A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier–Lebesgue Spaces

We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2021), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^\frac{1}{2}({\mathbb {T}})$$\end{document}H12(T). Here, we employ the method introduced by Deng et al. (Commun Math Phys 384(1):1061–1107, 2021) to prove local well-posedness of the second renormalized mKdV equation in the Fourier–Lebesgue spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}L^{s,p}({\mathbb {T}})$$\end{document}FLs,p(T) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\ge \frac{1}{2}$$\end{document}s≥12 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p <\infty $$\end{document}1≤p<∞. As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier–Lebesgue spaces with infinite momentum.

torus T = R/(2πZ): where u is a complex-valued function.The complex-valued mKdV equation (1.1) appears as a model for the dynamical evolution of nonlinear lattices, fluid dynamics and plasma physics (see [17,34], for example).This equation is a completely integrable complex-valued generalization of the usual mKdV equation also referred to as the mKdV equation of Hirota [19].Indeed, real-valued solutions of (1.1) are also solutions of (1.2).From the completely integrable structure, it follows that (1.1) has an infinite number of conservation laws.In particular, the mass and the momentum play an important role: Exploiting the conservation of mass µ u(t) = µ(u 0 ) for solutions u ∈ C R; L 2 (T) of (1.1), we can consider the first renormalized mKdV equation (mKdV1): 3) The mKdV1 equation (1.3) is equivalent to mKdV (1.1) in L 2 (T) in the following sense: u ∈ C R; L 2 (T) is a solution of (1.1) if and only if G 1 (u)(t, x) := u t, x∓µ(u)t is a solution of (1.3).In [2], Bourgain showed that (1.3) is locally well-posed in H s (T) for s ≥ 1 2 , using the Fourier restriction norm method.
In [5], we showed that the momentum plays an important role in the well-posedness of the complex-valued mKdV equation (1.1).In fact, outside of H 1 2 (T), the momentum is no longer conserved nor finite, preventing us from making sense of the nonlinearity.There, we established H 1 2 (T) as the limit for the well-posedness theory of the complex-valued mKdV equation (1.1).In particular, we proved the non-existence of solution for initial data with infinite momentum.Our analysis was based on the Fourier-Lebesgue spaces FL s,p (T), defined through the following norm where . More specifically, we are interested in FL s,p (T) for s ≥ 1 2 and 2 ≤ p < ∞, since H 1 2 (T) FL where P ≤N denotes the Dirichlet projection onto the spatial frequencies {|n| ≤ N }.Then, for any T > 0, there exists no distributional solution u ∈ C([−T, T ]; FL s,p (T)) to the complex-valued mKdV equation (1.1) satisfying the following conditions: (i) u| t=0 = u 0 , (ii) The smooth global solutions {u N } N ∈N of mKdV (1.1), with u N | t=0 = P ≤N u 0 , satisfy u N → u in C([−T, T ]; FL s,p (T)).
This ill-posedness result motivated establishing an alternative model for the complexvalued mKdV equation (1.1) at low regularity.Similarly to the first gauge transform G 1 which exploited the conservation of mass, we introduced in [5] a second renormalization of the equation through the following gauge transformation depending on the momentum G 2 (u)(t, x) = e ∓iP (u)t u(t, x). (1.6) Note that if u ∈ C R; H 1 2 (T) , the momentum is finite and conserved P u(t) = P (u 0 ), thus the gauge transformation G 2 is invertible and u solves (1.1) if and only if G 2 • G 1 (u) solves the second renormalized mKdV equation (mKdV2): The effect of the gauge transformation G 2 is to remove certain resonant frequency interactions in the nonlinearity which are responsible for the ill-posedness result in Theorem 1.1.This supports our choice of gauge transformation G 2 .Hence, we propose mKdV2 (1.7) as the correct model to study the complex-valued mKdV equation (1.1) outside of H 1 2 (T).Indeed, in [5], we proved that, unlike mKdV1 (1.3), mKdV2 (1.7) is locally well-posed in FL s,p (T) for s ≥ 1  2 and 1 ≤ p < 4. Our goal in this paper is to improve this well-posedness result without exploiting the complete integrability of the equation.
Remark 1.3.(i) The solutions constructed in Theorem 1.2 satisfy the Duhamel formulation: where S(t) denotes the linear propagator, N (u, u, u) corresponds to the right-hand side of (1.7), for some T > 0 depending only on u 0 F L 1 2 ,p (T).See Section 2 for further details.(ii) Theorem 1.2 is sharp with respect to the method, since local uniform continuity of the data-to-solution map is known to fail in FL s,p (T) for any s < 1  2 and 1 ≤ p < ∞ [5].Without imposing uniform continuous dependence on the initial data, we expect it to be possible to lower s in Theorem 1.2.In a forthcoming work, we intend to pursue the question of local well-posedness of mKdV2 (1.7) in FL s,p (T) for s < 1  2 by combining the method introduced by Deng-Nahmod-Yue [9] and the energy method in [28] and [27].(iii) To show Theorem 1.2, we apply the method introduced by Deng-Nahmod-Yue [9], which is based on constructing solutions u with a particular structure.As a consequence, uniqueness holds conditionally in a sub-manifold of X 1 2 , 1 2 p,2− (see Definition 2.1) determined by the structure imposed on u.In Sobolev spaces, unconditional uniqueness holds in H s (T) for s ≥ 1  3 (see [24,27]).It would be of interest to consider the problem of unconditional uniqueness of mKdV2 (1.7) in the Fourier-Lebesgue spaces.
Using the a priori bounds established by Oh-Wang [31], which generalize the result by Killip-Vişan-Zhang [23] to the Fourier-Lebesgue setting, we extend the solutions in Theorem 1.2 globally-in-time.
For real-valued solutions u, the momentum P (u) ≡ 0 which implies that G 2 (u) ≡ u.Consequently, the previous results on the complex-valued mKdV2 equation (1.3) in Theorems 1.2 and 1.4 also apply to the real-valued setting.
Corollary 1.5.The real-valued mKdV1 equation (1.3) is globally well-posed in FL s,p (T) for s ≥ 1  2 and 1 ≤ p < ∞.Remark 1.6.Corollary 1.5 (restricted to s ≥ 1  2 ) extends the result by Kappeler-Molnar to the defocusing case and also to the large data setting.Furthermore, our solutions satisfy the Duhamel formulation.
The real-valued mKdV equation (1.2) has garnered more attention than its complexvalued counterpart in the periodic setting.For context, we review some of the known well-posedness results of the former equation.Local well-posedness in H s (T) for s ≥ 1  2 is due to Bourgain [2].The result was shown through the Fourier restriction norm method and it is sharp if one requires Lipschitz continuity of the solution map [4,6].Colliander-Keel-Staffilani-Takaoka-Tao [7] showed that these solutions can be extended globally in time, by applying the I-method.By weakening the resonant contribution in the nonlinearity, Takaoka-Tsutsumi [36] and Takaoka-Nakanishi-Tsutsumi [28] extended the local wellposedness to H s (T), s > 1  3 .The end-point result was shown by Molinet-Pilod-Vento in [27].These solutions are global-in-time due to the a priori bounds by Killip-Vişan-Zhang [23].Exploiting the complete integrability of the equation, Kappeler-Topalov [21] showed global existence and uniqueness of solutions of the real-valued defocusing mKdV equation (with '+' sign) in H s (T) for s ≥ 0. These solutions are the unique limit of smooth solutions and are not known to satisfy the equation in the distributional sense (see [21,25,32] for more details).In [26], Molinet showed that these solutions are indeed distributional solutions and proved ill-posedness of the mKdV equation below L 2 (T), in the sense of failure of continuity of the solution map.See also the work of Schippa [35].
This ill-posedness result, motivated the study of the mKdV equation outside of Sobolev spaces, in particular, in spaces with similar scaling.To clarify, we briefly recall the scaling heuristics.The mKdV equation (1.1) on the real line enjoys the following scaling symmetry which induces the scaling critical regularity s crit = − 1 2 in the sense that the Ḣ− 1 2 (R)norm is invariant under the scaling (1.8).In the subcritical regime, in H s (T) for s > − 1  2 , one might expect well-posedness of (1.1) from the scaling heuristics.However, Molinet's result shows that there is no hope of covering this regime within the Sobolev scale.An alternative approach is to study (1.1) in spaces with similar scaling, namely the Fourier-Lebesgue spaces.In [20], Kappeler-Molnar showed local well-posedness of the real-valued defocusing mKdV equation in FL s,p (T), for s ≥ 0, 1 ≤ p < ∞ (see also [29]).Given that ḞL s,p (R) scales like Ḣσ (R) for σ = s + 1 p − 1 2 , this result covers the full subcritical regime, although the solutions are not yet known to be distributional solutions.
In the euclidean setting, the well-posedness of the complex-valued mKdV equation (1.1) has been a long standing topic of interest.Kenig-Ponce-Vega [22] showed local wellposedness of (1.1) in H s (R) for s ≥ 1  4 .This result is sharp with respect to the method, due to the failure of uniform continuity of the data-to-solution map in H s (R), s < 1  4 .In the Fourier-Lebesgue setting, Grünrock-Vega [13] proved local well-posedness of (1.1) in FL s,p (R) for s > 1 2p and 1 ≤ p < ∞; see also [11].These solutions can be extended globally in time due to the a priori bounds by Oh-Wang in [31].In a recent paper [16], Harrop-Griffiths-Killip-Vişan showed optimal global well-posedness of the complex-valued mKdV equation in H s (R) for s > − 1 2 by exploiting the complete integrability of the equation, giving a definite answer to the problem of local well-posedness of mKdV (1.1) on the real line, in Sobolev spaces.
We conclude this section by stating some further remarks.
Remark 1.7.(i) The ill-posedness result in Theorem 1.1 follows an argument by Guo-Oh [15].The proof combines the local well-posedness of mKdV2 (1.7) in Theorem 1.2 and the rapid oscillation of the phase in the gauge transformation G 2 (1.6), due to the assumption on the momentum (1.5).Thus, since Theorem 1.1 follows from [5,15], we omit the details.
(ii) The scaling heuristics described above can be transported to the Fourier-Lebesgue setting.The critical regularity is given by s crit (p) = − 1 p .We can also compare the scaling of the two families of spaces to conclude that ḞL s,p (R) scales like Ḣσ (R) for σ = s − 1 2 + 1 p .From these heuristics, we see that the results in Theorem 1.2 and Corollary 1.5 are at the scale of L 2 (T).
Remark 1.8.In [5], we showed that it is possible to recover solutions of mKdV (1.1) from solutions of mKdV2 (1.7) outside of H 1 2 (T) by imposing the following notion of finite momentum at low regularity.Definition 1.9.Suppose that Then, we say that f has finite momentum and denote the limit by P (f ).
By imposing this notion of finite momentum to initial data in FL s,p (T) for s ≥ 1 2 and 1 ≤ p < 3, we showed that the corresponding solutions of mKdV2 (1.7) have finite and conserved momentum.The restriction follows from an energy estimate, which we do not know how to improve at the moment.As a consequence of conservation of momentum, we proved existence of distributional solutions of mKdV (1.1) by using the gauge transformation G 2 and a limiting argument.We expect a similar result to hold for the full range of wellposedness, s ≥ 1 2 and 1 ≤ p < ∞, if the momentum of the initial data is finite.Since the main focus of this paper is on improving the previous well-posedness result of mKdV2 (1.7), we will not discuss further how to recover solutions of the complex-valued mKdV equation (1.1) from those of mKdV2 (1.7).
1.2.Outline of the strategy.In our previous work on the complex-valued mKdV equation [5], we proved local well-posedness of mKdV2 (1.7) in FL s,p (T) for s ≥ 1 2 and 1 ≤ p < 4 by using the Fourier restriction norm method.The solutions are constructed through a contraction mapping argument on the Duhamel formulation where S(t) denotes the linear propagator, N R(u) the nonlinearity of the mKdV2 equation and D the Duhamel operator.In particular, we look for solutions in the X s,b spaces adapted to the Fourier-Lebesgue setting defined by the following norm for b = 1 2 and q = 2.The main difficulty resides in controlling the derivative in the nonlinearity.Since the Duhamel operator has smoothing in time but not space, we want to exploit the multilinear dispersion by using the modulations, i.e., the weights τ − n 3 in the norm (1.10).The need to use the highest modulation to help control the derivative imposes the restriction 1 ≤ p < 4 and we do not know how to overcome it within this framework.
In this paper, we apply the method introduced by Deng-Nahmod-Yue [9] to the mKdV2 equation (1.7) extending our previous local well-posedness result to FL s,p (T) for s ≥ 1 2 and 4 ≤ p < ∞.Instead of running a contraction mapping argument on the Duhamel formulation (1.9), we want to establish a system of equations which imposes structure on the solution u but guarantees that it still satisfies the Duhamel formulation (1.9).The structure is appropriately chosen to avoid the nonlinear estimates (1.11) that we could not prove in our previous work.In particular, we want to avoid using the modulations to gain smoothing in space until strictly necessary.
The method was introduced to improve the local well-posedness results of Grünrock-Herr [12] for the derivative nonlinear Schrödinger equation (DNLS).Here, they showed DNLS is locally well-posed in FL s,p (T) for s ≥ 1 2 and 1 ≤ p < 4, using the Fourier restriction norm method on the gauged equation introduced by Herr [18].Moreover, they proved that the main nonlinear estimate fails for p ≥ 4, preventing the improvement of the well-posedness result within this framework.The new method by Deng-Nahmod-Yue is designed to bypass the problematic cases responsible for the failure of the nonlinear estimate.To that end, motivated by the probabilistic setting, one looks for solutions which are centered around a suitable object.
The idea of looking for solutions with a particular structure is not new in the context of probabilistic PDEs (with random initial data or stochastic forcing).Centering the solution around a suitably chosen object was seen in the works of Bourgain [3], Da Prato-Debussche [8] and Gubinelli-Imkeller-Perkowski [14], for example.In these works, the randomness of the centers introduces smoothing in space on the remainder nonlinear pieces, making it possible to improve purely deterministic results.The lack of randomness in our setting leads us to considering a moving 'center' w and the solution can be thought of as parameterized by w.The equation for w is chosen so that u with the imposed structure solves the Duhamel formulation (1.9).Unlike the previously discussed works, w is not smoother in space but in time.
Choosing the correct structure for u is the main difficulty in the method by Deng-Nahmod-Yue and it is strongly motivated by the difficulties in the nonlinear estimate (1.11).We want to find an appropriate functional F (u, w) such that we can solve the following system (1.12) There are three main points in establishing and solving the system (1.12): (i) choosing the frequency regions of the nonlinear terms in F (u, w); (ii) modifying the Duhamel operator to induce smoothing in space; (iii) using second iteration to solve the equation for w.
We start by choosing the spaces for u and w.In the following discussion, we will focus on the endpoint s = 1 2 .We want to construct solutions u ∈ X  ,p (T)).In order to choose the space for w, we look at the nonlinear estimate (1.11).In certain regions of frequency space, the estimate (1.11) holds for any fixed 2 ≤ p < ∞ and some b = 1− and q = ∞− (see Remark 6.3 for more details).This motivates the choice of X as the space where w lives.Now, we focus on (i).Our first guess for F (u, w) corresponds to the cubic terms in the Duhamel formulation for which we cannot show the trilinear estimate (1.11) in any X 1 2 ,b p,q ⊂ C(R; FL 1 2 ,p (T)) space, regardless of the choice of b and q.These contributions should appear in the equation for u, in order to be estimated in a weaker norm X 1 2 , 1 2 p,2− .In particular, the contributions essentially look like the following where P N and P ≪N denote the Dirichlet projections onto the spatial frequencies {|n| ∼ N } and {|n| ≪ N }, respectively.These terms can roughly be seen as 'paracontrolled' by w (see [14] for details on paracontrolled distributions).
Unfortunately, this first ansatz for F (u, w) is not sufficient.When estimating the above contributions in X 1 2 , 1 2 p,2− , the lack of smoothing in space of the Duhamel operator D forces us to use the modulations.This leads us to (ii) and to the introduction of a modified Duhamel operator which not only has smoothing in time but also in space.The modification is introduced through a convolution in time with a smooth time cutoff parameterized by the resonance relation (see Section 3 for more details).Choosing F (u, w) as the contributions in (1.13) with D replaced by the modified Duhamel operator establishes an equation for u which can be solved through a fixed point argument, given a fixed w ∈ X 1 2 ,1− p,∞− .From the first equation, we obtain a function u parameterized by w which is not yet a solution to the mKdV2 equation.This will only follow after we have found the correct center w ∈ X 1 2 ,1− p,∞− .The crucial idea behind obtaining w is to modify the second equation in (1.12) by substituting u = u[w] by the first equation when relevant, using a partial second iteration (iii).This strategy resembles the second iteration method used by Bourgain [4], Oh [30] and Richards [33], for example.In these works, by iterating the Duhamel formulation, it becomes possible to estimate problematic regions in frequency space at the cost of increasing the multilinearity of the analysis.In our case, we have a partial second iteration, as we are exploiting the structure imposed to u, not the full Duhamel formulation.This will lead to new cubic and quintic terms for which we can establish estimates (see Sections 4 and 6).
Recall that our underlying motivation to establish the system of equations (1.12) is to avoid using the modulations until strictly necessary.On the one hand, the nonlinear terms in the equation for u have smoothing in space induced by the modified Duhamel operator, which is sufficient to control the derivative.On the other hand, in order to solve the equation for w, we can use second iteration which introduces smoother in time terms or increases the multilinearity of the contributions.In both cases, the modification allows us to show the estimates for any 2 ≤ p < ∞, even when the modulations need to be used.1.3.Outline of the paper.In Section 2 we introduce the relevant function spaces and some auxiliary results.In Section 3 we introduce the modified Duhamel operator and show relevant kernel estimates.In Section 4 we establish equation for u.Moreover, we explain how to use second iteration to obtain the equation for w, which is included in detail in Appendix A. The relevant estimates for the nonlinear terms in the equation for u are shown in Section 5. Lastly, Section 6 is dedicated to solving the equation for w.

Preliminaries
2.1.Notations and function spaces.We start by introducing some useful notations.We will use ϕ : R → R to denote a smooth time cutoff function, equal to 1 on [−1, 1] and 0 outside of [−2, 2] and ϕ T (t) := ϕ(T −1 t), for 0 ≤ T ≤ 1. Lastly, let A B denote an estimate of the form A ≤ CB for some constant C > 0. Similarly, A ∼ B will denote A B and B A, while A ≪ B will denote A ≤ εB, for some small constant 0 < ε ≪ 1.The notations a+ and a− represent a + ε and a − ε for arbitrarily small ε > 0, respectively.
Our conventions for the Fourier transform are as follows.The Fourier transform of u : R × T → C with respect to the space variable is given by The Fourier transform of u with respect to the time variable is given by The space-time Fourier transform is denoted by F = F t F x .We will also use u to denote F x u, F t u and F t,x u, but it will become clear which one it refers to from context, namely from the use of the spatial and time Fourier variables n and τ , respectively.Now, we focus on the relevant spaces of functions.Let S(R × T) denote the space of functions In [1], Bourgain introduced the X s,b -spaces defined by the norm In the following, we define the X s,b -spaces adapted to the Fourier-Lebesgue setting (see Grünrock-Herr [12]).
p,q -spaces defined above reduce to the standard X s,b -spaces defined in (2.1).
Before proceeding, we introduce the relevant spaces of functions and associated parameters.Let 0 < δ ≪ 1 a small parameter to be chosen later, depending on 2 < p < ∞.We introduce the following parameters Note that b 0 < b 1 , q 1 < q 0 and r 2 < r 1 < r 0 .We will focus on showing the result for the endpoint s = 1 2 , see Remark 4.4 for more details.Consequently, we will conduct our analysis in the following X s,b p,q spaces: Recall the following embedding: for any 1 ≤ p < ∞, In particular, it holds that Z 0 ⊂ Y 0 ⊂ C(R; FL where n 123 = (n 1 , n 2 , n 3 ) and Φ denotes the resonance relation , where the factorization holds if n = n 1 + n 2 + n 3 .Consider the following trilinear operators Consequently, we can decompose the nonlinearity (2.2) into non-resonant and resonant contributions Note that if n j is the spatial frequency corresponding to u j , j = 1, 2, 3 in (2.3) and (2.4), then |n 2 | ≥ |n 3 |.Consequently, we want to consider the following subregions: For * ∈ {A, B, C, D}, let N R * ,≥ , N R * ,> denote the operators defined in (2.3) and (2.4), respectively, with sums localized to X * (n).We can write the non-resonant contributions of the nonlinearity as and equivalently for N R > .We also introduce the following notation, which will be useful in Sections 5 and 6: The following lemma clarifies the relation between the frequencies in the subregions introduced.(1  (6), the resonance relation will not play a crucial role when estimating this contribution.
Lastly, we recall our notion of solution to (1.7).Define the linear propagator S(t) as follows Also, define the Duhamel operator, D, and its truncated version, D: Since we are only concerned with local well-posedness, let 0 < T ≤ 1 and consider the truncated Duhamel formulation: Since solving (2.7) for t ∈ R is equivalent to solving (2.6) for t ∈ [−T, T ], we will focus on the truncated equation.

Auxiliary results.
The following proposition allows us to gain a small power of the time of existence T , needed to close the contraction mapping argument.
Lemma 2.4.Suppose that F is a smooth function such that F (0) = 0.Then, we have the following estimates for any 0 < θ ≤ δ 2 and 0 < T ≤ 1. Proof.We want to estimate the following quantity Both estimates follow once we show The estimate above follows once we prove the two following estimates: Note that the first inequality is equivalent to the following result (2.9) Using Young's inequality with 1 The estimate follows from To prove (2.8), using the fact that R f (τ ) dτ = 0, we note that For the first contribution in (2.10), we distinguish between the regions {τ : |τ | T −1 } and {τ : |τ | ≫ T −1 }, and use mean value theorem to obtain for any α > 0. For the second contribution in (2.10), we have Combining the estimates for the two contributions in (2.10), we obtain by using Hölder's inequality for the last step with 1 = 1 r + 1 q .Thus, we have given that b > 1 − 1 q and b < 1. Combining the two bounds, we obtain the intended power of T .Remark 2.5.Lemma 2.4 will only be applied to functions of the form F (t) = t 0 G(t ′ ) dt ′ which satisfy the assumption F (0) = 0, namely the Duhamel operator D and the operators G, B defined in Section 3.
The following lemma estimates the number of divisors of a given natural number.The result was adapted from [9].

Splitting the Duhamel operator
In this section, we make the smoothing-in-time of the Duhamel operator explicit by estimating its kernel.The lack of smoothing in space motivates us to introduce a modified version of the Duhamel operator localized to X A (n), X B (n), in order to control the derivative from the nonlinearity.Moreover, we show kernel estimates for this modified Duhamel operator as well as the remainder operator.
where the kernel K is given by the following expression and satisfies the following estimates where α is a large enough positive number.
Proof.We start by calculating the space-time Fourier transform of DF , Using the fact that Consequently, Calculating the convolution with ϕ, we get the intended expression.It remains to show the estimate on the kernel.Consider the 2 regions of integration {µ : |µ| > 1} and {µ : |µ| ≤ 1}.Then, for the first region, using Cauchy-Schwarz inequality and Lemma 2.7, we have for any α > 0. For the second region, we distinguish two cases for τ : |τ | 1 or |τ | ≫ 1.In both cases, using mean value theorem, we obtain Combining the estimates, we get To show (3.2), note that τ τ − λ λ and τ − λ τ λ .Thus, We want to split each of the non-resonant contributions DN R * ,≥ , DN R * ,> for * ∈ {A, B} into two components: G * ≥ , B * ,≥ and G * ,> , B * ,> , respectively.The G contributions will have sufficient smoothing in space to allow us to control the derivative from the nonlinearity.This follows by introducing a convolution with a smooth function η parameterized by the resonance relation Φ(n 123 ).
Consider a Schwartz function η satisfying where H denotes the Hilbert transform, i.e., principal value convolution with the function 1 τ .We define the operators G * ,≥ , B * ,≥ through their spatial Fourier transform as follows with equivalent definitions for G * ,> , B * ,> where we impose the condition |n 2 | > |n 3 | to the sum.
In order to control the contributions that depend on G * , for * ∈ {A, B}, we want to calculate and estimate the kernel of this convolution operator.Proposition 3.2.Let * ∈ {A, B}.Then, the convolution operators G * ,≥ , G * ,> have the following space-time Fourier transform where the kernel K G is given by the following expression and satisfies the following estimates where α is a large enough positive number.
Proof.Since the relation between |n 2 | and |n 3 | will not play an important role in the proof, we will use G * to denote both G * ,≥ , G * ,> .We want to calculate the following where F (t) = ϕ(t) 3 j=1 u j (t, n j ).Denote Φ(n 123 ) by Φ, for simplicity.Proceeding as in the proof of Proposition 3.1, we have Substituting F (µ) and H F (µ), we obtain the intended expression.It remains to show the kernel estimate.First, note that for a Schwartz function f , we have Considering the first contribution, using mean value theorem and distinguishing the cases |ξ| 1 and |ξ| ≫ 1 gives for any α > 0. The second contribution is equal to zero, so it only remains to control the third one.Using Lemma 2.7, it follows that Consequently, Since ϕ is a Schwartz function, using (3.6), Now, considering the kernel and the estimates for H η, H ϕ, we have the following From the first contribution I , from Lemma 2.7 we have For the second contribution II, applying Lemma 2.7 or Cauchy-Schwarz inequality give the following estimates Consequently, II , and the first estimate follows.For (3.5), we consider different cases max( Φ , λ ) max( Φ , τ ) or max( Φ , λ ) ≪ max( Φ , τ ).Note that for the latter, max( Φ , τ ) = τ and τ − λ ∼ τ .The estimate follows by choosing α ≥ 2.
Remark 3.3.For * ∈ {A, B}, consider the operators DN R * ,≥ (u 1 , u 2 , u 3 ) and G * ,≥ (u 1 , u 2 , u 3 ), and the kernel estimates in Propositions 3.1 and 3.2.Then, where the kernel K B is given by and satisfies the following estimate for any α > 0. Using the expressions (3.1) and (3.4) for the kernels, we obtain the following For I 1 , using mean value theorem gives for any α > 0.
For the following contribution, using (3.6) and Cauchy-Schwarz inequality gives Before estimating I 3 , note that since H η(−1) = −1 and using mean value theorem, we get Using the above estimate, it follows from previous arguments that for any α > 0.
In order to estimate I 4 , we start by showing a bound for 1 Φ η µ Φ .If |Φ| |µ|, we use the fact that η(−1) = 0 and mean value theorem.Otherwise, Using the above estimate and the fact that |H ϕ(µ − λ)| µ − λ −1 in (3.6), we have For the last contribution, we have for any α > 0. The estimate follows.
We want to further split the operators B * ,≥ , B * ,> , * ∈ {A, B}, to obtain better kernel estimates.We will split the kernel K B in two pieces: when we can estimate the multiplier directly, and when we also have to use where the kernels are defined below Thus, we have the following estimates for the kernels, for any 0 ≤ α ≤ 1, Considering the contribution corresponding to the kernel K 0 in B * ,≥ , B * ,> , it can be easily estimated.However, in order to estimate the one corresponding to K + , we will have to use the largest modulation.In particular, using (3.8), we see that for any 0 ≤ α ≤ 1, since λ = τ 1 + τ 2 + τ 3 and Thus, we can use σ max in order to estimate the numerator of the second contribution in (3.8), which motivates splitting the operators depending on which modulation is the largest.
In particular, we have where has kernel K 0 and B j * , ≥ has kernel K + localized to the region where

System of equations
Instead of running a contraction mapping argument on the integral equation (2.6), we will solve an ordered system of equations.The first equation imposes structure on u centered around a smoother-in-time function w, while the second guarantees that u solves the Duhamel formulation.In order to obtain a solution of mKdV2 (1.7), we start by solving the equation for u obtaining a function parameterized by w, u = u[w].Then, we apply a partial second iteration argument, using the additional structure of u, in order to obtain an appropriate center w.
In this section, we establish the relevant equations for u and w and the main results needed to show Theorem 1.2.For a fixed p in 2 ≤ p < ∞, we will only focus on showing local well-posedness in FL 1 2 ,p (T).The result for s > 1  2 follows from a persistence of regularity argument.See Remark 4.4 for more details.
For a fixed w ∈ Z 0 , we consider the following equation for u Now, we want to establish an equation for w such that if u = u[w] satisfies (4.1), then it also satisfies the Duhamel formulation (2.7).We see that w must satisfy the following equation The equation (4.1) imposes structure on u, which we want to exploit in order to modify the above equation for w.We want to use the idea introduced by Bourgain in [4] and apply a partial second iteration by replacing u by its equation (4.1)where appropriate.The decomposition on the operators DN R and B, introduced in Sections 2 and 3, explicitly identify which entries have the largest frequencies and the largest modulations, respectively.This information will guide how we apply second iteration.
For the terms DN R * ,≥ , DN R * ,> , * ∈ {A, B, C, D}, we replace the equation for u (4.1) from left to right to obtain only cubic and quintic terms, as in the following example This strategy prioritizes the entry corresponding to the derivative followed by the one with the largest frequency between the remaining two factors.For DN R * ,≥ , DN R * ,> with * ∈ {A, B} there will be no cubic terms after second iteration, due to the differences in (4.2).
For the terms B * ,≥ , B * ,> , with * ∈ {A, B}, we split the operators into four pieces B j * ,≥ , B j * ,> , j = 0, 1, 2, 3 as defined in (3.10).The contributions corresponding to j = 0 are easily estimated, but for j = 1, 2, 3 the largest modulation plays an important role in estimating the kernel.If the j−th entry corresponds to a u or u term, we replace it with the equation for u (4.1).For example, we have Proceeding as detailed above, we obtain a new equation for w.Due to its length, we have decided to only include it in Appendix A.

Proposition 4.2. For any
Remark 4.3.In order to show Proposition 4.2, we will not run a contraction mapping argument for the map defined by the right-hand side of (4.2) nor the equation (A.1) included in Appendix A. Some quintic terms in (A.1) require the use of the equation for u (4.1) once again, introducing new quintic terms but also new septic terms.Given the considerable number of new terms that this additional step introduces, we have decided to omit them when presenting the equation for w.The strategy for obtaining the new contributions is described in Section 6.2 along with the estimates needed for both the quintic and septic terms.
Remark 4.4.In Sections 5 and 6, we will establish the estimates needed to show Propositions 4.1 and 4.2, from which Theorem 1.2 follows for s = 1 2 .In order to extend this result to s > 1  2 , note that the required estimates in follow easily from the estimates shown by associating the extra weight n s− 1 2 in the norm to the function with the largest frequency.Consequently, by a persistence of regularity argument, we obtain a unique solution u ∈ C([−T, T ]; FL s,p (T)) where In the remaining of the paper, we will establish the nonlinear estimates needed to show Propositions 4.1 and 4.2.The results follow from a contraction mapping argument in Y 0 and Z 0 , respectively.From Lemma 2.4, it suffices to estimate the terms in equations ( 4 In this section, we establish the estimates needed to show Proposition 4.1.
Denoting the inner norm by I , we can rewrite the sum as follows ).Since the following bounds hold uniformly in σ and n, for any r > 1, r 2 and we can apply Schur's test 2 to obtain Let P N j denote the projection onto n ∼ N j and let Then, using Minkowski and Hölder's inequalities, we get 3 , so we use Lemma 2.6 to count the divisors |n|, for any 0 < ε ≤ 1  3 , we conclude that there are at most O(N ε j ) choices for d j , j = 2, 3. Since n is fixed, this determines the choices of n 2 , n 3 and consequently of n 1 .If * = B, then |Φ(n 123 )| |n 2 | 3 and we can use the standard divisor counting estimate to conclude that there are at most O(N ε 2 ) choices for n 2 , n 3 .
2 Schur's test: Let X, Y be measurable spaces, K : X × Y → R a non-negative Schwartz kernel and where we use Minkowski's inequality and the fact that r 2 < 2 ≤ p in the last inequality.Choosing ε < δ, we obtain by using Hölder and Minkowski's inequalities.
Remark 5.2.The change of variables from τ j to the modulation σ j = τ j − n 3 j , j = 1, 2, 3, in (5.1) is needed to guarantee that the quantity has an explicit dependence on the resonance relation Φ(n 123 ) and that when fixing its value, Φ(n 123 ) = µ, there is no longer dependence on the variables n 1 , n 2 , n 3 .Thus, one can consider the quantity inside the norm as a convolution operator in µ, depending on τ : This trick allows us to estimate the norm in τ and introduce a restriction on the value of the resonance relation.This strategy will be used throughout the paper.

Proof of Proposition 4.2
In this section, we show the multilinear estimates needed to prove Proposition 4.2 by a contraction mapping argument.In particular, we estimate the trilinear and quintilinear operators on the right-hand side of (A.1).
The quintic terms in (A.1) arise from substituting a u entry by an G # -operator, for # ∈ {A, B}.First, note that for any choice of s, b, p, q, we will omit the contributions that depend on G # , as they can be estimated analogously.
We start by calculating the space-time Fourier transform of the quintic contributions arising from DN terms.For example, for * ∈ {A, B, C, D} and # ∈ {A, B}, we have the following estimate for some 0 < θ < 1.Similar estimates can be obtained for the contributions The main difficulty is controlling the spatial multiplier defined as follows We will refer to the frequencies in X * (n) as the first generation of frequencies and those in X # (n 0 ) as the second when referring to the quintic terms.In Subsection 6.2, we will estimate the contributions for which α(n, n 0...5 ) 1, namely where * ∈ {A, B, C, D}, # ∈ {B, C, D} and u j ∈ {u, u}, w j ∈ {u, w}, j = 1, . . ., 5. The estimate for these contributions follows once we control Q(u 1 , . . ., u 5 ) defined by its spacetime Fourier transform as follows In Subsection 6.2, we establish an estimate for (6.3) under particular assumptions on the frequencies.Not all the contributions in (6.2) will satisfy these additional assumptions, which will forces us to use the equation for u once again, introducing new septic terms for which we also establish relevant estimates.
It remains to consider the following quintic contributions where u j ∈ {u, u}, w j ∈ {w, w}, j = 1, . . ., 5. The DN C contributions are not controlled by (6.3) and thus need a more refined approach.For the B j * contributions, not only does the j-th modulation play an important role, but also the largest modulation of the new functions in G # .Thus these contributions require a more careful analysis, detailed in Subsection 6.3.6.1.Cubic terms.We start by estimating the cubic terms in (6.1).Lemma 6.1.The following estimate holds Proof.Using the kernel estimate for D in (3.2) and Young's inequality, we have . Applying Hölder's inequality gives where Proof.Let * ∈ {C, D}, then n 123 ∈ X * (n) implies that n proceed as in (5.1).Using Minkowski's and Hölder's inequalities gives , since from the standard divisor counting estimate, we have that |X µ * (n)| ε |µ| ε , for any ε > 0. Choosing ε ≤ δ and applying Schur's test with 1 + 1 q 0 = 1 p + 1 q , we obtain for δ < 1 5p .Consequently, using Hölder's and Minkowski's inequalities, it follows that . This motivates the splitting of the Duhamel operator for these contributions, since we require a convolution operator whose kernel has a negative power of the resonance relation Φ, needed to control the loss of derivative from the nonlinearity, without using the largest modulation.
(ii) Consider the estimate  2) or (3) hold, we can always use the largest frequency which is not in a pairing to control the spatial weight from the norm n 1 2 .If the contributions do not satisfy any of the above conditions, then the largest frequency that is not in a pairing corresponds to a function u and we want to use the equation for u again.This leads to one quintic term that satisfies the assumptions above and two septic terms, which are easily estimated.
To further clarify, let Q ′ (u 1 , . . ., u 5 ) denote a contribution in (6.2), u j ∈ {u, u, w, w}.Let n j correspond to the spatial Fourier variable of u j , j = 1, . . ., 5. If n 1 is the largest frequency that is not in a pairing and u 1 ∈ {w, w}, then we keep the contribution as is.Otherwise, u 1 ∈ {u, u} and we will use the equation (4.1) to replace the first entry in Q ′ .For simplicity, assume that u 1 = u, then we have By carefully examining the frequencies and pairings of the terms in (6.2) and applying the above modification, we obtain the final equation for w.We have decided to not include this full equation for w due to its length.
All the resulting quintic and septic terms arising from (6.2), can be estimated by the two following propositions.Proposition 6.6.Let Q as defined in (6.3)where the first factor has the largest spatial Fourier frequency which is not in a pairing.Then, the following estimate holds Proof.Case 1: no pairing Let P N j denote the Dirichlet projection onto n j ∼ N j , j = 1, . . ., 5. Since there is no pairing we have |n| |n 1 |, therefore using Minkowski's inequality gives Using the change of variables σ j = τ j − n 3 j , j = 1, . . ., 5, and Schur's test, we get . . ., 5. Note that we can trivially restrict µ to the following region 2 for fixed n.Thus, it follows from Hölder's inequality in µ that Now, we consider two distinct cases depending on the size of the frequencies.
Subcase 1.1: Using Cauchy's inequality with α > 0, omitting the time dependence, we have In order to estimate |B(n, n 2 , n 3 , µ)|, |B(n, n 4 , n 5 , µ)|, we use Lemma 2.6.For the first one, to count the choices of (n 1 , n 4 , n 5 ) it suffices to count the number of divisors l − n 4 , l − n 5 , Using Cauchy-Schwarz inequality, we have where Applying the previous estimate to (6.5) and Hölder's inequality give It only remains to sum in the dyadics N j : Proof of Proposition 6.9.Due to the θ loss in the largest frequency when estimating α, we will distinguish two cases: when |n 1 |

10 A. CHAPOUTO 2 . 2 .
Nonlinearity and notion of solution.The nonlinearity of (1.7) has the following spatial Fourier transform

Proposition 3 . 1 .
The truncated Duhamel operator D has the following space-time Fourier transform

Proposition 3 . 4 .
n 3 )| (λ).Thus, for the modified Duhamel operators G * ,≥ we can 'exchange' the gain in time derivatives through τ − n 3 for spatial derivatives through Φ(n 123 ) , unlike the usual Duhamel operator.Now we estimate the kernel of the remainder operators B * ,≥ , B * ,> , * ∈ {A, B}.The assumptions on η in (3.3) play an important role in establishing the following kernel estimates.Let * ∈ {A, B}.Then, the convolution operators B * ,≥ , B * ,> have the following space-time Fourier transform

. 1 )Proposition 4 . 1 .
We start by showing that the equation (4.1) is locally well-posed via a fixed point argument, obtaining u = u[w], i.e., u parameterized by w.For any w ∈ Z 0 satisfying w Z 0 ≤ A 2 , there exists a unique u ∈ Y 0 with u Y 0 ≤ A 3 satisfying (4.1), for some T = T (A 2 ) > 0. The mapping w → u[w] is Lipschitz from the A 2 -ball of Z 0 to the A 3 -ball of Y 0 .