section 3, the multilinear operators S(t)N (j) 1 , and R (j) are autonomous. The discussion above shows that the normal form equation (1.15) expressed in terms of u(t) = S(t)u(t) is a better model to study than the renormalized cubic NLS (1.2) (and the cubic NLS (1.1)) in the low regularity setting, which can be viewed as a further renormalization to the (renormalized) cubic NLS.
Lastly, we point out that the terms on the left-hand side of (1.16) are indeed autonomous (unlike the non-autonomous multilinear terms in (1.14)). See Section 3.
Remark 1.11. A precursor to this normal form approach first appeared in the work of Babin-Ilyin-Titi [3] in the study of KdV on T, establishing unconditional well-posedness of the KdV in L 2 (T). See also [30]. In [22], the first author with Z. Guo and S. Kwon further developed this normal form approach and introduced an infinite iteration scheme of normal form reductions in the context of the cubic NLS on the circle. In this series of work, the viewpoint of unconditional well-posedness was first introduced in [30], while the viewpoint of the (Poincaré-Dulac) normal form reductions was first introduced in [22]. This normal form approach has also been used to prove nonlinear smoothing [13], improved energy estimates [32,35], and construct an infinite sequence of invariant quantities under the dynamics [11]. Remark 1.12. In a recent paper [14], the first author with Forlano studied the cubic NLS on R. In particular, by implemented an infinite iteration of normal form reductions, they proved analogues of Theorems 1.4, 1.5, and 1.9 in almost critical Fourier-Lebesgue spaces FL p (R), 2 ≤ p < ∞, and almost critical modulation spaces M 2,p (R), 2 ≤ p < ∞. Relevant multilinear estimates were studied based on the idea introduced in [29], namely, successive applications of basic trilinear estimates (called localized modulation estimates).

Proof of the main results
In this section, we present the proofs of the main results (Theorems 1.4, 1.5, and 1.9), assuming the validity of the transformation of the equation (1.11) to the normal form equation (1.12) and the boundedness of the multilinear operators in (1.12) (Proposition 2.1).
2.1. Series expansion of regular solutions. In Section 3, we implement an infinite iteration of normal form reductions and transform the equation (1.9) into the normal form equation (1.12) for regular solutions. The following proposition summarizes the properties of the multilinear operators in (1.12). Given R > 0, we use B R to denote the ball of radius R centered at the origin in various function spaces.
such that any regular solution u ∈ C([−T, T ]; FL 3 2 (T)) to (1.9) with u(0) ∈ B R ⊂ FL p (T) satisfies the following normal form equation: for some absolute constant C p > 0 depending only on p.
In Proposition 2.1, we imposed a strong regularity assumption: u ∈ C([−T, T ]; FL 3 2 (T)). This regularity assumption can be easily relaxed. In view of the estimates (2.2), (2.3), and (2.4), we see that the right-hand side of (2.1) is convergent for u ∈ C([−T, T ]; FL p (T)). See also the proof of Theorem 1.9 below. By using the multilinearity of the operators, we only need to estimate the difference such as N This, however, does not cause any issue thanks to the fast decay (2.5) of the coefficients C 0,j and C 1,j . Since the proof of Corollary 2.2 is straightforward computation with (2.2), (2.3), and (2.4), we omit details.
We postpone the proof of Proposition 2.1 to Section 3. In the remaining part of this section, we present the proofs of Theorems 1.4, 1.5, and 1.9, assuming Proposition 2.1. In Subsection 2.4, we discuss the case of the (unrenormalized) NLS (1.1). 9 Here, we view N We first present the proof of Theorem 1.9.
Proof of Theorem 1.9. Given 1 ≤ p < ∞, let u 0 ∈ FL p (T). With K = K( u 0 F L p ) ≥ 1 (to be chosen later), define the map Γ u 0 by where the multilinear terms on the right-hand side (depending on the choice of K ≥ 1) are as in Proposition 2.1. Let T > 0. Then, by Proposition 2.1, we have for any u ∈ B 2R ⊂ C([−T, T ]; FL p (T)). The series in (2.6) are obviously convergent for any K ≥ 1 thanks to the fast decay in j but by choosing K = K(R, p) ≫ 1 sufficiently large, we can guarantee that Note that the third series in (2.6) has non-negative powers of K for 1 ≤ j < 4 p ′ −1 , while a power of K does not appear in the fourth series when j = 1. These terms can be controlled by choosing T = T (K, R) = T (R) > 0 sufficiently small. As a result, we obtain for any u ∈ B 2R ⊂ C([−T, T ]; FL p (T)). A similar argument also yields the following difference estimate: In establishing the difference estimate (2.7), we need to estimate the differences such as N Lastly, by taking the difference of two solutions u, v ∈ C([−T, T ]; FL p (T)) with different initial data u 0 and v 0 , we have for all m, n ≥ 1 and any T > 0. This shows that {u m } m∈N is a Cauchy sequence in C(R; FL p (T)) endowed with the compact-open topology (in time) and hence converges to some u ∞ in C(R; FL p (T)). Now, we prove uniqueness of the limit u ∞ , independent of smooth approximating solutions. Given u 0 ∈ FL p (T), let {u m } m∈N and {v n } n∈N be two sequences of smooth solutions such that u m (0), v n (0) → u 0 in FL p (T) as m, n → ∞. Then, by the argument above, there exist u ∞ , v ∞ ∈ C(R; FL p (T)) such that u m → u ∞ and v n → v ∞ in C(R; FL p (T)) as m, n → ∞. Then, by the triangle inequality with (1.13) and (2.8), we obtain Lastly, combining this convergence with (1.2), we obtain in the distributional sense as m, n → ∞. Therefore, we conclude that (1.2) is globally well-posed in the sense of sensible weak solutions.

2.4.
On the cubic NLS. We conclude this section by discussing the situation for the cubic NLS (1.1). By writing we see that the third term III is the only difference from the case for the renormalized cubic NLS (1.2). By taking an interaction representation, we can write (1.1) as where R 2 (u)(n) is given by ˆT |u| 2 dx u(n).
As compared to (1.9), R 2 (u) is the only difference. Note that this extra term R 2 (u) imposes the restriction p ≤ 2. As in the case of the renormalized NLS (1.2), we prove the following proposition in Section 3.
With Proposition 2.3, we can proceed as in the proof of Theorem 1.9 and prove the following unconditional well-posedness of the normal form equation Then, Theorem 1.4 for 1 ≤ p ≤ 2 and Theorem 1.5 for the cubic NLS (1.1) follow from arguments analogous to those presented above. We omit details.

Normal form reduction: Proof of Proposition 2.1
In this section, we implement an infinite iteration of normal form reductions in the Fourier-Lebesgue space FL p (T), 1 ≤ p < ∞, and prove Proposition 2.1. The argument is presented in an inductive manner. More precisely, we start with the formulation (1.9) and refer to this case as the first step (J = 1). Define where N 1 (u)(n) and R(u)(n) are as in (1.9). In what follows, we view N 1 and R as trilinear operators.
For notational convenience, we set R (1) := R and N (1) := N 1 . While we keep the resonant part R (1) as it is, we divide the non-resonant part N (1) into a "good" part N (1) 1 (nearly resonant part) and a "bad" part N (1) 2 (highly non-resonant part), depending on the size of the phase function Φ(n). On the one hand, the restriction on the phase function Φ(n) allows us to establish an effective estimate on the good part N 0 , R (2) , and N (2) in the second generation (J = 2). We can easily estimate the terms N (2) 0 and R (2) . As in the first step, we divide N (2) into a good part N 2 , where the threshold is now given by the phase function for the quintilinear term N (2) . While the good part N allows for an effective quintilinear estimate, we apply a normal form reduction to the bad part N tends to 0 as J → ∞. In order to carry out the strategy described above, we need to address the following four issues: • How do we separate N (J) into "good" and "bad" parts?
• How do we estimate these good terms in the FL p (T) when p ≫ 1? As we see below, N We address these issues in the remaining part of this section. In the following, we fix 1 ≤ p < ∞. The major part of this section is devoted to studying the renormalized cubic NLS (1.2). As for the (unrenormalized) cubic NLS (1.1), see Subsection 3.6.
3.1. Base case: J = 1. Define the trilinear operators N (1) and R (1) by where Φ(n) is as in (1.10). For notational simplicity, we set N (1) (u) = N (1) (u, u, u), etc. when all the three arguments coincide. Note that this notation is consistent with (1.9) and (3.1). Then, we can write (1.9) as The resonant part satisfies the following trivial estimate.
Remark 3.2. (i) In the following, we establish various multilinear estimates. To simply notations, we only state and prove estimates when all arguments agree with the understanding that they can be easily extended to multilinear estimates. Under this convention, (3.4) is written as We also use u n = u n (t) to denote u(n, t). Moreover, given a multilinear operator M, we simply use M(u)(n) to denote the Fourier coefficients of M(u).
(ii) The multilinear operators that appear below are non-autonomous, i.e. they depend on a parameter t ∈ R. They, however, satisfy estimates uniformly in time and hence we simply suppress their time dependence. See (3.10) for example.
Next, we consider the non-resonant part N (1) in (3.2). As it is, we can not establish an effective estimate and hence we divide it into two parts. Given K ≥ 1 (to be chosen later) and 1 ≤ p < ∞, let ε = ε(p) > 0 be a small positive number such that In the following, we simply set such that (3.5) is satisfied. Furthermore, we set We write N (1) in (3.2) as A 1 (n) := (n, n 1 , n 2 , n 3 ) : n = n 1 − n 2 + n 3 , n 1 , n 3 = n, and N (1) 2 1 . Then, the "good" part N (1) 1 satisfies the following trilinear estimate.
As in the p = 2 case studied in [22], the following divisor estimate [23] plays an important role in the following. Given an integer n, let d(n) denote the number of divisors of m. Then, we have d(n) e c log n log log n (= o(n δ ) for any δ > 0). Proof. Fix n, µ ∈ Z with |µ| ≤ (3K) θ . Then, it follows from the divisor estimate (3.11) that there are at most (3K) 0+ many choices for n 1 and n 3 (and hence for n 2 from n = n 1 −n 2 +n 3 ) satisfying µ = 2(n − n 1 )(n − n 3 ).
(3.12) 11 Clearly, the number 3 θ in (3.9) does not make any difference at this point. However, we insert it to match with (3.25). See also (3.18).
Hence, we have sup n |µ|≤(3K) θ n=n 1 −n 2 +n 3 n 2 =n 1 ,n 3 µ=Φ(n) Then, by Hölder's inequality, we have The boundary term N can be estimated in a straightforward manner. Using the equation (1.9), we can express N (2) (u)(n) as a quintilinear form: (3.14) In view of (3.2), we regard R (2) (u)(n) and N (2) (u)(n) on the right-hand side as quintilinear forms. As in the first step, we will need to divide N (2) into good and bad parts and apply another normal form reduction to the bad part. Before proceeding further, we first recall the notion of ordered trees introduced in [22]. This allows us to express multilinear terms in a concise manner.
Remark 3.5. We formally exchanged the order of the sum and the time differentiation in the first term at the third equality. This can be easily justified in the distributional sense (see Lemma 5.1 in [22]) and also in the classical sense if u ∈ C([−T, T ]; FL 3 2 (T)) ⊂ C([−T, T ]; L 3 (T)). See [22].

3.2.
Notations: index by trees. In this subsection, we recall the notion of ordered trees and relevant definitions from [22]. Definition 3.6. (i) Given a partially ordered set T with partial order ≤, we say that b ∈ T with b ≤ a and b = a is a child of a ∈ T , if b ≤ c ≤ a implies either c = a or c = b. If the latter condition holds, we also say that a is the parent of b.
(ii) A tree T is a finite partially ordered set satisfying the following properties.
• Let a 1 , a 2 , a 3 , a 4 ∈ T . If a 4 ≤ a 2 ≤ a 1 and a 4 ≤ a 3 ≤ a 1 , then we have a 2 ≤ a 3 or a 3 ≤ a 2 , • A node a ∈ T is called terminal, if it has no child. A non-terminal node a ∈ T is a node with exactly three children denoted by a 1 , a 2 , and a 3 , • There exists a maximal element r ∈ T (called the root node) such that a ≤ r for all a ∈ T . We assume that the root node is non-terminal, • T consists of the disjoint union of T 0 and T ∞ , where T 0 and T ∞ denote the collections of non-terminal nodes and terminal nodes, respectively.
The number |T | of nodes in a tree T is 3j + 1 for some j ∈ N, where |T 0 | = j and |T ∞ | = 2j + 1. Let us denote the collection of trees in the jth generation by T (j): T (j) := {T : T is a tree with |T | = 3j + 1}.
Note that T ∈ T (j) contains j parental nodes.
(iii) (ordered tree) We say that a sequence {T j } J j=1 is a chronicle of J generations, if • T j ∈ T (j) for each j = 1, . . . , J, • T j+1 is obtained by changing one of the terminal nodes in T j into a non-terminal node (with three children), j = 1, . . . , J − 1.
Given a chronicle {T j } J j=1 of J generations, we refer to T J as an ordered tree of the Jth generation. We denote the collection of the ordered trees of the Jth generation by T(J). Note that the cardinality of T(J) is given by |T(J)| = 1 · 3 · 5 · · · · · (2J − 1) = (2J − 1)!! =: c J . (3.15) The notion of ordered trees comes with associated chronicles; it encodes not only the shape of a tree but also how it "grew". This property will be convenient in encoding successive applications of the product rule for differentiation. In the following, we simply refer to an ordered tree T J of the Jth generation but it is understood that there is an underlying chronicle {T j } J j=1 .
Given a tree T , we associate each terminal node a ∈ T ∞ with the Fourier coefficient (or its complex conjugate) of the interaction representation u and sum over all possible frequency assignments. In order to do this, we introduce the index function n assigning frequencies to all the nodes in T in a consistent manner.
Definition 3.7 (index function). Given an ordered tree T (of the Jth generation for some J ∈ N), we define an index function n : T → Z such that, (i) n a = n a 1 − n a 2 + n a 3 for a ∈ T 0 , where a 1 , a 2 , and a 3 denote the children of a, (ii) {n a , n a 2 } ∩ {n a 1 , n a 3 } = ∅ for a ∈ T 0 , (iii) |µ 1 | := |2(n r − n r 1 )(n r − n r 3 )| > (3K) θ , 12 where r is the root node, where we identified n : T → Z with {n a } a∈T ∈ Z T . We use N(T ) ⊂ Z T to denote the collection of such index functions n. Remark 3.8. Note that n = {n a } a∈T is completely determined once we specify the values n a for a ∈ T ∞ .
Given an ordered tree T J of the Jth generation with the chronicle {T j } J j=1 and associated index functions n ∈ N(T J ), we use superscripts to denote "generations" of frequencies.
Fix n ∈ N(T J ). Consider T 1 of the first generation. Its nodes consist of the root node r and its children r 1 , r 2 , and r 3 . We define the first generation of frequencies by 3.3. Second generation: J = 2. With the ordered tree notion introduced in the previous subsection, we now rewrite (3.14) as In the first equality, we used (1.9) and replace ∂ t u n b by R (1) (u)(n b ) and N (1) (u)(n b ). Strictly speaking, the new phase factor may be µ 1 − µ 2 when the time derivative falls on the complex conjugate. However, for our analysis, it makes no difference and hence we simply write it as µ 1 + µ 2 . We apply the same convention for subsequent steps. Putting (3.13) and (3.17) together, we have The boundary term N 0 (u) and the "resonant" term R (2) can be bounded in a straightforward manner.
For the proof of Lemma 3.10, see Lemma 3.12 and 3.13 with J = 2.
With θ > 0 as in (3.7), we decompose the frequency space 13 of N (2) for fixed T 2 ∈ T(2) into and its complement A c 2 . Then we decompose N (2) as where N 1 := N (2) | A 2 is defined as the restriction of N (2) on A 2 and N 1 . Thanks to the restriction (3.18) on the frequencies, we can estimate the first term N Lemma 3.11. Let 1 ≤ p < ∞. Then, we have For the proof of Lemma 3.11, see Lemma 3.14 with J = 2. 13 If we fix T2 ∈ T(2), then the frequency space of N (2) for this fixed T2 in (3.17) is given by In view of Remark 3.8, we can then identify the frequency space of N (2) for this fixed T2 with N(T2).
As we do not have a good control on the operator N 2 , we apply another normal form reduction to N (2) 2 . On the support of N (2) 2 , we have By applying differentiation by parts once again, we have (3.21) where the summations are restricted to (3.20). As for the last term N (3) (u), we need to decompose it into N The first two terms on the right-hand side admit good estimates; see Lemmas 3.1 and 3.3. We then applied the first step of normal form reductions to the troublesome term N See (3.13), (3.17), and (3.19). Note that only the last term N (2) 2 (u) can not be estimated in a direct manner. By applying a normal form reduction once again, we obtained (3.23) See (3.21). Once again, all the terms in (3.23), except for the last term N 2 (u), admit good estimates; see Lemmas 3.12, 3.13, and 3.14 below. We then apply the third step of normal form reductions to N 2 (u). We can formally iterate this process. In particularly, after applying normal form reductions J − 1 times, we would arrive at (3.24) In the following, we define each term on the right-hand side of (3.24) properly. With µ j as in (3.16), define µ j by We then set where θ > 0 is as in (3.7). Given j ∈ N, we define N 2 , and N 2 that we saw in the previous subsections. By applying a normal form reduction to (3.26) with (3.3), we obtain Here, we formally exchanged the order of the sum and the time differentiation, which can be justified. See Remark 3.5. As in Subsections 3.1 and 3.3, we divide N (j+1) into (u). This allows us to define all the terms appearing in (3.24) in an inductive manner by applying a normal form reduction to N (j+1) 2 .
In the remaining part of this subsection, we estimate the multilinear operators N Lemma 3.12. Let 1 ≤ p < ∞. Then, there exists C p > 0 such that for any integer j ≥ 2 and K ≥ 1.
Proof. From (3.27) (with j + 1 replaced by j), we have Then, by Hölder's inequality with (3.15), we have In the last step, we used n n∈N(T j−1 ) nr=n We claim that where B p > 0 is a constant depending only on p. Then, by setting we see that (3.29) follows from (3.30) and (3.31). It remains to prove (3.31). First, note that given any small ε > 0, there exists C = C(ε) > 0 such that Recalling that ε in (3.6) depends only on p, we see that B p and hence C p depend only on 1 ≤ p < ∞. This completes the proof of Lemma 3.12.
As a consequence of Lemma 3.12 with Lemma 3.1, we obtain the following estimate on R (j) . Lemma 3.13. Let 1 ≤ p < ∞. Then, there exists C p > 0 such that for any j ∈ N and K ≥ 1.
Proof. When j = 1, this is precisely Lemma 3.1. Let j ≥ 2. Note that R (j) (u) is nothing but N (j) 0 (u) by replacing u n b with R (1) (u)(n b ) for b ∈ T ∞ j and summing over b ∈ T ∞ j . Then, (3.33) follows from Lemma 3.12 with Lemma 3.1 and noting that given T j ∈ T(j − 1), we have #{b : b ∈ T ∞ j−1 } = 2j − 1. This extra factor 2j − 1 does not cause a problem thanks to the fast decaying constant in (3.33).
Lastly, we estimate N (j) Lemma 3.14. Let 1 ≤ p < ∞. Then, there exists C p > 0 such that for any j ∈ N and K ≥ 1.
Proof. From (3.27) (with j + 1 replaced by j), we have Proceeding as in (3.30) with Hölder's inequality, we have We claim that there exists B p > 0 such that Then, the desired estimate (3.34) follows from (3.35) and (3.36) by setting It remains to prove (3.36). As compared to (3.31) in the proof of Lemma 3.12, the main difference is that the summation in (3.36) is over n ∈ N(T j ) rather than n ∈ N(T j−1 ). Note that .
We briefly discuss the proof of Proposition 2.1.