The profile decomposition for the hyperbolic Schrödinger equation
Benjamin Dodson Jeremy L. Marzuola Benoit Pausader Daniel Spirn


35Q55; 35Q35





In this note, we prove the profile decomposition for hyperbolic Schrödinger (or mixed signature) equations on ^2 in two cases, one mass-supercritical and one mass-critical.  First, as a warm up,  we show that the profile decomposition works for the Ḣ^1/2 critical problem. Then, we give the derivation of the profile decomposition in the mass-critical case  based on an estimate of Rogers-Vargas <cit.>.


[
    Shaoming Guo
    December 30, 2023
=====================







§ INTRODUCTION


We will consider the hyperbolic (or mixed signature) Schrödinger equation on 𝐑^2, which is given by

    i ∂_t  u + ∂_x∂_y u = |u|^p u,  u(x, y,0) = u_0(x, y).

In particular, we will focus on the cases p=4 and p=2.  The case p=2
arises naturally in the study of modulation of wave trains in gravity water waves, see for instance <cit.>; it is also a natural component of the Davey-Stewartson system <cit.>.  As can be observed quickly from the nature of the dispersion relation, the linear problem

    i ∂_tu + ∂_x∂_y u = 0,  u(x,y,0) = u_0(x, y).

satisfies the same Strichartz estimates and rather similar local smoothing estimates[See <cit.> for a general treatment of smoothing estimates for dispersive equations with non-elliptic symbols.] as its elliptic counterpart, the standard Schrödinger equation. In particular, there exists a constant C<∞ such that

    ‖ e^it∂_x∂_yf‖_L^4_x,y,t≤ C‖ f‖_L^2_x,y.

Hence, large data local in time well-posedness and global existence for small data with p≥2 can be observed using standard methods that can be found in classical texts such as <cit.>.  For quasilinear problems with mixed signature, some local well-posedness results have been developed recently, see <cit.>.  Non-existence of bound states was established in <cit.> and a class of bound states that are not in L^2 were constructed in <cit.>.

Long time low regularity theory for this equation at large data remains unknown however.  Recently, an approach to global existence for sufficiently regular solutions was taken in <cit.>, but it is conjectured  that (<ref>) should have global well-posedness and scattering for all initial data in L^2. Much progress has been made recently in proving global well-posedness and scattering for various critical and supercritical dispersive equations by applying concentration compactness tools, which originated with the works of Lions <cit.>.  One major step in applying modern concentration compactness tools to dispersive equations is the profile decomposition, see <cit.>.  The idea is that given a small data global existence result, one proves that if the large data result is false then there is a critical value of norm of the initial data at which for instance a required integral fails to be finite.  Then, the profile decomposition ensures that failure occurs because of an almost periodic critical element, which may then be analyzed further and in ideal settings ruled out completely.  See <cit.> and references therein for applications of this idea in the setting of focusing and defocusing Schrödinger equations for instance.

A major breakthrough in profile decompositions arose in the works of Gérard <cit.>, Merle-Vega <cit.>, Bahouri-Gérard <cit.>, Gallagher <cit.> and Keraani <cit.>.  Those results have then been used to understand how to prove results about scattering, blow-up and global well-posedness in many settings, see <cit.> for some examples. We also mention the recent work by Fanelli-Visciglia <cit.>, where they consider profile decompositions in mass super-critical problems for a variety of operators, including (<ref>).

As can be seen in <cit.>, the profile decomposition follows from refined bilinear Strichartz estimates. Using refined Strichartz estimates from <cit.> and bilinear Strichartz estimates, Bourgain <cit.> proved concentration estimates and global well-posedness in H^3/5 + ϵ for the defocusing, cubic elliptic nonlinear Schrödinger equation in ^2. Building on this work, Merle-Vega <cit.> proved a profile decomposition for the mass-critical elliptic nonlinear Schrödinger equation in two dimensions.

For the hyperbolic NLS, Lee, Vargas and Rogers-Vargas <cit.> have provided refined linear and bilinear estimates, drawing  on results of Tao <cit.> for the elliptic Schrödinger equation. In particular <cit.> gives an improved Strichartz estimate similar to our Proposition <ref> and uses it to prove lower bounds on concentration of mass at blow-up. An improved Strichartz estimate is also the key element in our profile decomposition, following the standard machinery described in <cit.>. For completeness, we provide a proof of Proposition <ref>, which, although drawing on similar ideas as in <cit.>, outlines more explicitly the additional orthogonality of rectangles with skewed ratios.


The major issue in following the standard proof of the profile decomposition is that the mixed signature nature of (<ref>) means that an essential bilinear interaction estimate that holds in the elliptic case fails.  This is compensated for in <cit.> by making a required orthogonality assumption for the refined bilinear Strichartz to hold (see the statement in Lemma <ref> below). To overcome this difficulty, we use a double Whitney decomposition to precisely identify the right scales, which introduces many different rectangles that are controlled using the fact that functions with support on two rectangles of different aspect ratios have small bilinear interactions.  We note that while we here focus on analysis in 2 dimensions to keep the technical computations focused and directed, we expect many of the calculations to be generalizable to other dimensions as in <cit.>.  


The paper is structured as follows: in Section <ref>, we set up the problem, discuss some basic symmetries and establish some important bilinear estimates; in Section <ref> we establish the result in the mass-supercritical case using the extra compactness that comes from the Sobolev embedding; in Section <ref>, we establish the main precise Strichartz estimate in the paper and in Section <ref>, we obtain the profile decomposition for the mass-critical problem and deduce the existence of a minimal blow-up solution. Finally, we conclude with a short Appendix in which we correct an error in the published version of this article.


Acknowledgments. The first author was supported in part by
  U.S. NSF Grant DMS - 1500424  The second author was supported in
  part by U.S. NSF Grants DMS–1312874 and DMS-1352353. The third author was supported in part by
  U.S. NSF Grant DMS–1558729 and a Sloane Research fellowship.  The fourth author was supported in part by
  U.S. NSF Grant DMS-1516565.  We wish to
  thank Andrea Nahmod, Klaus Widmayer, Daniel Tataru, Nathan Totz for helpful conversations during the
  production of this work. Part of this work was initiated when some of the authors were at the Hausdorff Research Institute for Mathematics in Bonn, then progressed during visits to the Institut des Hautes Études in Paris, the Mathematical Sciences Research Institute in Berkeley and the Institute for Mathematics and Applications in Minneapolis.  The authors would like to thank these institutions for hosting subsets of them at various times in the last several years.  
  
We thank also an anonymous referee who pointed out the results of Rogers-Vargas <cit.> and its relevance for this work and Emanuel Carneiro for pointing out that error in the original appendix and sharing his manuscript on optimizers for (<ref>), <cit.>.
  



§ PROPERTIES OF (<REF>)




Observe that a solution to 


    i ∂_tu + ∂_x∂_y u = |u|^2 u,  u(x, y,t) = u_0(x, y),


has a number of symmetries:



  * Translation: for any (x_0, y_0) ∈𝐑^2,

    u(x,y,t) ↦ u(x - x_0, y - y_0,t),



  * Modulation: for any θ∈𝐑,

    u(x,y,t) ↦ e^i θ u(x,y,t).



  * Scaling: for any λ_1, λ_2 > 0,

    u(x,y,t) ↦√(λ_1λ_2) u(λ_1 x, λ_2 y,λ_1λ_2 t),
 


  * Galilean symmetry: for (ξ_1, ξ_2) ∈𝐑^2,

    u(x,y,t) ↦ e^-it ξ_1ξ_2 e^i[x ξ_1 + y ξ_2] u(x - ξ_1 t, y - ξ_2 t,t).



  * Pseudo-conformal symmetry:

    u(x,y,t) ↦e^ixy/t/itu(x/t,y/t,1/t).




These symmetries all preserve the L^2_x,y norm. The first two symmetries (<ref>)-(<ref>), as well as the scaling symmetry properly redefined, also preserve the Ḣ_h^s norm for any s ∈𝐑, where


    ‖ f‖_Ḣ_h^s^2   =‖|∂_x|^s/2|∂_y|^s/2f‖_L^2^2.




Note that this norm has similar scaling laws as the more usual Ḣ^s norm.  Other examples of anisotropic equations have appeared in for instance <cit.>. For example, for the Ḣ^1/2 - critical problem


    i ∂_tu + ∂_x∂_y u = |u|^4 u,  u(x,y,0) = u_0(x, y),


the symmetries are thus:




  * Translation: for any (x_0, y_0) ∈𝐑^2,


    u(x,y,t) ↦ u(x - x_0, y - y_0,t),



  * Scaling: for any λ_1, λ_2 > 0,


    u(x,y,t) ↦ (λ_1λ_2)^1/4 u(λ_1 x, λ_2 y,λ_1λ_2 t),



  * Modulation: for any θ∈𝐑,


    u(x,y,t) ↦ e^i θ u(x,y,t).



We will treat the profile decomposition for (<ref>) as a warm-up, before tackling the profile decomposition for the mass-critical problem (<ref>).





 §.§ Notations


Let φ be a usual smooth bump function such that φ(x)=1 when | x|≤ 1 and φ(x)=0 when | x|≥ 2. We also let

    ψ(x)=φ(x)-φ(2x).


We will often consider various projections in Fourier space. Given a rectangle R=R(c,ℓ_x,ℓ_y), centered at c=(c_x,c_y) and with sides parallel to the axis of length 2ℓ_x and 2ℓ_y, we define

    φ_R(x,y)=φ(ℓ_x^-1(x-c_x))φ(ℓ_y^-1(y-c_y)).

We define the operators

    Q_M,Nf(ξ,η)   =ψ(M^-1ξ)ψ(N^-1η)f(ξ,η),
    P_Rf(ξ,η)   =φ_R(ξ,η)f(ξ,η).

The first operator is only sensitive to the scales involved, while the second also accounts for the location in Fourier space.
We also let | R|=4ℓ_xℓ_y denote its area.



 §.§ Some Preliminary Estimates


We start with a nonisotropic version of the Sobolev embedding.


There holds that

    ‖ f‖_L^q_x,y≲‖|∂_x|^s/2|∂_y|^s/2f‖_L^p_x,y

whenever 1<p≤ q<∞, 0≤ s<1 and

    1/q=1/p-s/2





The proof, although easy, highlights the need to treat each direction independently. Using Sobolev embedding in 1d, Minkowski inequality and Sobolev again, we obtain that

    ‖ f‖_L^q_x(𝐑,L^q_y(𝐑))   ≲‖|∂_x|^s/2f‖_L^q_y(𝐑,L^p_x(𝐑))≲‖|∂_x|^s/2f‖_L^p_x(𝐑,L^q_y(𝐑))
       ≲‖|∂_y|^s/2|∂_x|^s/2f‖_L^p_x,y

which is what we wanted.


We have two basic refinements of (<ref>). Note the difference in orthogonality requirements between Lemma <ref> and Lemma <ref>.

Assume that f=P_R_1f and g=P_R_2g where R_i=R(c^i,ℓ_x,ℓ_y) and | c^1_x-c^2_x| = N≥ 4ℓ_x, and let u (resp. v) be a solution of (<ref>) with initial data f (resp. g). Then

    ‖ uv‖_L^2_x,y,t≲(ℓ_x/N)^1/2‖ f‖_L^2_x,y‖ g‖_L^2_x,y.







Assume that f=P_R_1f and g=P_R_2g where R_i=R(c^i,ℓ_x,ℓ_y), | c^1_x-c^2_x|≥ 4  ℓ_x and | c^1_y-c^2_y|≥ 4ℓ_y, and let u (resp. v) be a solution of (<ref>) with initial data f (resp. g). Then

    ‖ uv‖_L^q_x,y,t≲ (ℓ_xℓ_y)^1-2/q‖ f‖_L^2_x,y‖ g‖_L^2_x,y

whenever q>5/3.


Lemma <ref> is the main refined bilinear estimate and appears essentially in <cit.> when dealing with cubes.   The result as stated here follows by scaling rectangles to cubes. 



 We simply write that

    u^2(ξ,η,t)   =I(ξ,η,t),
    
    I(ξ,η,t)   =∬_𝐑e^-itωφ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1)  
       ×f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)dξ_1 dη_1,
    ω   =ξ_1η_1+(ξ-ξ_1)(η-η_1)

we may now change variable in the integral

    (ξ_1,η_1)↦ (ξ_1,ω),     J:=∂(ξ_1,ω)/∂(ξ_1,η_1)=[      1      0; 2η_1-η 2ξ_1-ξ ]

and in particular, we remark that

    | J|=| (ξ-ξ_1)-ξ_1|≃ N,

so that

    I(ξ,η,t) =    ∬_𝐑e^-itωφ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1) 
       ×f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)· J^-1 dξ_1 dω,
    η_1   =η_1(ξ_1,ω;ξ,η).

Taking into consideration the Fourier transform in time and using Plancherel, followed by Cauchy-Schwarz, we find that

    ‖ I(ξ,η,·)‖_L^2_t^2=  
       ∫_𝐑|∫_𝐑φ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1)f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)· J^-1 dξ_1|^2 dω
       ≤sup_ξ,η,η_1∫_𝐑φ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1)dξ_1×
         ∫_𝐑∫_𝐑φ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1)|f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)|^2· J^-2 dξ_1 dω.

Now, we use the fact that R_1 has width ℓ_x, together with (<ref>) to obtain, after undoing the change of variables, that

    ‖ I(ξ,η,·)‖_L^2_t^2   ≲ℓ_x/N∫_𝐑∫_𝐑φ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1) 
       |f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)|^2· J^-1 dξ_1 dω
       ≲ℓ_x/N∫_𝐑∫_𝐑|f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)|^2 dξ_1 dη_1.

Integrating with respect to (ξ,η), we then obtain (<ref>).




We will in fact use the following consequence of Lemma <ref>.


Under the assumptions of Lemma <ref>, it holds that

    ‖ uv  ‖_L^40/21_x,y,t≲ (ℓ_xℓ_y)^-3/20‖f‖_L^20/11_x,y‖g‖_L^20/11_x,y.




 Indeed, using Lemma <ref>, we find that

    ‖ e^it ∂_x ∂_yf· e^it ∂_x ∂_y g‖_L^12/7_x,y,t≲ (ℓ_xℓ_y)^-1/6‖f‖_L^2_x,y‖g‖_L^2_x,y,

while a crude estimate gives that

    ‖ e^it∂_x ∂_yf· e^it∂_x ∂_yg‖_L^∞_x,y,t≲‖f‖_L^1_x,y‖g‖_L^1_x,y.

Interpolation gives (<ref>).


Another tool we will need in the profile decomposition is the following local smoothing result which is essentially equivalent to Lemma <ref>:


Let ϕ∈ L^2_x,y. There holds that

    sup_x‖ Q_M,Ne^it∂_x∂_yϕ(x,·)‖_L^2_y,t   ≲ N^-1/2‖ϕ‖_L^2_x,y,
    sup_y‖ Q_M,Ne^it∂_x∂_yϕ(·,y)‖_L^2_x,t   ≲ M^-1/2‖ϕ‖_L^2_x,y.





The proof is similar to the one in the elliptic case and follows from Plancherel after using a change of variable similar to (<ref>).  An equivalent statement with proof occurs in <cit.>.  See also <cit.> for a general statement of Local Smoothing Estimates for Dispersive Equations.




§ MASS-SUPERCRITICAL HNLS



In this section, we observe that Ḣ_h^s has similar improved Sobolev inequalities as the Ḣ^1/2 Sobolev norm. A typical example is the following:



Let f∈ C^∞_c(𝐑^2). There holds that

    ‖ f‖_L^6_x,y   ≲(sup_M,N(MN)^-1/6‖ Q_M,Nf‖_L^∞)^1/3‖ f‖_Ḣ_h^2/3^2/3≲‖ f‖_Ḣ_h^2/3,

and consequently,

    ‖ f‖_L^4_x,y≲(sup_M,N(MN)^-1/4‖ Q_M,Nf‖_L^∞)^1/6‖ f‖_Ḣ_h^1/2^5/6≲‖ f‖_Ḣ_h^1/2.





This is essentially a consequence of the following simple inequalities

    ‖ Q_M,Nf‖_L^∞_x,y   ≲ N^1/2‖ Q_M,Nf‖_L^∞_xL^2_y≲ N^1/2‖ Q_M,Nf‖_L^2_yL^∞_x
       ≲ (MN)^1/2‖ Q_M,Nf‖_L^2_x,y,

and similarly after exchanging the role of x and y.

Proof of Lemma <ref>:


Indeed, we may simply develop

    ‖ f‖_L^6_x,y^6   ≲∑_M_1,…, M_6,
     N_1,…, N_6∬_𝐑×𝐑Q_M_1,N_1f· Q_M_2,N_2f… Q_M_6,N_6f dxdy

without loss of generality, we may assume that

    M_5,M_6≲μ_2=max_2{M_1,M_2,M_3,M_4},
    
    N_5,N_6≲ν_2=max_2{N_1,N_2,N_3,N_4},

where max_2(S) denotes the second largest element of the set S, and then using Hölder's inequality and summing over M_5,M_6 and N_5,N_6, we obtain

    ‖ f‖_L^6_x,y^6 ≲(sup_M,N(MN)^-1/6‖ Q_M,Nf‖_L^∞)^2×
           ∑_M_1,…, M_4,   M_5,M_6≤μ_2
     N_1,…, N_4,   N_5,N_6≤ν_2(M_5M_6N_5N_6)^1/6∬_𝐑×𝐑| Q_M_1,N_1f|…| Q_M_4,N_4f| dxdy
       ≲(sup_M,N(MN)^-1/6‖ Q_M,Nf‖_L^∞)^2   ×
       ∑_M_1,…, M_4
     N_1,…, N_4(μ_2ν_2)^1/3∬_𝐑×𝐑| Q_M_1,N_1f|…| Q_M_4,N_4f| dxdy.

Now, using (<ref>) and estimating the norms corresponding to the two lower frequencies in each direction in L^∞, and the two highest ones in L^2, one quickly finds that

    ∑_M_1,…, M_4
     N_1,…, N_4(μ_2ν_2)^1/3∬_𝐑×𝐑| Q_M_1,N_1f|…| Q_M_4,N_4f|  dxdy≲‖ f‖_Ḣ^2/3_h^4,

which finishes the proof. Inequality (<ref>) then follows by interpolation.





At this point, the usual profile decomposition follows easily from the following simple Lemma <ref> below.

There exists δ>0 such that

    ‖ e^it∂_x ∂_yf‖_L^8_x,y,t≲(sup_M,N,t,x,y(MN)^-1/4|(e^it∂_x ∂_yQ_M,Nf)(x,y)|)^δ‖ f‖_Ḣ^1/2_h^1-δ.





We use Hölder's inequality, Sobolev embedding Lemma <ref>, Strichartz estimates and (<ref>) to get for u=e^it∂_x∂_yf

    ‖ u‖_L^8_x,y,t   ≲‖ u‖_L^6_tL^12_x,y^3/4‖ u‖_L^∞_tL^4_x,y^1/4
       ≲‖|∂_x|^1/4|∂_y|^1/4u‖_L^6_tL^3_x,y^3/4·(sup_M,N,t(MN)^-1/4‖ Q_M,Nu(t)‖_L^∞_x,y)^1/24‖ f‖_Ḣ^1/2_h^5/24
       ≲‖ f‖_Ḣ^1/2_h^23/24·(sup_M,N,t(MN)^-1/4‖ Q_M,Nu(t)‖_L^∞_x,y)^1/24.











 §.§ The mass-supercritical profile decomposition






















































Let us take the group action  on functions given by g_n^j = g(x_n^j, y_n^j, λ_1, n^j, λ_2, n^j) such that

    (g_n^j)^-1 f = (λ_n, 1^jλ_n, 2^j)^1/4  [f](λ_n, 1^j x + x_n^j, λ_n, 2^j y + y_n^j).


We can now state the Ḣ^1/2_h-profile decomposition for (<ref>).



Let u_n_Ḣ^1/2_h ≤ A be a sequence that is bounded Ḣ^1/2_h. Then possibly after passing to a subsequence, for any 1 ≤ j < ∞ there exist ϕ^j∈Ḣ^1/2_h, (t_n^j, x_n^j, y_n^j) ∈𝐑^3, λ_n, 1^j, λ_n, 2^j∈ (0, ∞) such that for any J,


    u_n = ∑_j = 1^J g_n^j e^i t_n^j∂_x∂_yϕ^j + w_n^J,



    lim_J →∞lim sup_n →∞ e^it ∂_x ∂_y   w_n^J_L_x,y,t^8 = 0,


such that for any 1 ≤ j ≤ J,


    e^-i t_n^j∂_x ∂_y (g_n^j)^-1 w_n^J⇀ 0,


weakly in Ḣ^1/2_h, 


    lim_n →∞( u_n_Ḣ^1/2_h^2 - ∑_j = 1^Jϕ^j_Ḣ^1/2_h^2 -  w_n^J_Ḣ^1/2_h^2) = 0,


and for any j ≠ k,


    lim sup_n→∞[  |ln( λ_n, 1^j/λ_n, 1^k)| + |ln( λ_n, 2^j/λ_n, 2^k)|   + |x_n^j - x_n^k |/(λ_n, 1^jλ_n, 1^k)^1/2   + |y_n^j - y_n^k |/(λ_n, 2^jλ_n, 2^k)^1/2.  
       . + |t_n^j (λ_n,1^jλ_n,2^j) - t_n^k (λ_n, 1^kλ_n, 2^k)|/(λ_n, 1^jλ_n, 2^jλ_n, 1^kλ_n, 2^k)^1/2]=∞.






The proof of Proposition <ref> of this follows by simple adaptation of the techniques in <cit.>, as originally introduced in <cit.>. We note that a similar statement also appears in works of Fanelli-Visciglia <cit.>.



§ PROFILE DECOMPOSITION FOR THE MASS-CRITICAL HLS



In this section, we focus on the mass-critical case. This case is more delicate for two reasons. First we need to account for the Galilean invariance symmetry in (<ref>) and second, we cannot use a simple Sobolev estimate as in (<ref>) to fix the frequency scales. We follow closely the work in <cit.> with a small variant in the use of modulation orthogonality and an additional argument for interactions of rectangles with skewed aspect ratios.




 §.§ A precised Strichartz inequality


The main result in this section is the following proposition from which it is not hard to obtain a good profile decomposition. We need to introduce the norm

    ‖ϕ‖_X_p:=(∑_R∈ℛ| R|^-p/20‖ϕ1_R‖_L^20/11^p)^1/p

where ℛ stands for the collection of all dyadic rectangles.    That is, rectangles with both sides parallel to an axis, of possibly different dyadic size, whose center is a multiple of the same dyadic numbers, given by the form

    ℛ   := {R_k,n,p,m : k, n, m, p ∈ℤ},
    
    R_k,n,p,m   := {  (x,y) :   n-1 ≤ 2^-kx ≤ n+1,    m-1 ≤ 2^-py ≤ m+1  }.

Note in particular that these spaces are nested: X_p⊂ X_q whenever p≤ q.  The motivation for the space L^20/11 in (<ref>) can be motivated by the X^q_p Strichartz estimate in Theorem 4.23 from <cit.>.


Let ϕ∈ C^∞_c(𝐑^2), then, there holds that for all p>2,

    ‖ϕ‖_X_p≲_p ‖ϕ‖_L^2

and in addition, there exists p>2 such that

    ‖ e^it∂_x ∂_yϕ‖_L^4_x,y,t^4≲(sup_R| R|^-1/2sup_x,y,t| e^it∂_x ∂_y(P_Rϕ)(x,y)|)^4/21‖ϕ‖_X_p^80/21.




We refer to <cit.> for a different proof of a slightly stronger estimate. Let us first recall the Whitney decomposition.



There exists a tiling of the plane minus the diagonal

    𝐑^2∖ D   =⋓ I× J,      D ={(x,x),   x∈𝐑},

made of dyadic intervals such that | I|=| J| and

    6| I|≤dist(I× J,D)≤ 24 | I|.





We will consider two independent Whitney decompositions of 𝐑×𝐑:

    1_{𝐑^2×𝐑^2∖ D}(ξ_1,η_1,ξ_2,η_2):=∑_I_1∼ I_2,   J_1∼ J_21_I_1(ξ_1)1_J_1(η_1)1_I_2(ξ_2)1_J_2(η_2),

where I_i and J_j are dyadic intervals of 𝐑 and ∼ is an equivalence relation such that, for each fixed I, there are only finitely many J's such that I∼ J, uniformly in I (i.e. equivalence classes have bounded cardinality) and if I∼ J, then | I|=| J | and dist(I,J)≃| I|. We also extend the equivalence relation to rectangles in the following fashion:

    I× J ∼ I'× J'     if and only if      I ∼ I'  and J ∼ J'.




We would like to follow the argument in <cit.> for the profile decomposition for the elliptic nonlinear Schrödinger equation. However, it is at this point where we reach the main technical obstruction to doing this. Recall that to estimate the L_x,y,t^2 norm of [e^it Δ f]^2, it was possible to utilize Plancherel's theorem, reducing the L_x,y,t^2 norm to an l^2 sum over pairs of Whitney squares.

This was because Plancherel's theorem in frequency turned the sum over all pairs of equal area squares to an l^2 sum over squares centered at different points in frequency space, and then Plancherel's theorem in time separated out pairs of squares with different area. Because there is only one square with a given area and center in space, this is enough. However, there are infinitely many rectangles with the same area and the same center. Thus, to reduce the L_x,y,t^2 norm of [e^it ∂_x∂_y f]^2 to a l^2 sum over pairs of rectangles, that is rectangles whose sides obey the equivalence relation in both x and y, it is necessary to deal with the off - diagonal terms, that is terms of the form


    [e^it ∂_x∂_y P_R_1 f][e^it ∂_x∂_y P_R_2 f][e^it ∂_x∂_y P_R_1' f] [e^it ∂_x∂_y P_R_2' f] _L_x,y,t^1,
  

where R_1∼ R_2 and R_1' ∼ R_2' are Whitney pairs of rectangles which have the same area, but very different dimensions in x and y. In this case, Lemma <ref> gives a clue with regard to how to proceed, since it leads to the generalized result that


    [e^it ∂_x∂_y P_R_1 f] [e^it  ∂_x∂_y P_R_1' f] _L_x,y,t^2≪ P_R_1 f _L^2_x,y P_R_1' f _L^2_x,y.


Thus, it may be possible to sum the off diagonal terms. We will not use Lemma <ref> specifically, but we will use the idea that rectangles of the same area but very different dimensions have very weak bilinear interactions. 

Before we turn to the details, we first present the main orthogonality properties we will use. For simplicity of notation, given a dyadic rectangle R, let

    ϕ_R(x,y):=ϕ(x,y)1_R(x,y)     and      u_R(x,y,t):=(e^it∂_x∂_yϕ_R)(x,y)

and set u=e^it∂_x∂_yϕ. Also we will consider rectangles R_1=I_1× J_1, R_2=I_2× J_2, R_1'=I_1'× J_1', R_2'=I_2' × J_2'.


Proceeding with the above philosophy in mind, using (<ref>), we have that


    ‖ u‖_L^4_x,y,t^4   =‖ u^2‖_L^2_x,y,t^2=‖∑_R_1∼ R_2u_R_1· u_R_2‖_L^2_x,y,t^2
       =‖∑_Ω∑_R_1∼ R_2,
    | R_1|=| R_2|=Ωu_R_1· u_R_2‖_L^2_x,y,t^2=‖∑_Ω I_Ω‖_L^2_x,y,t^2.



Using the polarization identity for a quadratic form,

    Q(x_1,y_1)+Q(x_2,y_2)=1/2[Q(x_1+x_2,y_1+y_2)+Q(x_1-x_2,y_1-y_2)],

 we compute that

    e^it/2ξηI_Ω   (ξ,η,t)=∑_R_1∼ R_2,
    | R_1|=| R_2|=Ω∫_𝐑^41_R_1(ξ_1,η_1)1_R_2(ξ_2,η_2)e^-it/2(ξ_1-ξ_2)(η_1-η_2)
       ×f(ξ_1,η_1)f(ξ_2,η_2)δ(ξ-ξ_1-ξ_2)δ(η-η_1-η_2)dξ_1 dξ_2 dη_1 dη_2.

Now we observe that since

    | I_1|=| I_2|≃dist(I_1,I_2),    | J_1|=| J_2|≃dist(J_1,J_2),

it holds that, on the support of integration,

    | (ξ_1-ξ_2)(η_1-η_2)|≃| I_1|·| J_1|≃Ω.

Therefore, we have the following orthogonality in time

    ‖∑_ΩI_Ω(ξ,η,·)‖_L^2_t^2 =‖ e^it/2ξη∑_ΩI_Ω(ξ,η,·)‖_L^2_t^2 ≲∑_Ω‖I_Ω(ξ,η,·)‖_L^2_t^2.




To continue, we need to control I_Ω uniformly in Ω. We write that

    ℐ_Ω   =‖∑_R_1∼ R_2,
    | R_1|=| R_2|=Ω u_R_1· u_R_2‖_L^2_x,y,t^2
       =∑_R_1∼ R_2,   R_1^'∼ R_2^',
    | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ω∫_𝐑^3_x,y,t u_R_1· u_R_2·u_R_1^'· u_R_2^'   dx dy dt
       =∑_R_1∼ R_2,   R_1^'∼ R_2^'
    | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ωℐ_R_1∼ R_2,R_1^'∼ R_2^'.

To any rectangle R=I× J, we associate its center c = (c_x, c_y) and its scales ℓ_x(R)=| I| and ℓ_y(R)=| J|=Ω/| I|. For 2 rectangles R and R^' of equal area, we define their relative discrepancy by

    δ(R,R^')=min{ℓ_x(R)/ℓ_x(R^'),ℓ_y(R)/ℓ_y(R^')}.

We want to decompose ℐ_Ω according to the discrepancy of R_1=I_1× J_1 and R_1^'=I_1^'× J_1^'. Using scaling relation (<ref>), we may assume that Ω=1, ℓ_x(R_1)=ℓ_x(R_2)=1 and that ℓ_x(R_1^')≤ℓ_y(R_1^'), so that R_1^' is a δ×δ^-1 rectangle, where δ=δ(R_1,R_1^').



We first notice that, if ℐ_R_1∼ R_2,R_1^'∼ R_2^' 0, we must have that

    | c_x(R_1)-c_x(R_1^')|     ≲  ℓ_x(R_1)+ℓ_x(R_1'),
    | c_y(R_1)-c_y(R_1^')|     ≲  ℓ_y(R_1)+ℓ_y (R_1').

and therefore, for any fixed R_1 and δ≳ 1, there can be only a bounded number of choices for R_1^', so that

    ℐ_1   ≲∑_R_1∼ R_2,
    | R_1|=| R_2|=1‖ u_R_1u_R_2‖_L^2_x,y,t^2.

At this stage, we are in a similar position as in the elliptic case and we may follow the proof in <cit.>. From now on, we will focus on the case δ≪1.



In the case δ≪1, we may in fact strengthen (<ref>). Indeed for ℐ_R_1∼ R_2,R_1^'∼ R_2^' to be different from 0, we must have that

    | c_x(R_1)-c_x(R_1^')|     ≃  ℓ_x(R_1)+ℓ_x(R_1'),
    | c_y(R_1)-c_y(R_1^')|     ≃  ℓ_y(R_1)+ℓ_y (R_1').

This follows from the fact that (say)

    c_x(R_1)+c_x(R_2)-c_x(R_1^')-c_x(R_2^')
    
    =   2[c_x(R_1)-c_x(R_1^')]-[c_x(R_1)-c_x(R_2)]+[c_x(R_1^')-c_x(R_2^')],

and the last bracket is bounded by 24δ, while the second to last is bounded below by 6; however, for ℐ_R_1∼ R_2,R_1^'∼ R_2^' to be nonzero, there must exists (ξ_1,ξ_2,ξ_1^',ξ_2^')∈ R_1× R_2× R_1^'× R_2^' such that

    ξ_1+ξ_2-ξ_1^'-ξ_2^'=0    and
       | (ξ_1+ξ_2-ξ_1^'-ξ_2^')-(c_x(R_1)+c_x(R_2)-c_x(R_1^')-c_x(R_2^'))|≤ 2+2δ.

We will keep note of this by writing R_1≃ R_1^' (or sometimes c(R_1)≃ c(R_1^')) whenever (<ref>) holds for rectangles of equal area.

Recall that R_1^' is a δ×δ^-1 rectangle; we can decompose all rectangles into δ× 1 rectangles. We may then partition

    R_1   =⋃_a=1^δ^-1I_1,a× J_1=⋃_a=1^δ^-1R_1,a,     R_2=⋃_a=1^δ^-1 I_2,a× J_2=⋃_a=1^δ^-1R_2,a,
    
    R_1^'   =⋃_b=1^δ^-1I_1^'× J_1,b^'=⋃_b=1^δ^-1R_1,b^',     R_2^'=⋃_b=1^δ^-1I_2^'× J_2,b^'=⋃_b=1^δ^-1R_2,b^'

and by orthogonality, we see that

    ℐ_R_1∼ R_2,R_1^'∼ R_2^'   =∑_a∼a,   b∼bℐ_R_1,a∼ R_2,a,R_1,b^'∼ R_2,b^'

where 

    a∼a    if and only if     | c_x (R_1,a) + c_x(R_2,a)  - c_x (R_1') - c_x (R_2') | ≲δ

and comparably in y for b∼b. Thus, for fixed R_1, R_2, R_1^' R_2^', this gives two equivalence relations with O(δ^-1) equivalence classes of (uniformly) bounded cardinality.

And proceeding as in (<ref>), we can easily see that

    | c_x(R_1,a)-c_x(R^'_1,b)|≳ 1,    | c_y(R_1,a)-c_y(R^'_1,b)|≳δ^-1,
    | c_x(R_2,a)-c_x(R^'_2,b)|≳ 1,    | c_y(R^'_1,a)-c_y(R^'_2,b)|≳δ^-1.



At this point, we have extracted all the orthogonality we need and we are ready to proceed with the proof of Proposition <ref>.



 §.§ Proof of (<ref>)


Using rescaling, we may assume that

    1=sup_R| R|^-1/2‖ e^it∂_x ∂_yϕ_R‖_L^∞_x,y,t.

From the considerations above, we obtain the expression

    ‖ e^it∂_x ∂_yϕ‖_L^4_x,y,t^4   ≲∑_Ω∑_R_1∼ R_2,  R_1^'∼ R_2^',
    | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ωℐ_R_1∼ R_2,R_1^'∼ R_2^',

where the rectangles satisfy the condition (<ref>). In addition, for fixed rectangles R_1∼ R_2, R_1^'∼ R_2^' of equal area Ω, let δ=δ(R_1,R_1^'). As explained above, for fixed δ=δ_0=O(1), we are in a position similar to the elliptic case and we may follow <cit.> to get

    ∑_Ω∑_R_1∼ R_2,  R_1^'∼ R_2^',
    | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ω,
    δ(R_1,R_1^')=δ_0|ℐ_R_1∼ R_2,R_1^'∼ R_2^'|
    ≲   ∑_Ω∑_R_1∼ R_2,
    | R_1|=| R_2|=Ω‖ u_R_1u_R_2‖_L^2_x,y,t^2
    ≲   (sup_R| R|^-1/2‖ u_R‖_L^∞_x,y,t)^4/21∑_R_1∼ R_2| R_1|^2/21‖ u_R_1u_R_2‖_L^40/21_x,y,t^40/21
    ≲   ∑_R_1∼ R_2{| R_1|^-1/20‖ϕ_R_1‖_L^20/11_x,y·| R_2|^-1/20‖ϕ_R_2‖_L^20/21_x,y,t}^40/21,

where we have used Cauchy-Schwarz in the first inequality, Hölder's inequality in the second and (<ref>) together with Lemma <ref> in the last inequality. 
This gives a bounded contribution as in (<ref>) for any p≤ 80/21.




We need to adjust the above scheme when δ≪1. In the following, we let

    T_≪1:=∑_δ≪1∑_Ω∑_R_1∼ R_2,  R_1^'∼ R_2^',
    | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ω,
    δ(R_1,R_1^')=δ|ℐ_R_1∼ R_2,R_1^'∼ R_2^'|

and to conclude the proof of (<ref>), we need to prove that, for some p>2,

    T_≪1≲‖ϕ‖_X_p^80/21.

We can now use the finer decomposition (<ref>) to write

    ℐ_R_1∼ R_2,R_1^'∼ R_2^'   =∑_a∼a,  b∼bℐ_R_1,a∼ R_2,a,R_1,b^'∼ R_2,b^'

where the new rectangles satisfy (<ref>). Using Cauchy-Schwartz, then Hölder's inequality with (<ref>), we have that

    |ℐ_R_1,a∼ R_2,a,R_1,b^'∼ R_2,b^'|   ≲‖ u_R_1,a· u_R^'_1,b‖_L^2_x,y,t·‖ u_R_2,a· u_R^'_2,b‖_L^2_x,y,t
       ≲ (δΩ)^2/21‖ u_R_1,a· u_R^'_1,b‖_L^40/21_x,y,t^20/21·‖ u_R_2,a· u_R^'_2,b‖_L^40/21_x,y,t^20/21 .

Now, using Lemma <ref> with (<ref>), we obtain that

    |ℐ_R_1,a∼ R_2,a,R_1,b^'∼ R_2,b^'|
    ≲   (δΩ)^2/21· (δ^-1Ω)^-6/21‖ϕ_R_1,a‖_L^20/11_x,y^20/21‖ϕ_R_2,a‖_L^20/11_x,y^20/21‖ϕ_R^'_1,b‖_L^20/11_x,y^20/21‖ϕ_R^'_2,b‖_L^20/11_x,y^20/21.

Since 20/11 <40/21 and since for fixed a, there are only a bounded number a such that a∼a, we can sum over a to get

    ∑_a∼a‖ϕ_R_1,a‖_L^20/11_x,y^20/21‖ϕ_R_2,a‖_L^20/11_x,y^20/21   ≲‖ϕ_R_1‖_L^20/11_x,y^20/21‖ϕ_R_2‖_L^20/11_x,y^20/21

and similarly for b, so that

    |ℐ_R_1∼ R_2,R_1^'∼ R_2^'|   ≲δ^8/21Ω^-4/21‖ϕ_R_1‖_L^20/11_x,y^20/21‖ϕ_R_2‖_L^20/11_x,y^20/21‖ϕ_R_1^'‖_L^20/11_x,y^20/21‖ϕ_R_2^'‖_L^20/11_x,y^20/21.

In addition, for rectangles of fixed areas and sizes | R_1|=| R_2|=| R_1^'|=| R_2^'|, ℓ_x(R_1)=ℓ_x(R_2), ℓ_x(R_1^')=ℓ_x(R_2^') also satisfying (<ref>), we may use Cauchy Schwartz in the summation over the centers to get

    ∑_R_1∼ R_2,
     R_1^'∼ R_2^'‖ϕ_R_1‖_L^20/11_x,y^20/21‖ϕ_R_2‖_L^20/11_x,y^20/21‖ϕ_R_1^'‖_L^20/11_x,y^20/21‖ϕ_R_2^'‖_L^20/11_x,y^20/21   ≲∑_R_1≃ R_1^'‖ϕ_R_1‖_L^20/11_x,y^40/21‖ϕ_R_1^'‖_L^20/11_x,y^40/21

where the sum is taken over all rectangles R_1≃ R_1^' of the given sizes satisfying (<ref>).



We can now get back to (<ref>) and use (<ref>) and the inequality above to get

    T_≪1   ≲∑_Ω∑_A∑_δ≤ 1δ^8/21Ω^-4/21·∑_c_1≃ c_1^'‖ϕ_R_1‖_L^20/11_x,y^40/21‖ϕ_R_1^'‖_L^20/11_x,y^40/21,

where we have parameterized the lengths of the rectangles by Ω=| R_1|=| R_1^'|, A=ℓ_x(R_1) and δ=ℓ_x(R_1^')/ℓ_x(R_1), and their centers by c_1, c_1^'.

Now for any p > 2 choose 0 < θ(p) < 1 such that

    2 θ/p + 1 - θ/p = 21/40.

and observe that θ(p) ↘1/20 as p ↘ 2. Then by interpolation,

    ∑_Ω,A,
    
    c_1≃ c_1^'Ω^-4/21‖ϕ_R_1‖_L^20/11_x,y^40/21‖ϕ_R_1^'‖_L^20/11_x,y^40/21
    ≲   (∑_Ω,A,
     c_1≃ c_1^'Ω^-p/10‖ϕ_R_1‖_L^20/11_x,y^p‖ϕ_R_1^'‖_L^20/11_x,y^p)^40/211-θ/p(∑_Ω,A,
     c_1≃ c_1^'Ω^-p/20‖ϕ_R_1‖_L^20/11_x,y^p/2‖ϕ_R_1^'‖_L^20/11_x,y^p/2)^40/212θ/p.

Now, on the one hand, we observe that for a fixed choice of scales (Ω, A and δ)  and for each fixed c_1, there are at most O(δ^-1) choices of c_1^' satisfying (<ref>) so we obtain that

    ∑_Ω,A,
     c_1≃ c_1^'Ω^-p/20‖ϕ_R_1‖_L^20/11_x,y^p/2‖ϕ_R_1^'‖_L^20/11_x,y^p/2   ≲δ^-1∑_Ω,A,c_1Ω^-p/20‖ϕ_R_1‖_L^20/11_x,y^p

and the other sum can be handled in an easier way: using Hölder's inequality and forgetting about the relationship c_1≃ c_1^', we obtain that

    ∑_Ω∑_A∑_c_1≃ c_1^'Ω^-p/10‖ϕ_R_1‖_L^20/11_x,y^p‖ϕ_R_1^'‖_L^20/11_x,y^p
    ≲   ∑_Ω∑_A(∑_c_1{Ω^-1/20‖ϕ_R_1‖_L^20/11_x,y}^p)·(∑_c_1^'{Ω^-1/20‖ϕ_R_1^'‖_L^20/11_x,y}^p)
    ≲   (∑_R| R|^-p/20‖ϕ_R‖_L^20/11_x,y^p)^2.

Recall the definition (<ref>). Combining (<ref>) and (<ref>), we obtain

    T_≪1   ≲∑_δ≤ 1δ^8/21·(δ^-1‖ϕ‖_X_p^p)^80/21θ/p(‖ϕ‖_X_p^2p)^40/211-θ/p≲∑_δ≤ 1δ^8/21(1-10θ/p)‖ϕ‖_X_p^80/21

and this is summable in δ for 2<p<40/17 small enough. The proof of (<ref>) is thus complete and it remains to prove (<ref>) which we now turn to.



 §.§ Proof of (<ref>)


We first state and prove the following simple result we will need in the proof.


Let 𝒟 denote the set of dyadic intervals (on 𝐑) and let p>2. For any g∈ C^∞_c(𝐑), there holds that

    ∑_I∈𝒟| I|^-p/20‖ g1_I‖_L^20/11_x^p≲‖ g‖_L^2_x^p.




We may assume that ‖ g‖_L^2_x=1. For fixed A, we let 𝒟_A denote the set of dyadic intervals of length A and we decompose

    g=g^++g^-,     g^+(x)=g(x)1_{| g(x)|>A^-1/2},     g^-(x)=g(x)1_{| g(x)|≤ A^-1/2}.

On the one hand, using that ℓ^20/11⊂ℓ^p,

    ∑_A∑_I∈𝒟_A| I|^-p/20‖ g^+1_I‖_L^20/11_x^p   ≲(∑_AA^-1/11∑_I∈𝒟_A‖ g^+1_I‖_L^20/11_x^20/11)^11 /20p
       ≲(∑_AA^-1/11∫_𝐑| g|^20/111_{| g(x)|>A^-1/2}dx)^11 /20p
       ≲(∫_𝐑| g|^20/11·(∑_A>| g(x)|^-2A^-1/11)dx)^11 /20p≲ 1,

while, for the other sum, we use Hölder's inequality to get

    ∑_A∑_I∈𝒟_A A^-p/20‖ g^-1_I‖_L^20/11_x^p
       ≲∑_A∑_I∈𝒟_A A^-p/20‖ g^-1_I‖_L^p^p·| I|^(11/20-1/p)p
       ≲∫_𝐑| g(x)|^p ·∑_{A<| g(x)|^-2}A^p-2/2dx
       ≲∫_𝐑| g(x)|^2dx≲ 1

and the proof is complete.


Now, we proceed to prove (<ref>).

 
Recall 𝒟 stand for the set of dyadic intervals and 𝒟_A for the set of dyadic intervals of length A. We want to prove that

    ∑_I∈𝒟| I|^-p/20∑_J∈𝒟| J|^-p/20‖ f 1_I× J‖_L^20/11_x,y^p ≲‖ f‖_L^2_x,y^p.

We claim that, for any fixed interval I,

    ∑_J∈𝒟| J|^-p/20‖ f1_I× J‖_L^20/11_x,y^p ≲‖ f1_I×𝐑‖_L^20/11_xL^2_y^p.

Once this is proved, we may simply apply Lemma <ref> to the function

    g(x):=‖ f(x,·)‖_L^2_y

to finish the proof.



From now on I denotes a fixed interval and f is a function supported on {x∈ I}, i.e. f=f1_I×𝐑. The proof of (<ref>) is a small variation on the proof of Lemma <ref>. Fix a dyadic number B and let

    c_B=c_B(x)=B^-1/2‖ f(x,·)‖_L^2_y

and decompose accordingly[Note that f(x,y)=0 whenever ‖ f(x,·)‖_L^2_y=0, so that c_B(x)>0 on the support of f^+.]

    f=f^++f^-,   f^+=f1_{| f(x,y)|>c_B(x)},   f^-=f1_{| f(x,y)|≤ c_B(x)}.

We then compute that

    ∑_B∑_J∈𝒟_B B^-p/20‖ f^+1_I× J‖_L^20/11_x,y^p  ≲(∑_B∑_J∈𝒟_B B^-1/11‖ f^+1_I× J‖_L^20/11_x,y^20/11)^11 /20p
       ≲(∑_B B^-1/11∫_I_x∫_𝐑_y| f^+|^20/11 dy  dx)^11 /20p
       ≲(∫_I_x∫_𝐑_y| f^+|^20/11·∑_{B:| f(x,y)|≥ c_B(x)}B^-1/11 dy  dx)^11/20p
       ≲(∫_I_x∫_𝐑_y| f(x,y)|^20/11·(| f(x,y)|/‖ f(x,·)‖_L^2_y)^2/11 dy  dx)^11/20p
       ≲(∫_I_x‖ f(x,·)‖_L^2_y^-2/11∫_𝐑_y| f(x,y)|^2dy  dx)^11/20p
       ≲‖ f‖_L^20/11_xL^2_y^p ,

 in the penultimate line, we note that though there is a negative power of the L^2_y norm, the product of the two quantities is well-defined, especially as we can assume f ∈ C^∞_c.  Also, we have used the embedding ℓ^1 ⊂ℓ^11/20p in the first inequality, the fact that dyadic intervals of a fixed length tile 𝐑 in the second inequality, and we have summed a geometric series in the fourth inequality. Now for the second part, we compute that

    ∑_B∑_J∈𝒟_B   B^-p/20‖ f^-1_I× J‖_L^20/11_x,y^p  ≲∑_B∑_J∈𝒟_B B^-p/20B^p(11/20-1/p)‖ f^-1_I× J‖_L^p_y(J:L^20/11_x(I))^p
       ≲∑_B B^p -2 /2∫_𝐑_y(∫_I_x| f^-(x,y)|^20/11dx)^11/20pdy
       ≲∫_𝐑_y∑_B (B^p -2 /220/111/p∫_I_x| f^-(x,y)|^20/11dx)^11/20pdy
       ≲∫_𝐑_y(∫_I_x∑_BB^p -2 /p10/11| f^-(x,y)|^20/11dx)^11/20pdy,

where we have used Hölder's inequality in the first line and the inclusion ℓ^1⊂ℓ^11/20p in the fourth line. Now, since f^- is supported where

    B≤(‖ f(x,·)‖_L^2_y/| f(x,y)|)^2,

summing in B gives

    ∑_B∑_J∈𝒟_B B^-p/20‖ f^-1_I× J‖_L^20/11_x,y^p
    ≲   ∫_𝐑_y(∫_I_x‖ f(x,·)‖_L^2_y^20/11p-2/p| f^-(x,y)|^20/112/pdx)^11/20pdy.

Using Minkowski inequality on the function

    h(x,y)=‖ f(x,·)‖_L^2_y^20/11p-2/p| f^-(x,y)|^20/112/p,

we obtain

    ∑_B∑_J∈𝒟_B B^-p/20‖ f^-1_I× J‖_L^20/11_x,y^p
    ≲   (∫_I_x(∫_𝐑_yh^11/20pdy)^20/111/pdx)^11/20p
    ≲   (∫_I_x(∫_𝐑_y| f(x,y)|^2dy)^20/111/p‖ f(x,·)‖_L^2_y^20/11p-2/p  dx)^11/20p
    ≲   (∫_I_x‖ f(x,·)‖_L^2_y^20/11dx)^11/20p,

which proves (<ref>). Thus the proof is complete.









§ THE PROFILE DECOMPOSITION AND APPLICATIONS


The profile decomposition then follows from Proposition <ref> in the usual way following the techniques in the proof of Theorems 4.25 (the Inverse Strichartz Inequality) and 4.26 (Mass Critical Profile Decomposition) from <cit.>, for instance.  We note that it is the proof of the Inverse Strichartz Inequality that requires the local smoothing estimates as in Lemma <ref> to establish pointwise a.e. convergence of profiles to an element of L^2_x,y through compactness considerations, otherwise the proof follows mutatis mutandis.  Once the Inverse Strichartz Inequality is established, the proof of the Profile Decomposition follows verbatim.  

Suppose g_n^j = g(x_n^j, y_n^j, λ_1, n^j, λ_2, n^j, ξ_n^j) is the group whose action on functions is given by

    (g_n^j)^-1 f = (λ_n, 1^jλ_n, 2^j)^1/2 e^-i ξ_n, 1^j (λ_n, 1^j x + x_n^j) e^-i ξ_n, 2^j (λ_n, 2^j x + y_n^j)
       × [f_n](λ_n, 1^j x + x_n^j, λ_n, 2^j y + y_n^j).


The profile decomposition gives the following.


Let u_n_L^2_x,y(𝐑^2)≤ A be a sequence that is bounded L^2_x,y(𝐑^2). Then possibly after passing to a subsequence, for any 1 ≤ j < ∞ there exist ϕ^j∈ L^2_x,y(𝐑^2), (t_n^j, x_n^j, y_n^j) ∈𝐑^3, ξ_n^j∈𝐑^2, λ_n, 1^j, λ_n, 2^j∈ (0, ∞) such that for any J,


    u_n = ∑_j = 1^J g_n^j e^i t_n^j∂_x∂_yϕ^j + w_n^J,



    lim_J →∞lim sup_n →∞ e^it ∂_x ∂_y   w_n^J_L_x,y,t^4 = 0,


such that for any 1 ≤ j ≤ J,


    e^-i t_n^j∂_x ∂_y (g_n^j)^-1 w_n^J⇀ 0,


weakly in L^2_x,y(𝐑^2), 


    lim_n →∞( u_n_L^2_x,y^2 - ∑_j = 1^Jϕ^j_L^2_x,y^2 -  w_n^J_L^2_x,y^2) = 0,


and for any j ≠ k,


    lim_n →∞[ |ln( λ_n, 1^j/λ_n, 1^k)| + |ln( λ_n, 2^j/λ_n, 2^k)| 
    + |t_n^j (λ_n,1^jλ_n,2^j) - t_n^k (λ_n, 1^kλ_n, 2^k)|/(λ_n, 1^jλ_n, 2^jλ_n, 1^kλ_n, 2^k)^1/2. 
        + (λ_n, 1^jλ_n, 1^k)^1/2 |ξ_n, 1^j - ξ_n, 1^k| 
    + (λ_n, 2^jλ_n, 2^k)^1/2 |ξ_n, 2^j - ξ_n, 2^k|
        + |x_n^j - x_n^k - 2 t_n^j (λ_n, 1^jλ_n, 2^j)(ξ_n, 1^j - ξ_n, 1^k)|/(λ_n, 1^jλ_n, 1^k)^1/2
       .  + |y_n^j - y_n^k - 2 t_n^j (λ_n, 1^jλ_n, 2^j)(ξ_n, 2^j - ξ_n, 2^k)|/(λ_n, 2^jλ_n, 2^k)^1/2]   = ∞.






 §.§ Minimal mass blow-up solutions


As an application of the profile decomposition, we turn to a calculation that for instance originated in <cit.>.  Namely we construct a minimal mass solution to (<ref>) which is a solution u of minimal mass such that there exists a time T^* such that

    ∫_-T^*^T^*∫_𝐑^2_x,y |u |^4   dx dy dt= + ∞.

In other words, it is a solution of least mass for which the small data global argument fails. 

It turns out that if u is a minimal mass blowup solution to (<ref>) then u lies in a compact subset of L^2_x,y(𝐑^2) modulo the symmetry group g; more precisely, following <cit.>, we can establish the following theorem.



Suppose u is a minimal mass blowup solution to (<ref>) on a maximal time interval I that blows up in both time directions.  That is, I is an open interval and for any t_0∈ I,


    ∫_t_0^sup(I)∫ |u(x,y,t)|^4 dx dy dt, ∫_inf(I)^t_0∫ |u(x,y,t)|^4 dx dy dt = ∞.


Then there exist λ_1, λ_2 : I → (0, ∞), ξ : I →𝐑^2, x, y : I →𝐑, such that for any η > 0 there exists C(η) < ∞ such that


    ∫_|x - x(t)| > C(η)/λ_1(t) |u(x,y,t)|^2 dx dy + ∫_|y - y(t)| > C(η)/λ_2(t) |u(x,y,t)|^2 dx dy 
     + ∫_|ξ_1 - ξ_1(t)| > C(η) λ_1(t) |û(ξ,t)|^2 dξ + ∫_|ξ_2 - ξ_2(t)| > C(η) λ_2(t) |û(ξ,t)|^2 dξ < η.



 Take a sequence t_n∈ I. Then conservation of mass implies that after passing to a subsequence we may make a profile decomposition of u(t_n) = u_n. If there exists j such that, along a subsequence, t_n^j→±∞, say t_n^j→∞, then


    lim_n →∞ e^it ∂_x∂_y (g_n^j e^i t_n^j∂_x ∂_yϕ^j) _L_x,y,t^4([0, ∞) ×𝐑^2) = 0,


so combining perturbative arguments, (<ref>), and the fact that u is a blowup solution with minimal mass then u scatters forward in time to a free solution. Thus, we may assume that for each j, t_n^j converges to some t^j∈𝐑. Then taking e^it^j∂_x ∂_yϕ^j to be the new ϕ^j, we may assume that each t_n^j = 0.

Now suppose that


    sup_jϕ^j_L^2_x,y(𝐑^2) <  u(t) _L^2_x,y(𝐑^2).


Then if v^j is the solution to (<ref>) with initial data ϕ^j, since u(t) _L^2 is the minimal mass for blowup to occur, each v^j scatters both forward and backward in time, with


    v^j_L_x,y,t^4(𝐑×𝐑^2)^2 ≲‖ϕ^j‖_L^2_x,y^2 < ∞,    uniformly in  j.


Then if v_n^j is the solution to (<ref>) with initial data g_n^jϕ^j, v_n^j = g_n^j (v^j( (λ_n,1^jλ_n,2^j)^-1  t)).
We note that, for v either a profile v_n^ℓ or the remainder w_n^J,

    v_n^j v_n^k v _L_x,y,t^4/3≤‖ v_n^jv_n^k‖_L^2_x,y,t‖ v ‖_L^4_x,y,t.

In addition, ‖ v‖_L^4_x,y,t remains bounded either by (<ref>) (for v_n^ℓ) or as a consequence of the small data theory and (<ref>) (for w_n^J).

By approximation by compactly supported functions, it is easy to see that, if j k,

    ‖ v_n^jv_n^k‖_L^2_x,y,t→0

when n →∞ as a consequence of (<ref>).

As a result, using simple perturbation theory, we obtain that, for J large enough,

    ‖ u(t_n+t)-∑_j=1^Jv_n^j(t)‖_L^4_x,y,t≲ 1

and using again (<ref>)-(<ref>), we obtain that

    ‖∑_j=1^Jv_n^j(t)‖_L^4_x,y,t^4≲∑_j=1^J‖ v_n^j‖_L^4_x,y,t^4≲∑_j=1^J‖ϕ^j‖_L^2_x,y^2<∞.

which, together with (<ref>) contradicts (<ref>).

































Thus, after reordering we should have ϕ^1_L^2_x,y =  u(t) _L^2_x,y and ϕ^j = 0 for any j ≥ 2. But this holds if and only if u(t) lies in a set G K, where G is the group generated by g_n^j and K is a compact set in L^2. This completes the proof of the theorem.











§ EXTREMIZERS FOR STRICHARTZ ESTIMATES FOR (<REF>)




We end with a few considerations on extremizers for the Strichartz inequality (<ref>), that is functions f of unit L^2_x,y-norm and constant C such that

    ‖ e^it∂_x ∂_yf‖_L^4_x,y,t=C:=sup{‖ e^it∂_x ∂_yg‖_L^4_x,y,t:  ‖ g‖_L^2_x,y=1}.

The original version of this paper stated erroneously that Gaussians were optimizers for the Strichartz norm above, and gave a corresponding numerical value for C. In fact, it was later proved in <cit.> that Gaussians are not critical points of the Strichartz norm and thus cannot be optimizers. One can however use the profile decomposition for Theorem <ref> to prove existence of an extremizer (see <cit.> for a similar proof).


There exists f∈ L^2_x,y(𝐑^2) such that

    ‖ e^it∂_x ∂_yf‖_L^4_x,y,t = C‖ f‖_L^2_x,y.

where C is given in (<ref>).


The exact nature of the extremizer f above remains mysterious.

Preliminary numerical investigations confirm that the optimizer should be a genuine function of x and y and suggests that it has nice decay and smoothness properties.

We reproduce some plots of the amplitude of an extremizer and its Fourier transform below. These are obtained by maximizing the Strichartz norm among functions in the span of the first 25 Hermite functions with unit L^2 norm. Some care has been given to obtain accurate computations and minimize boundary effects, but we do not mean that the result is anything but suggestive. However, it seems to indicate that extremizers are indeed smooth and localized and that they can be chosen real valued, which is why we have not reproduced their phase.













































































































































































MA




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