Low regularity well-posedness for KP-I equations: the dispersion-generalized case

We prove new well-posedness results for dispersion-generalized Kadomtsev--Petviashvili I equations in $\mathbb{R}^2$, which family links the classical KP-I equation with the fifth order KP-I equation. For strong enough dispersion, we show global well-posedness in $L^2(\mathbb{R}^2)$. To this end, we combine resonance and transversality considerations with Strichartz estimates and a nonlinear Loomis--Whitney inequality. Moreover, we prove that for small dispersion, the equations cannot be solved via Picard iteration. In this case, we use an additional frequency dependent time localization.


Introduction and main results
We consider the Cauchy problem for the fractional Kadomtsev-Petviashvili I (fKP-I) equation ( 1) where 2 < α < 4, and the operator D α x is given by (D α x f ) ∧ (ξ) = |ξ| α f (ξ).For 2 < α 5  2 , we only consider real-valued solutions; for α > 5 2 we also treat complexvalued solutions.Note that the solution stays real-valued provided that the initial data is real-valued.In this paper, we consider initial data from anisotropic Sobolev spaces H s1,s2 (R 2 ), which are defined by The following quantities are conserved for real-valued solutions: Hence, the natural energy space is given by We prefer to study the solutions in the scale of anisotropic Sobolev spaces.We believe that adapting the present analysis will yield global well-posedness in the energy space, which is a smaller space, as well.Here we focus on the much larger anisotropic Sobolev spaces.For further remarks on the connection between Sobolev spaces and the energy spaces, we refer to [22].Moreover, if u solves the problem (1) with initial data φ, then u λ given by u λ (t, x, y) = λ −α u(λ −(α+1) t, λ −1 x, λ − α+2 2 y) also solves the same with scaled initial data (4) φ λ = λ −α φ(λ −1 x, λ − α+2 2 y).
By local well-posedness, we refer to existence, uniqueness, and continuity of the data-to-solution mapping locally in time.
The range of dispersion considered in this paper starts with the classical KP-I equation ( 6) which has been extensively studied (see [15,9,6] and references therein).Ionescu-Kenig-Tataru [9] proved global well-posedness in the energy space, and Z. Guo et al. [6] showed improved local well-posedness in the anisotropic Sobolev space H 1,0 (R 2 ).The derivative loss in case of unfavorable resonance makes the equation quasilinear.This means it is not amenable to Picard iteration in standard Sobolev spaces as observed by Molinet-Saut-Tzvetkov [15].In the works [9,6], short-time Fourier restriction was used to overcome the derivative loss in the nonlinearity.We refer to the PhD thesis of the second author for an overview of short-time Fourier restriction [20].Since short-time Fourier restriction also involves energy estimates, the results in [9,6] require real-valued solutions.Likewise, the results we prove for small dispersion require real-valued initial data: Theorem 1.1.Let 2 < α 5 2 .Then, (1) is locally well-posed in H s,0 (R 2 ) for s > 5 − 2α and real-valued initial data.
We give a technically more detailed version of the above theorem in Section 5.
However, the data-to-solution mapping constructed in the proof of Theorem 1.1 is not analytic.Indeed, we show that for α < 7  3 , the data-to-solution mapping cannot be of class C 2 .Previously, Molinet-Saut-Tzvetkov [15] showed that the data-to-solution mapping cannot be C 2 for the KP-I equation (see also [12]).This result was generalized by Linares-Pilod-Saut [14] for α < 2. It turns out that the argument extends to α < 7  3 : Theorem 1.2.Let α < 7  3 , (s 1 , s 2 ) ∈ R 2 .Then, there exists no T > 0 such that there is a function space X T ֒→ C([−T, T ]; H s1,s2 (R 2 )), in which (1) admits a unique local solution such that the flow-map for (1) given by The problematic nonlinear interaction is a resonant High × Low-interaction in which a free solution with high x frequencies interacts with a solution at low x frequencies.With the dispersion relation for the fractional KP-I equation given by (7) ω α (ξ, η) we find the resonance function to be Due to opposite signs of the terms (η1ξ2−η2ξ1) 2 ξ1ξ2(ξ1+ξ2) , the resonance function can become much smaller than the first term, which we refer to as resonant case.However, we shall see that in the resonant case, we can argue that the interaction between the two nonlinear waves and the dual factor with low modulation is strongly transverse, which we quantify via a nonlinear Loomis-Whitney inequality.This transversality was already observed in [9], while in the proof in [9] this is not related to nonlinear Loomis-Whitney.We believe that pointing out the connection with nonlinear Loomis-Whitney inequalities makes the proof more systematic.Nonlinear Loomis-Whitney inequalities were first investigated by Bennett-Carbery-Wright [3] and quantitative versions suitable for application to PDEs were proved by Bejenaru-Herr-Tataru [1,2].These were all local though.We use a global version to simplify the argument, which is a result of Kinoshita and the second author [11].We also refer to references in [11] for further discussion of nonlinear Loomis-Whitney inequalities.
The crucial ingredient in the resonant case of low modulation is to use the nonlinear Loomis-Whitney inequality to show a genuinely trilinear estimate, which improves on the bilinear estimate.Let f i ∈ L 2 (R 3 ; R + ) denote functions dyadically localized in spatial frequency in the x direction around N i ∈ 2 Z and in modula- Then, we show the estimate Clearly, for α 2 and N 3 1, this ameliorates the derivative loss.The observation is that for for some κ > 0, this estimate still suffices to overcome the derivative loss, whereas for N 3 N −κ 1 , the bilinear Strichartz estimate gains additional powers of N −1 1 .The bilinear Strichartz estimate is another consequence of transversality in case of resonance.It reads for free solutions in the resonant case with N 1 ≫ N 2 : By combining the nonlinear Loomis-Whitney inequality and the bilinear Strichartz estimate, we note that the fractional KP-I equations are semilinearly wellposed for α > 5  2 .In this range we solve the equations by applying the contraction mapping principle in suitable function spaces.This suggests the choice for frequency dependent time localization obtained by interpolating between (α, T (N )) = (2, N −1 ) and (α, T (N )) = 5 2 +, 1 , which suggests T (N ) = N −(2α−5)−ε .We shall choose ε = ε(α).
Theorem 1.3.Let 5 2 < α < 4.Then, (1) is analytically locally well-posed in H s,0 (R 2 ) for s > 5  4 − α 2 .The analyticity of the data-to-solution mapping is a consequence of applying the contraction mapping principle and the analyticity of the nonlinearity.By conservation of mass and persistence of regularity, we have the following: Corollary 1.4.Let s 0, and 5 2 < α < 4.Then, (1) is globally well-posed in H s,0 (R 2 ) for real-valued initial data.
We remark that it was well-known that the fifth order KP-I equation ( 8) can be solved via Picard iteration as pointed out by Saut-Tzvetkov [18,19].Their result was improved by B. Guo et al. [5] using short-time Fourier restriction and Yan et al. [23] (see also [13] for an earlier result) recovered the same local wellposedness result without using frequency dependent time localization.
In the limiting cases of α presently considered, we recover the currently best local well-posedness results in anisotropic Sobolev spaces.For α ↓ 2 we recover the result from [6] and for α ↑ 4 we arrive at the result from [23].We note that there is still a mismatch between the range of dispersion, for which we can show failure of Picard iteration and for which we actually use frequency-dependent time localization.It is unclear whether one has to improve the counterexample or the argument to show semilinear local well-posedness.
Moreover, in the companion paper [8], we consider the dispersion-generalized KP-I equation (2 < α < 4) in three dimensions in non-periodic, periodic, and mixed settings.1.0.1.Organization.In Section 2, we introduce the notation and function spaces.For the proof of Theorem 1.1, we use short-time Fourier restriction spaces introduced by Ionescu-Kenig-Tataru [9] and for the proof of Theorem 1.3, we use standard Fourier restriction spaces (cf.[4]).We also recall linear Strichartz estimates.In Section 3, we show that the data-to-solution mapping fails to be C 2 for α < 7/3 as stated in Theorem 1.2.In Section 4, we quantify the transversality in case of resonant interaction.This allows for the proof of bilinear Strichartz estimates and a trilinear estimate based on the nonlinear Loomis-Whitney inequality.In Section 5, we prove Theorem 1.1 by showing short-time nonlinear estimates and energy estimates in short-time function spaces.In Section 6, we show Theorem 1.3.In the Appendix, we provide details of the proof of the trilinear estimate as a consequence of the nonlinear Loomis-Whitney inequality.

Notation and function spaces
We use a± to denote a ± ǫ for ǫ > 0 sufficiently small.Also, we use notation A B for A CB with C a harmless constant, which is allowed to change from line to line.Dyadic numbers are denoted by capital letters N, L, . . .∈ 2 Z .2.1.Fourier transform.Spatial variables are denoted by (x, y) ∈ R 2 , and the time variable by t ∈ R. The corresponding Fourier variables are denoted by (ξ, η) ∈ R 2 and τ , respectively.We use the following convention for the space-time Fourier transform: e −i(tτ +xξ+yη) u(t, x, y)dtdxdy.
We shall also use notation û = F x,y u for the purely spatial Fourier transform, which should be clear from context.The Fourier transform is inverted by 2.2.Function spaces.We introduce the short-time X s,b spaces now and state their properties.The proofs of the forthcoming results can be found in [9], and we refer to [20,Section 2.5] for an overview of the properties. Let Let N 0 := N ∪ {0}.We define Littlewood-Paley projections: For f ∈ S ′ (R d ) and For N ∈ 2 N , let with the obvious modification for A 1 .Moreover, for N ∈ 2 Z , we let Additionally, for N ∈ 2 N0 , L ∈ 2 N , we define with the obvious modification for L = 1.
In the following we write for notational convenience, in order to distinguish modulation and spatial frequencies, η L (τ ) = φ L (τ ) for L ∈ 2 N0 , and η L (τ ) = and we record the estimate We find for Schwartz functions For α ∈ (2, 5/2] and dyadic frequency N ∈ 2 N0 , we choose the time localization as The dependence on α is suppressed.We place the solution into these short-time function spaces after dyadic frequency localization.For the nonlinearity, we consider correspondingly We localize the spaces in time by the usual means: For T ∈ (0, 1], let . We assemble the spaces F s,0 (T ), N s,0 (T ), and E s,0 (T ) via Littlewood-Paley decomposition: We state the multiplier properties of admissible time-multiplication.For N ∈ 2 N0 , we define the set S N of N -acceptable time multiplication factors: We have, for any s 0 and T ∈ (0, 1] (10) Next, recall the embedding F s,0 (T ) ֒→ C([−T, T ]; H s,0 ) and the linear energy estimate for short-time X s,b spaces.The following statements were proved for the KP-I equation in [9] with the proofs carrying over to the present setting.11) sup ) and Then, the following estimate holds: (12) u F s,0 (T ) u E s,0 (T ) + f N s,0 (T ) .

Linear Strichartz estimates.
We define the linear propagator U α (t) as a Fourier multiplier acting on functions φ ∈ S(R 2 ) whose Fourier transform is supported away from the origin Since U α (t) is a linear isometric mapping on H s1,s2 , the above extends by density.We state the linear Strichartz estimates.These enable us to handle the non-resonant interactions.Furthermore, we observe the smoothing effect pertaining to the higher dispersion for α > 2. The following Strichartz estimates are due to Hadac [7] for dispersion-generalized KP-II equations, but it is easy to see that the argument transfers to KP-I equations, as pointed out for α = 2 by Saut [17].
Then, we have We record a second linear Strichartz estimate for low x frequencies whose proof is simpler: Lemma 2.4 (Strichartz estimates for low frequencies).Let N, K ∈ 2 Z , I ⊆ R be an interval of length |I| ∼ K, and |ξ| ∼ N for any ξ ∈ I. Suppose that û0 (ξ, η) = 0, if ξ / ∈ I.Then, the following estimate holds: Proof.We use Bernstein's inequality in x, Plancherel's theorem, and Minkowski's inequality to find Hence, it suffices to prove By a change of variables supposing ξ > 0 without loss of generality and Hölder in time, we find e i(yη+tη 2 ) û0 (ξ, η)dη e i(yη+tη 2 ) û0 (ξ, η)dη The ultimate estimate is an application of the L 8 t L 4 y -Strichartz estimate for the one-dimensional Schrödinger equation (cf.[21,Section 2.3]).
As a consequence of the transfer principle (cf.[21, Lemma 2.9]), we have the following:

C 2 ill-posedness
In this section, we prove that (1) cannot be solved via Picard iteration for α close to 2 as stated in Theorem 1.2.This is a consequence of the derivative nonlinearity in case of resonance.
Recall that the resonance function is given by This will quantify the time oscillation in the Duhamel integral.To estimate the size of the resonance function, we separate Ω α as In the following we denote H s = H s1,s2 .
Proof of Theorem 1.2.We define the functions φ 1 and φ 2 via their Fourier transform φ1 (ξ where D i = Di ∪ (− Di ) and Di are defined as follows: Here N, γ > 0 are real numbers such that N ≫ 1, γ ≪ 1 and will be chosen later.A simple computation gives as N → ∞.We show the above for the contribution, which comes from the interaction of a high with a low frequency.This is denoted by u 2 below.Here we are using that the Fourier support is disjoint from the Fourier support of other possible interactions like low-low-or high-high-interaction.For more details, we refer to the proof of [14,Theorem 3.2] for fractional KP-I equations with weaker dispersion (see also [15] for KP-I).We can write the Fourier transform of We estimate the size of the resonance function as follows.
Proof.We carry out a case-by-case analysis: This is the same form as obtained in case (i).Hence, we can conclude the same for this case.
Remark 3.2.The above argument can be used to determine the size of the resonance function in other cases.

Resonance, transversality, and the nonlinear Loomis-Whitney inequality
In this section, we analyze the resonance function and use it to obtain trilinear estimates via the nonlinear Loomis-Whitney inequality.Moreover, we employ transversality in the resonant case to obtain genuinely bilinear estimates.We recall that the dispersion relation for the fKP-I equation is given by and the resonance function is given by We say that we are in the resonant case, if Suppose that we have then from the computation done in Lemma 3.1, we get that the right-hand side in the above equation has size N α max N min .We find in the resonant case This can be further simplified to max N min .We consider the gradient of the dispersion relation next: Using (20), we have The above relation shall be employed to obtain precise multilinear estimates via the nonlinear Loomis-Whitney inequality and bilinear Strichartz estimates.
4.1.Nonlinear Loomis-Whitney inequality.In this section, we state the setting and prove the trilinear estimate in the resonant case via the nonlinear Loomis-Whitney inequality from [11].We recall the assumptions on the parametrizations.

Assumption:
(2) the unit normal vector field n i on S i satisfies the Hölder condition sup σ,σ∈Si i=1 satisfies Assumption 4.1.1.Then, for ǫ > 0, we find the following estimate to hold: where the implicit constant is independent of β and b.
In the following, we apply Theorem 4.1 in the resonant case to obtain a trilinear estimate: Proof.Taking into account the localization of the functions, for N 2 ≪ N 1 ∼ N 3 , we write the left-hand side of ( 23) as We shall estimate the above in the resonant case, where for some c ∈ Z.To lighten the notation, we still denote the decomposed pieces by f i,Ni,Li , i = 1, 2, 3.After a harmless translation, we can suppose that these are supported in the unit neighborhood of the characteristic surface.Then it suffices to prove (23) with L i , i = 1, 2, 3 replaced by 1 because the sum over the additional decomposition is handled by the Cauchy-Schwarz inequality.We consider the characteristic surface S i , i = 1, 2 given by with surface normals (not necessarily of unit length) Hence, we are in the resonant case |Ω 1 α | ∼ |Ω 2 α | and obtain (20).
We use the anisotropic rescaling , which leaves the dispersion relation invariant and normalizes the large ξ-frequency.(20) becomes Recall that we suppose by additional decomposition that the functions have modulations of size 1.After rescaling, the functions have modulation thickness ε = N −(α+1) 1 . We obtain To apply the nonlinear Loomis-Whitney inequality, we need to find a lower bound for the determinant of the normals and check the regularity conditions Assumption 4.1.1(2).To this end, we shall carry out several almost orthogonal decompositions.It should be mentioned that these decompositions already play a crucial role in the proof of [9, Lemma 5.1 (a)].
For the decompositions (25) is the key.This allows us to decompose η ′ i /ξ ′ i into intervals of length 1, and we find that η ′ 2 /ξ ′ 2 is correspondingly decomposed into intervals of length 1.We can suppose that and then argue by perturbation.With the unnormalized normals given in (24) we observe that since |η ′ i /ξ ′ i | 1, the surface normals from (24) are already essentially normalized.We compute with details given in the Appendix: For this reason, and that we can decompose the range of (ξ ′ i , η ′ i ) almost orthogonally into balls of radius c(N 2 /N 1 ) for some c ≪ 1, we can extend the lower bound from the convolution constraint to the general case in (27).
To invoke Theorem 4.1, we still need to ensure the Hölder regularity conditions for the unit normals of the surface.With the constant in ( 22) not depending on b and β, we can carry out a very crude analysis.In the following let β = 1.Since after our reductions the surface normals (24) are essentially normalized, it suffices to verify Assumption 4.1.1(2) with normals given by (24).Let σ = (ω α (ξ, η), ξ, η) and σ = (ω α ( ξ, η), ξ, η).It follows We turn to the second size condition, which is reduced likewise: by the mean value theorem.This verifies the Hölder condition.
We continue with the estimate of ( 26).An application of the nonlinear Loomis-Whitney inequality provided by Theorem 4.1 with ε = N Taking into account the scaling factor from (26), we finish the proof.

Bilinear Strichartz estimates.
We employ transversality in the resonant case to derive bilinear estimates.We first note a trivial result.
Lemma 4.3.Let I, J be intervals and f : J → R be a smooth function.Then, Proof.The estimate is a consequence of the mean value theorem.Let x 1 , x 2 ∈ J be such that f (x 1 ), f (x 2 ) ∈ I.Then, for ξ ∈ (x 1 , x 2 ), have their Fourier supports in DN1,L1 and DN2,L2 , respectively, and that for (τ 1 , ξ 1 , η 1 ) ∈ supp(û) and (τ 2 , ξ 2 , η 2 ) ∈ supp(v), the resonance condition holds.Then, . Using Plancherel's identity and Cauchy-Schwarz inequality, we have where the set E is given by The measure of this set can be estimated by Fubini's theorem.From ( 21), Lemma 4.3 and almost orthogonality, we have Substituting this in (29), we obtain Remark 4.5.The estimate (28) remains true if we replace the functions on the left-hand side of (28) by their complex conjugates.
The next lemma allows us to handle the non-resonant case when the smallest frequency has size 1.
Proof.The proof is a generalization of the proof of [6, Lemma 3.1] to the case α > 2. We provide the details for the sake of completeness.Define Then, for i = 1, 2, 3, The left-hand side of (30) can be bounded by By using Cauchy-Schwarz inequality, it is sufficient to prove where g i : R 2 → R + are L 2 functions supported in ÃNi , i = 1, 2 and g : R 3 → R + is an L 2 function supported in [−L max , L max ] × ÃN3 .After a change of variables, and using the Cauchy-Schwarz inequality, we find that the left-hand side of (31) is dominated by We have 2 N 1 and using we have Using |ξ 2 | ∼ |ξ 2 − ξ 1 | and Fubini, we get This completes the proof.

Quasilinear well-posedness
This section is devoted to the proof of the theorem below, which yields Theorem 1.1.
Theorem 5.1.Let α ∈ (2, 5  2 ], u 0 ∈ H ∞,0 (R 2 ), and s > 5 − 2α.Then, there exists continuous T = T ( u 0 H s,0 (R 2 ) ) > 0 such that there is a unique solution Moreover, the mapping given by (32) extends uniquely to a continuous mapping ). Existence of local-in-time solutions for initial data in H 2,0 to the KP-I equation was proved by Molinet-Saut-Tzvetkov [16].The proof is a non-trivial variant of the energy method, which relies on commutator estimates.Also, persistence of regularity is discussed in [16].These arguments transpire to the fKP-I case and show the existence of a mapping S ∞ T .
5.1.Short-time bilinear estimates.In this subsection, we prove short-time bilinear estimates which we need to control the nonlinearity.
There is ε = ε(α) such that for the time localization T (N ) = N −(5−2α)−ε , such that for s ′ 0, and u, v ∈ F s ′ ,0 (T ), the following estimate holds: Remark 5.3.As a particular case of the above proposition, we obtain Proposition 5.2 will be proved by means of dyadic estimates which we prove in the following.We first consider the High × Low → High interaction.In this case, we can choose the time localization T (N ) = N −(5−2α)−ε for any ε > 0 to prove a favorable estimate.
Lemma 5.4.Let ε > 0 and the time localization be given by T Then, the following estimate holds for some c(ε) > 0: (34) Proof.Using the definition of the N N norm, we can bound the left-hand side of (34) by Using the properties ( 9) and (10), it suffices to prove that if L 1 , L 2 N (5−2α)+ε and f N1,L1 , g N2,L2 : R × R 2 → R + are functions supported in D N1,L1 and D N2,L2 and for We also note that by duality, it suffices to prove: (36) where h N,L is supported in D N,L .
Let L max = max(L 1 , L 2 , L).In case N 2 = 1, we make an additional dyadic decomposition in the low frequencies.Now we abuse notation, and let N 2 ∈ 2 Z denote the dyadic frequency.We consider two cases: , using the estimate (23), the left-hand side of (36) can be bounded by To decrease the power of L by ε 3 , we use that L N α+1 1 , which yields , using the bilinear Strichartz estimate (28), we have We obtain LHS of (35) , we interpolate with the estimate (note that the power of N 2 is positive, whereas the power of N is negative) 2 g N2,L2 L 2 for c 1 , c 2 > 0, which is acceptable for N 2 1 because the additional factor N c1(ε) 2 can be used to carry out the summation in N 2 .
• L max N α 1 N 2 : In the case N 2 1, we assume that L N α 1 N 2 (other cases give improved estimates).We use (17) as follows: 1, we assume L max = L (the other cases are similar).We use the estimate (30): which is sufficient to obtain (34) after summing up.
Next, we consider the High × High → Low interaction.In this case we have to increase time localization to match the localization of the input frequencies.This will give a constraint on ε because the larger ε becomes, the more we lose when adding time localization.
• L max N α+1 1 : For L N α+1 1 , using the L 4 Strichartz estimate and the size of L, we have LHS of (42 For N , we find the above estimate up to N δ 1 by two L 4 Strichartz estimates involving the dual function and a logarithmic summation loss.
Finally, we consider the very low frequency case: Lemma 5.7.Let ε > 0 and the time localization given by Proof.This estimate is a direct application of (17).Using the definitions of the function spaces, it is sufficient to prove that for L 1 , L 2 1 and f N1,L1 , g N2,L2 : R × R 2 → R + , supported in DN1,L1 , DN2,L2 , respectively, we have Using ( 17), we have LHS of (44) N L 2 g N2,L2 L 2 , which is sufficient.
Proof of Proposition 5.2.Given α ∈ (2, 5  2 ], we choose ε = ε(α) such that the estimate from Lemma 5.5 is valid.Note that the High×High → Low interaction is the only interaction, which imposes a constraint on time localization.We decompose the nonlinearity ∂ x (uv) as follows: Of the first two summands above, it is sufficient to consider the first by making the assumption that the derivative hits the high frequency.Each of the terms can be then separately handled by Lemma 5.4, Lemma 5.5, Lemma 5.6, and Lemma 5.7, respectively.We multiply each of the estimates in the lemmata by N 2s ′ and sum up dyadically over the spatial frequencies to obtain the required estimates.5.2.Energy estimates.We prove the energy estimates for the solution and the difference of the solutions in this section.The former is crucial to conclude an a priori estimate for the solution while the latter is required to prove the continuity of the data-to-solution map.
To begin, we assume that T ∈ (0, 1], ).Since in the estimates below, we can spare a small power of L max and gain a factor N 0+ = |A|, we can also handle the contribution of A c .We shall focus on n ∈ A in the following.

5.2.1.
Energy estimate for the solution.In this section we shall prove energy estimates for solutions to (1) for some s ′ s 0 with s = s(α).If α is large enough, we can reach s ′ = 0. Also, the time localization will depend on α.
• 24 11 < α After proving suitable bounds for the last term, (49) follows from multiplying (50) with N 2s ′ and summation in N .We consider the integrand: Using the notation from ( 45) and (46), we define and consider following cases: Strichartz estimate (17), we obtain Note that for ε < 21α 8 − 21 4 the exponent of N 1 is negative.For N 1, we have easy summation in N for any s ′ s 0. In the following let N 1.For 24 11 < α < 4 and ε according to the assumptions, we have Since 6 − 11α 4 + ε < 0, we have easy summation for s ′ s 0 and N 1.For 2 < α < 24  11 , we estimate with easy summation.
Case (ii) can be handled in a similar way as case (i) as the derivative hits the low frequency.For case (iii), we use a commutator argument, see [9,Lemma 6.1] and [10,Remark 5.9] to transfer the derivative to the low frequency.We can then use the same argument as in case (i) to obtain the required estimate.Case (iv) can be handled similarly.
Note that in this case the small frequency N 1 can have size 1.
(63) N N (5−2α)+ε N 1 4 This suffices if N 1 1.If N 1 1, we can interpolate with the prior estimate to find with straight-forward summation The other assumptions, namely L max = L 1 or L max = L 2 lead to the same conclusion.
The proof of (55) follows by substituting the obtained estimates in (57) and carrying out a summation in the x frequencies.For (56), we multiply the same by N 2s and sum up.Noting that u 1 = v + u 2 leads to (56).
5.3.Proof of Theorem 5.1.We conclude the proof of Theorem 1.1 in this section.
In the first step, we show a priori estimates.
Since s = 0 for α > 24 11 , by the above a priori estimates and the conservation of mass (2), we can show global existence of solutions.
Theorem 5.11 (Global existence for smooth solutions).Let α ∈ ( 24 11 , 5 2 ], and u 0 ∈ H ∞,0 (R 2 ).For any s 0 we have a solution u ∈ C(R; H s,0 ) to (1), and there exist Proof.It is enough to consider t > 0 by time-reversal.Firstly, we can rescale the initial data u 0 to u 0λ , which satisfies u 0λ L 2 = ε ≪ 1.By the local well-posedness result in H 2,0 due to Molinet-Saut-Tzvetkov [16] we have for the corresponding solution u λ ∈ C([0, T ], H s,0 ) with T = T ( u 0λ H 2,0 ).Let T ′ ≤ 1 ∧ T .We have the following set of estimates: Secondly, we have Consequently, u λ (t) H 2,0 remains bounded for t ≤ 1 ∧ T and by the local wellposedness result u λ exists until t = 1: We have u λ ∈ C([0, 1], H s,0 ) with u λ (1) H s,0 u 0λ H s,0 .However, u 0λ L 2 = u 0λ L 2 = ε ≪ 1.For this reason, the argument can be iterated and we find u λ ∈ C(R; H s,0 ) with u λ (t) H s,0 ≤ C 1 e C2t u 0λ H s,0 , which follows from iterating (68).Hence, u(t) H s,0 ≤ C 1 (λ)e C2(λ)t u 0 H s,0 with λ = λ( u 0 L 2 ).The proof is complete.Now we prove the continuity of the data-to-solution map.In the first step, we show Lipschitz continuous dependence of the solutions in L 2 for small initial data of higher regularity.Lipschitz continuous dependence in L 2 : Let s > 5 − 2α and u 1 , u 2 denote two localin-time solutions with initial data u i (0) H s,0 ε 0 .By the above argument, we have for s ′ s ).This enables us to conclude since u i F s,0 (1) ε 0 are chosen sufficiently small.
Continuity of the data-to-solution mapping: Also, from Lemma 2.2, Proposition 5.2, and Proposition 5.10, we have From the above set of estimates, we can conclude a priori estimates for v F s,0 (T ) : We use the smallness of u i H s,0 to absorb the term from the nonlinear estimate into the left-hand side.
For s > 5 − 2α, let φ ∈ H s,0 be fixed and By rescaling and subcriticality, we can again assume that φ H s,0 ε 0 ≪ 1 and φ n H s,0 2ε 0 ≪ 1 for all n ∈ N. Let u 1 be the solution corresponding to initial data φ n , and u 2 be the solution corresponding to initial data P N φ n .We construct the data-to-solution mapping as an extension of the data-to-solution mapping for smooth initial data.Let ) denote the solution corresponding to smooth initial data.We can take the existence time as 1 by the a priori estimates and persistence property argued above.
To prove the continuity of the data-to-solution map, we need to show that the sequence S ) is a Cauchy sequence in the space C([−1, 1]; H s,0 ), s > 5 − 2α.Hence, it suffices to show that for any δ > 0, there exists M δ ∈ N such that The third term can be handled by using the continuity of the data-to-solution map for smooth data in H 2,0 : We observe that v is the solution corresponding to initial data P >K φ n .From (71), we have Combining the above with (73), we conclude an a priori estimate for v which now depends on the profile of the initial data, namely on P >K φ n .We have ];H s,0 ) P >K φ n H s,0 + P >K φ m H s,0 + C(m, n, K).
By the convergence of φ n and choosing K large enough so that P >K φ n H s,0 + P >K φ m H s,0 < ε, we conclude that {S ∞ T (φ n )} n∈N is a Cauchy sequence in C([−1, 1]; H s,0 ).This shows that S ∞ T extends to a continuous map S T : H s,0 → C([−1, 1]; H s,0 ).

Semilinear well-posedness
For α > 5 2 , we observe via estimates ( 23) and (28) that we can remedy the derivative loss completely without having to use frequency-dependent time localization.We show local well-posedness through a fixed point argument.This we carry out within the standard Fourier restriction spaces as our auxiliary spaces.Let s, b ∈ R and ω α (ξ, η) = |ξ| α ξ + η 2 ξ .The space X s,b corresponding to the fractional KP-I equation ( 1) is defined as the closure of Schwartz functions with respect to the norm With the function spaces introduced, we give a precise version of Theorem 1.3.
The section is devoted to the proof of Theorem 6.1.We begin with a reminder on the basic properties of X s,b spaces, which show that for the proof of the theorem, it suffices to show the bilinear estimate for some b > 1/2.The bilinear estimate is proved in Subsection 6.2.
6.1.Properties of X s,b spaces.Proofs of the following basic properties can be found in [21, Section 2.5].First, recall that free solutions are in X s,b locally in time.Recall the linear propagator of (1) from ( 13).
Lemma 6.2 (cf.[21, Lemma 2.8]).Let s ∈ R, u 0 ∈ H s,0 (R 2 ) and η ∈ S(R).Then, the following estimate holds: This yields the following transfer principle for b > 1/2, stating that properties of free solutions are inherited by X s,b -functions: For the frequency and modulation localization operators we use same notations like in Section 2. 6.2.Bilinear estimate.To prove Theorem 1.3 via the fixed point theorem, we require to control the nonlinearity in the X s,b−1 norm which we do in the following.We prove the estimate in a fixed time interval [0, 1] so that we do not have to keep track of additional decomposition in modulation or gain of small powers in T .For brevity, we also omit the subscript 1 for the length of the time interval in the X s,b norms.Proposition 6.5.Let 5 2 < α < 4.Then, for s > 5 4 − α 2 , there is some b > 1 2 such that the following estimate holds: For s ≥ 0, there is some b > 1 2 such that the following holds: Proof.By duality and Plancherel's theorem, we can reduce the above to proving (79) Let N i ∈ 2 Z , L j ∈ 2 N0 .For functions f N1,L1 , g N2,L2 and h N,L supported in DN1,L1 , DN2,L2 and DN,L , respectively, we focus on dyadic estimates (80) This gives From (85), we have that the first factor is For the second factor observe that sgn((α + 1)(|ξ ).
Moreover, by the resonance condition it holds that The first factor is estimated by the mean-value theorem: Hence, the size of this determinant becomes in the resonant case.
where U i denote open and convex sets in R 2 and G i ∈ O(3) such that (1) the oriented surfaces S i are given by
By this symmetry we can suppose that the intervals in which η ′ i /ξ ′ i are supported, are essentially centered at the origin.Since |η ′ 2 | N 2 /N 1 , we can now further decompose η ′ i into intervals of size N 2 /N 1 by almost orthogonality.Moreover, ξ ′ i can be decomposed into intervals of length N 2 /N 1 because |ξ ′ 2 | ∼ N 2 /N 1 .Now we can establish the lower bound for the unit normals