Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we previously used to study analogous problems on $S^1$, and on $\mathbb{R}$.

The assumptions that we have on V are the following: (i) V ∈ L 1 (T 2 ), or V ∈ L 1 (R 2 ), respectively.
The Hartree equation arises naturally in the dynamics of large quantum systems. It occurs in the context of the mean-field limit of N -body dynamics when we take V to be the interaction potential [26,39]. It makes physical sense to consider this equation both in the periodic, and in the non-periodic setting.
The region of integration is either T 2 or R 2 , depending whether we are considering the periodic or the non-periodic setting. The fact that mass is conserved follows from the fact that V is real-valued. The fact that energy is conserved follows from integration by parts, by using the fact that V is even [14].
By using the two conservation laws, and by arguing as in [29], we can deduce global existence of (1) in H 1 and a priori bounds on the H 1 norm of a solution, in the non-periodic setting. By persistence of regularity, we obtain global existence in H s , for s > 1. Hence, it makes sense to analyze the behavior of u(t) H s . A similar argument holds in the periodic setting, whereas here, we need to use periodic variants of Strichartz estimates [3].
Given a real number x, we denote by x+ and x− expressions of the form x + ǫ and x − ǫ respectively, where 0 < ǫ ≪ 1. With this notation, the result that we prove for (1) on T 2 is: Theorem 1.1. (Bound for the Hartree equation on T 2 ) Let u be the global solution of (1) on T 2 . Then, there exists a function C s , continuous on H 1 (T 2 ) such that for all t ∈ R : Similarly, in the non-periodic setting one has: Theorem 1.2. (Bound for the Hartree equation on R 2 ) Let u be the global solution of (1) on R 2 . Then, there exists a function C s , continuous on H 1 (R 2 ) such that for all t ∈ R : Heuristically, we expect to get a better bound in the non-periodic setting, due to the presence of stronger dispersion.
In the non-periodic setting, let us formally take V = δ. Then, (1) becomes: The Cauchy problem (4) is also known to be globally well-posed in H s [28]. We will see that the proof of Theorem 1.2 holds when we formally take V = δ. Hence, we also deduce the following: Corollary 1.3. (Bound for the Cubic NLS on R 2 ) Let u be the global solution of (4). Then, there exists a function C s , continuous on H 1 (R 2 ) such that for all t ∈ R : (5) u(t) H s (R 2 ) ≤ C s (Φ)(1 + |t|) 4 7 s+ Φ H s (R 2 ) .
This improves the previously known bound u(t) H s (1 + |t|) 2 3 s+ Φ H s , for all s ∈ N. This bound was proved in [16]. Similarly, we can take V = δ in the periodic setting. However, in this way, we obtain the bound u(t) H s (1 + |t|) s+ Φ H s , which had been proved for all s ∈ N in [46].

1.2.
Motivation for the problem and previously known results: The growth of high Sobolev norms has a physical interpretation in the context of the Low-to-High frequency cascade. In other words, we see that u(t) H s weighs the higher frequencies more as s becomes larger, and hence its growth gives us a quantitative estimate for how much of the support of | u| 2 has transferred from the low to the high frequencies. This sort of problem also goes under the name weak turbulence [1,2,45].
By local well-posedness theory [6,14,29,44], it can be observed that there exist C, τ 0 > 0, depending only on the initial data Φ such that for all t: (6) u(t + τ 0 ) H s ≤ C u(t) H s .
For a wide class of nonlinear dispersive equations, the analogue of (7) can be improved to a polynomial bound, as long as we take s ∈ N, or if we consider sufficiently smooth initial data. This observation was first made in the work of Bourgain [4], and was continued in [42,43].
The crucial step in the mentioned works was to improve the iteration bound (6) to: (8) u(t + τ 0 ) H s ≤ u(t) H s + C u(t) 1−r H s .
As before, C, τ 0 > 0 depend only on Φ. In this bound, r ∈ (0, 1) satisfies r ∼ 1 s . One can show that (8) implies that for all t ∈ R: In [4], (8) was obtained by using the Fourier multiplier method. In [42,43], the iteration bound was obtained by using multilinear estimates in X s,b -spaces. Similar estimates were used in [36] in the study of well-posedness theory. The key was to use a multilinear estimate in an X s,b -space with negative first index. Such a bound was then used as a smoothing estimate. A slightly different approach, based on the analysis in [11], is used to obtain (8) in the context of compact Riemannian manifolds in [13], and [46].
An alternative iteration bound, based on the use of the upside-down I-method, which was used in our previous work [40,41], gave better polynomial bounds for solutions of nonlinear Schrödinger equations on S 1 and R. The main idea was to consider the operator D, related to D s such that Du L 2 was slowly varying. This is the technique which we will apply in the present paper as well.
In the case of the linear Schrödinger equation with potential on T d , better results are known. In [7], Bourgain studies the equation: (10) iu t + ∆u = V u.
The potential V is taken to be jointly smooth in x and t with uniformly bounded partial derivatives with respect to both of the variables. It is shown that solutions to (10) satisfy for all ǫ > 0 and all t ∈ R: The proof of (11) is based on separation properties of the eigenvalues of the Laplace operator on T d . of smooth solutions of the cubic defocusing nonlinear Schrödinger equation on T 2 , whose H s norm is arbitrarily small at time zero, and is arbitrarily large at some large finite time. One should note that behavior at infinity is still an open problem.
1.3. Techniques of the proof. As was mentioned in the previous section, the main idea is to define D to be an upside-down I-operator. This operator is defined as a Fourier multiplier operator. By construction, we will be able to relate u(t) H s to Du(t) L 2 , so we consider the growth of the latter quantity. Following the ideas of the construction of the standard I-operator, as defined by Colliander, Keel, Staffilani, Takaoka, and Tao [17,18,19], our goal is to show that the quantity Du(t) 2 L 2 is slowly varying. This is done by applying a Littlewood-Paley decomposition and summing an appropriate geometric series. Let us remark that a similar technique was applied in the low-regularity context in [18].
As in our previous work [40,41], we will use higher modified energies, i.e. quantities obtained from Du(t) 2 L 2 by adding an appropriate multilinear correction. In this way, we will obtain E 2 (u(t)) ∼ Du(t) 2 L 2 , which is even more slowly varying. Due to more a more complicated resonance phenomenon in two dimensions, the construction of E 2 is going to be more involved than it was in one dimension. In the periodic setting, E 2 is constructed in Subsection 3.3. In the non-periodic setting, E 2 is constructed in Subsection 4.3.
We prove Theorem 1.1 and Theorem 1.2 for initial data Φ, which we assume lies only in H s (T 2 ) and H s (R 2 ), respectively. We don't assume any further regularity on the initial data. However, in the course of the proof, we work with Φ which is smooth, in order to make our formal calculations rigorous. The fact that we can do this follows from an appropriate Approximation Lemma (Proposition 3.2 and Proposition 4.2).

Organization of the paper:
In Section 2, we give some notation, and we recall some facts from Harmonic Analysis. In Section 3, we prove Theorem 1.1. Section 4 is devoted to the proof of Theorem 1.2. In the Appendix, we prove local-in-time bounds for (1) on the torus. The techniques mentioned in the Appendix apply to prove analogous bounds for (1) on the plane.

Acknowledgements:
The author would like to thank his Advisor, Gigliola Staffilani for suggesting this problem, and for her help and encouragement. He would also like to thank Hans Christianson and Antti Knowles for several useful comments and discussions.

Notation and known facts.
In our paper, we denote by A B an estimate of the form A ≤ CB, for some C > 0. If C depends on d, we write A d B. We also write the latter condition as C = C(d).
We are taking the convention for the Fourier transform on T 2 to be: On R 2 , we define the Fourier transform by: Here n ∈ Z 2 and ξ ∈ R 2 .
On T 2 × R, we define the spacetime Fourier transform by: On R 2 × R, we define it by: Let us take the following convention for the Japanese bracket · : x := 1 + |x| 2 .
Let us recall that we are working in Sobolev Spaces H s (T 2 ) on the the torus, and H s (R 2 ) on the plane, whose norms are defined for s ∈ R by: Let us define: and An important tool in our work will also be X s,b spaces. We recall that these spaces come from the norm defined for s, b ∈ R: When there is no confusion, we write these spaces just as H s and X s,b .
In our proofs, we will frequently have to use Littlewood-Paley decompositions. Given a function u ∈ L 2 (T 2 ) and a dyadic integer N , we define by u N the function obtained from u by restricting its Fourier transform to the dyadic annulus |n| ∼ N . Hence, we have: We analogously define v N for v ∈ L 2 (R 2 ).
Having defined the spaces in which we will be working, let us recall some estimates which we will use in our analysis.
2.1. Estimates on T 2 . By Sobolev embedding on T 2 , we know that, for all 2 ≤ q < ∞, one has: From [33], we know that on T 2 : (A similar local-in-time estimate was earlier noted in [3].) By definition, one has: From Sobolev embedding, it follows that: If we take the 1 2 + in (13) to be very close to 1 2 , we can interpolate between (13) and (14) to deduce: Similarly, we can interpolate between (13) and (15) to obtain: Let c < d be real numbers, and let us denote by χ = χ(t) = χ [c,d] (t). One then has, for all s ∈ R, and for all b < 1 2 : The proof of (18) is the same as the proof of Lemma 2.1. in [40] (see also [12,20]). From the proof, we note that the implied constant is independent of c and d. We omit the details.
We can interpolate between (14) and (15) to deduce that, for M ≫ 1, one has: Furthermore, from Sobolev embedding in time, we know that: We can interpolate between (14) and (20) to obtain: x u X 0, 1

+
An additional estimate we will use is: The estimate (22) is a consequence of the following: Lemma 2.1. Suppose that Q is a ball in Z 2 of radius N , and center n 0 . Suppose that u satisfies supp u ⊆ Q. Then, one has: 1 is proved in [6] by using the Hausdorff-Young inequality and Hölder's inequality. We omit the details.
To deduce (22), we write u = N u N . By the triangle inequality and Lemma 2.1, we obtain: We can now interpolate between (13) and (22) to deduce: By using an appropriate change of summation, as in [6], we see that (24) implies: Suppose that u is as in the assumptions of Lemma 2.1, and suppose that b 1 , s 1 ∈ R satisfy 1 4 < b 1 < 1 2 +, s 1 > 1 − 2b 1 . Then, one has: Estimates on R 2 . We note that all the mentioned estimates in the periodic setting carry over to the non-periodic setting. However, there are some estimates which hold only in the non-periodic setting, which express the fact that the dispersion phenomenon is stronger on R 2 than on T 2 . Such estimates allow us to get a better bound in Theorem 1.2 than the one we obtained in Theorem 1.1.
The first modification is that, on the plane, (13) is improved to: Consequently, one can improve (16) to: On the plane, we will use the following estimate: follows from (26), the fact that u L 2 t,x = u X 0,0 , and interpolation. Furthermore, a key fact is the following result, which was first noted by Bourgain in [5]: Proposition 2.3. (Improved Strichartz Estimate) Suppose that N 1 , N 2 are dyadic integers such that N 1 ≫ N 2 , and suppose that u, v ∈ X 0, 1 2 + (R 2 × R) satisfy, for all t: supp u(t) ⊆ {|ξ| ∼ N 1 }, and supp v(t) ⊆ {|ξ| ∼ N 2 }. Then, one has: An alternative proof (in the 1D case) is given in [17].
Let us note the following corollary of Proposition 2.3.
be as in the assumptions of Proposition 2.3. Then one has: Proof. We observe that: In order to deduce this bound, we used Bernstein's inequality, and the non-periodic analogue of (20).
In our analysis, we will have to work with χ = χ [t0,t0+δ] (t), the characteristic function of the time interval [t 0 , t 0 + δ]. It is difficult to deal with χ directly, since this function is not smooth, and since its Fourier transform doesn't have a sign. Instead, we will decompose χ as a sum of two functions which are easier to deal with. This goal will be achieved by using an appropriate approximation to the identity. We will use the following decomposition, which is originally found in [17]: Given φ ∈ C ∞ 0 (R), such that: 0 ≤ φ ≤ 1, R φ(t) dt = 1 , and λ > 0, we recall that the rescaling φ λ of φ is defined by: We observe that such a rescaling preserves the L 1 norm: Having defined the rescaling, we write, for the scale N > 1: (33) χ(t) = a(t) + b(t), for a := χ * φ N −1 .
In Lemma 8.2. of [17], the authors note the following estimate: On the other hand, for any M ∈ (1, +∞), one obtains: which is by Young's inequality: If we now define: Then the previous bound on b L M t and the Littlewood-Paley inequality [25] imply: To explain the fact that C(M, χ) = C(M, Φ), we note that χ is defined as the characteristic function of an interval of size δ, and δ, in turn, depends only on Φ.
We will frequently use the following consequence of Proposition 2.3 satisfy the assumptions of Proposition 2.3. Suppose that N 1 N . Let u 1 , v 1 be given by: Then one has: The same bound holds if Proposition 2.5 follows from Proposition 2.3, Corollary 2.4, the decomposition (33), and the estimates associated to this decomposition. We omit the details of the proof. An analogous statement is proved in one dimension in [41]. The only difference is that on R 2 , the coefficient on the right-hand side of (29) is , and hence we obtain the coefficient We also must consider estimates on the product uv, when u and v are localized in dyadic annuli as before, but when we no longer assume that N 1 ≫ N 2 .
By using Hölder's inequality and (26), it follows that: We note that (31) still holds. We now interpolate between (31) and (38) to deduce: An additional form of a bilinear Strichartz Estimate that we will have to use will be the following bound, which was first observed in [22]: Then the function F defined by: obeys the bound: For the proof of Proposition 2.6, we refer the reader to the proof of Lemma 8.2. in [22].
Let us give some useful notation for multilinear expressions, which can also be found in [17,21]. Let us first consider the periodic setting. For k ≥ 2, an even integer, we define the hyperplane: As in [17], we adopt the notation: We will also sometimes write n ij for n i + n j . In the non-periodic setting, we analogously define: In this case, the measure on Γ k is induced from Lebesgue measure dξ 1 · · · dξ k−1 on (R 2 ) k−1 by pushing forward under the map: Finally, let us recall the following Calculus fact, which is often referred to as the Double Mean Value Theorem: Suppose that x, η, µ ∈ R 2 are such that: |η|, |µ| ≪ |x|. Then, one has: The proof of Proposition 2.7 follows from the standard Mean Value Theorem.
3. The Hartree equation on T 2 .

3.1.
Definition of the D-operator. As in our previous work [40,41], we want to define an upsidedown I operator. We start by defining an appropriate multiplier: Suppose N > 1 is given. Let θ : Z 2 → R be given by: We observe that: Our goal is to then estimate Du(t) L 2 , from which we can estimate u(t) H s by (45). In order to do this, we first need to have good local-in-time bounds.

3.2.
Local-in-time bounds. Let u denote the global solution to (1) on T 2 . One then has: which are continuous in energy and mass, such that for all t 0 ∈ R, there exists a globally defined function v : T 2 × R → C such that: Proposition 3.1 is similar to local-in-time bounds we had to prove in [40,41]. Since we are working in two dimensions, the proof is going to be a little different. Our proof of Proposition 3.1 is similar to the proof of Theorem 2.7. in Chapter V of [6]. For completeness, we present it in the Appendix.
As in [40], Proposition 3.1 implies the following: If Φ satisfies: and if the sequence (u (n) ) satisfies: where Φ n ∈ C ∞ (T 2 ) and Φ n H s −→ Φ, then, one has for all t: The mentioned approximation Lemma allows us to work with smooth solutions and pass to the limit in the end. Namely, we note that if we take initial data Φ n as earlier, then u (n) (t) will belong to H ∞ (T 2 ) for all t. This allows us to rigorously justify all of our calculations. Now, we want to argue by density. For this, we first need to know that energy and mass are continuous on H 1 The fact that mass is continuous on H 1 is obvious. To see that energy is continuous on H 1 , let 1 = 1 1+ + 1 M . Then, by Hölder's inequality, Young's inequality, and (12), we obtain: Continuity of energy on H 1 follows from (51).
Now, by continuity of mass,energy, and the H s norm on H s , it follows that: Suppose that we knew that Theorem 1.1 were true in the case of smooth solutions. Then, for all t ∈ R, it would follow that: The claim for u would now follow by applying the continuity properties of C and the Approximation Lemma. So, from now on, we can work with Φ ∈ C ∞ (T 2 ).
Let us choose: (65) β 0 ∼ 1 N The reason why we choose such a β 0 will become clear later. For details, see Remark 3.6.
Hence Lemma 3.3 implies: The bound from (66) allows us to deduce the equivalence of E 1 and E 2 . We have the following bound: Proposition 3.4. One has that: Here, the constant is independent of N as long as N is sufficiently large.
Proof. We estimate E 2 (u) − E 1 (u) = λ 4 (M 4 ; u). By construction, one has: |M 4 (n 1 , n 2 , n 3 , n 4 )|| u(n 1 )|| ū(n 2 )|| u(n 3 )|| ū(n 4 )| Let us dyadically localize the n j , i.e., we find N j dyadic integers such that |n j | ∼ N j . We consider the case when N 1 ≥ N 2 ≥ N 3 ≥ N 4 . The other cases are analogous. We know that the nonzero contributions occur when: Let us denote the corresponding contribution to λ 4 (M 4 ; u) by I N1,N2,N3,N4 . We use Parseval's identity and (66) to deduce that: Let us define F j : j = 1, . . . , 4 by: By Parseval's identity, one has: Furthermore, we use Sobolev embedding, and the fact that taking absolute values in the Fourier transform doesn't change Sobolev norms to deduce that this expression is: Here, we used the fact that u H 1 x 1. We now recall (68) and sum in the N j to deduce that: The claim now follows.
Let δ > 0, v be as in Proposition 3.1. For t 0 ∈ R, we are interested in estimating: The iteration bound that we will show is: For all t 0 ∈ R, one has: Arguing similarly as in [40,41], Theorem 1.1 will follow from Lemma 3.5. We recall the proof for completeness.
Proof. (of Theorem 1.1 assuming Lemma 3.5) The point is that we can iterate the following bound (obtained from Lemma 3.5): ) ∼ N 1− times without getting any exponential growth. We hence obtain obtain that for T ∼ N 1− , one has: By recalling (45), it follows that: This proves Theorem 1.1 for times t ≥ 1. The claim for times t ∈ [0, 1] follows by local wellposedness theory. The claim for negative times holds by time-reversibility.
We now have to prove Lemma 3.5.

Proof. (of Lemma 3.5)
Let us WLOG consider t 0 = 0. The general claim will follow by time translation, and the fact that all of the implied constants are uniform in time. Let v be the function constructed in Proposition 3.1, corresponding to t 0 = 0. By (59), and with notation as in this equation, we need to estimate: We now have to estimate A and B separately. Throughout our calculations, let us denote by χ = χ(t) = χ [0,δ] (t). Terms). By symmetry, we can consider WLOG the contribution when: |n 1 | ≥ |n 2 |, |n 3 |, |n 4 |, and |n 2 | ≥ |n 4 |. We note that when all |n j | ≤ N , one has: (θ(n 1 )) 2 − (θ(n 2 )) 2 + (θ(n 3 )) 2 − (θ(n 4 )) 2 = 0. Hence, we need to consider the contribution in which one has:
Let us assume that: The other cases are dealt with similarly.
Arguing similarly as in Case 1, it follows that: x Hölder inequality and argue as earlier to see that this term is: (76), and (67), it follows that: We now use (77), sum in the N j , and recall (74) to deduce that: The Lemma now follows from (73) and (78).

Further remarks on the equation.
Remark 3.6. The quantity β 0 was chosen as in (65) in order to get the same decay factor in the quantities A and B. We note that the quantity β 0 only occurred in the bound for B, whereas in the bound for A, we only used the fact that the terms corresponding to the largest two frequencies in the multiplier (θ(n 1 )) 2 − (θ(n 2 )) 2 + (θ(n 3 )) 2 − (θ(n 4 )) 2 appear with an opposite sign. As we will see, in the non-periodic setting, the quantity β 0 will occur both in the bound for A and in the bound for B. For details, see (111) and (119).
Remark 3.7. Let us observe that, when s is an integer, or when Φ is smooth, essentially the same bound as in Theorem 1.1 can be proved by using the techniques of [46]. The approach is more complicated due to the presence of the convolution potential, but the proof for the cubic NLS can be shown to work for the Hartree equation too. The reason why one uses the fact that s is an integer is because one wants to use exact formulae for the (Fractional) Leibniz Rule for D s . By using an exact Leibniz Rule, one sees that certain terms which are difficult to estimate are in fact equal to zero. We omit the details here.  [40], given α ∈ C and n ∈ Z 2 , the function: is a solution to (1) on T 2 with initial data Φ = αe i n,x . A similar construction was used in [10] to prove instability properties in Sobolev spaces of negative index.

4.1.
Definition of the D-operator. Let us now consider (1) on R 2 . The proof of Theorem 1.2 will be based on the adaptation of the previous techniques to the non-periodic setting. We start by defining an appropriate upside-down I-operator.
Let N > 1 be given. Similarly as in the periodic setting, we define θ : R 2 → R to be given by: We then extend θ to all of R 2 so that it is radial and smooth. Arguing similarly as in the 1D setting [41], it follows that, for all ξ ∈ R 2 \ {0}, one has: Then, if f : R 2 → C, we define Df by: We also observe that:

4.2.
Local-in-time bounds. Let u denote the global solution of (1) on R 2 . As in the periodic setting, our goal is to estimate Du(t) L 2 .
We start by noting: Furthermore, we have: If u satisfies: and if the sequence (u (n) ) satisfies: where Φ n ∈ C ∞ (R 2 ) and Φ n H s −→ Φ, then, one has for all t: The proofs of Propositions 4.1 and 4.2 are analogous to the proofs of Propositions 3.1 and 3.2. The main point is that all the auxiliary estimates still hold in the non-periodic setting. As before, we can assume WLOG that Φ ∈ S(R 2 ). We omit the details.

4.3.
A higher modified energy and an iteration bound. As in the periodic setting, we will apply the method of higher modified energies. We will see that we can obtain better estimates in the non-periodic setting due to the fact that we can apply the improved Strichartz estimate (Proposition 2.3), and the angular improved Strichartz estimate (Proposition 2.6).
In the non-periodic setting, we will see that we can choose a larger β 0 from which we can get a better bound. Let us choose: (98) β 0 ∼ 1 N α Here, we take α ∈ (0, 1). We determine α precisely later (see (123)). For now, we notice: (99) β 0 ≥ 1 N We observe that Lemma 4.3 and (99) imply: The bound from (100) allows us to deduce the equivalence of E 1 and E 2 . We have the following bound: Proposition 4.4. One has that: Here, the constant is independent of N as long as N is sufficiently large.
Lemma 4.5. For all t 0 ∈ R, one has: Arguing as in the case of (1) on T 2 , Theorem 1.2 will follow from Lemma 4.5.
We now prove Lemma 4.5 Proof. It suffices to consider the case when t 0 = 0. As on T 2 , we compute that E 2 (u(δ)) − E 2 (u(0)) equals: We now have to estimate A and B separately.

4.3.1.
Estimate of A (Quadrilinear Terms). By symmetry, we can consider WLOG the contribution when: |ξ 1 | ≥ |ξ 2 |, |ξ 3 |, |ξ 4 |, and |ξ 2 | ≥ |ξ 4 |. Hence, we are considering the contribution in which one has: We dyadically localize the frequencies: |ξ j | ∼ N j ; j = 1, . . . , 4. We order the N j to obtain N * j ≥ N * 2 ≥ N * 3 ≥ N * 4 . As in the periodic setting, we have: We denote the corresponding contribution to A by A N1,N2,N3,N4 . In other words: As in the periodic setting, we have: Using Parseval's identity in time, it follows that: We now consider two subcases: We observe that: Let us define F j ; j = 1, . . . , 4 by: Consequently, by Parseval's identity: We use an L 4+ t,x , L 4− t,x , L 4 t,x , L 4 t,x Hölder inequality, and argue as earlier to deduce that, in this subcase: In the last step, we used Proposition 4.1.
In this subcase, we need to consider two sub-subcases. Let γ ∈ (0, 1) be fixed. We will determine γ later. (in equation (121)) Sub-subcase 1: Let the functions F j ; j = 1, . . . , 4 be defined as in (105). We use an L 2 t,x , L 2 t,x Hölder inequality, and we argue as before to deduce that: We use Proposition 2.3 and Proposition 2.5 to deduce that this expression is: In this sub-subcase, we have to work a little bit harder. The crucial estimate will be Proposition 2.6. We suppose that (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) is a frequency configuration occurring in the integral defining A N1,N2,N3,N4 . We argue as in [22]. We note the elementary trigonometry fact that in this frequency regime, one has: ∠(ξ 1 , ξ 14 . Furthermore, one can use Lipschitz properties of the cosine function to deduce that: We now use an L 2 t,x , L 2 t,x Hölder inequality, and recall (105) to deduce that one now has: which by using Proposition 2.6 and Proposition 2.5 is: We combine (106), (107), and (109) to deduce that: We then sum in the N j , use (103), and Proposition 4.4 to deduce that: Estimate of B (Sextilinear Terms). Let us consider just the first term in B coming from the summand M 4 (ξ 123 , ξ 4 , ξ 5 , ξ 6 ) in the definition of M 6 . The other terms are bounded analogously. In other words, we want to estimate: The bounds that we will prove for B (1) will also hold for B, with different constants. We now dyadically localize ξ 123 , ξ 4 , ξ 5 , ξ 6 , i.e., we find N j ; j = 1, . . . , 4 such that: Let us define: We now order the N j to obtain: N * 1 ≥ N * 2 ≥ N * 3 ≥ N * 4 . As before, we know the following localization bound: In order to obtain a bound on the wanted term, we have to consider two cases.
Case 1: As in the periodic case, we consider the case when: The other cases are analogous.
We use Parseval's identity together with the Fractional Leibniz Rule for D, and argue as in the periodic case to deduce that it suffices to bound the quantity: τ 4 )|dξ j dτ j We must consider several subcases: Let us define the functions F j ; j = 1, . . . , 6 by: x Hölder inequality to deduce that: By Proposition 2.5, (19), (26), (17) adapted to the non-periodic setting, by the fact that taking absolute values in the spacetime Fourier transform, and since N 1 ∼ N 2 , it follows that this expression is: We use localization in frequency to deduce that this is: which by Proposition 4.1 is: x Hölder inequality, and we argue as in the periodic case to deduce that: We assume as in the periodic case that N 1 = N * 3 . Let's also suppose that N 3 = N * 1 , N 2 = N * 2 . The other contributions are bounded analogously. Arguing as in the periodic case, we have to bound: We consider two subcases: Let us estimate L N1,N2,N3,N4 . We define F j , j = 1, . . . , 4 by: x Hölder inequality, (19) adapted to the non-periodic setting, Proposition 2.5, and (28) to deduce that: For the last inequality, we used Proposition 4.1.
We argue similarly as in Subcase 2 of Case 1 to deduce that: We use (114), (115), (116), and (117) to deduce that: We sum in N j . Using (112) and (118), it follows that: Proposition 4.4, and by construction of B (1) , we deduce that: Choice of the optimal parameters. By (102), (111), and (119), it follows that: Hence, we choose: (121) γ := 1 2 One then has that: Let us now choose β 0 . We recall that by (98), one has: β 0 ∼ 1 N α , α ∈ (0, 1). In order to have − , we should take: α ≤ 3 4 . Consequently, we take: From the preceding, we may conclude that: Remark 4.6. Let us observe that Theorem 1.2 would follow immediately if we knew that the equation (1) on R 2 scattered in H s . To the best of our knowledge, this result isn't available, and it can't be deduced from currently known techniques used to prove scattering. Some scattering results for the Hartree equation were previously studied in [30,31,32]. In [30,31], the asymptotic completeness step was proved by using techniques from [38], which work in dimensions n ≥ 3. In [32], the one and two-dimensional equations are studied. In this case, scattering results are deduced for a subset of solutions with initial data which belongs to a Gevrey class.
Further scattering results for the Hartree equation are noted in [27,34]. In these papers, one assumes that the initial data lies in an appropriate weighted Sobolev space. The implied bounds depend on the corresponding weighted Sobolev norms of the initial data. Hence, uniform bounds on appropriate Sobolev norms of solutions whose initial data doesn't lie in the weighted Sobolev spaces can't be deduced by density. Finally, the techniques used to prove [37] and similar results are restricted to dimensions n ≥ 5.
Following [6], we write: Here, Q α are balls of radius 2 k2 . We can choose this cover so that each element of D k1 lies in a fixed finite number of Q α . This number is independent of k 1 and k 2 .
time, it follows that v ∈ L ∞ t H s x . Consequently, there exist A, B > 0 such that, for all t ∈ R, one has: We observe: We take L 2 norms in x and use Minkowski's inequality to deduce: x dt ′ In order to bound the integral, we need the two following bounds, which follow from Hölder's inequality, Young's inequality, and Sobolev embedding 5 .
Arguing as in [40], we note that all the implied constants depend on s, energy, and mass, and that they are continuous in energy and mass. This proves Proposition 3.1.