Almost sure scattering for the energy critical nonlinear wave equation
Bjoern Bringmann


[1]


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    December 30, 2023
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MSC2010: 35L05, 42B20, 42B37
Keywords: nonlinear wave equation, probabilistic well-posedness, scattering, wave packet decomposition 




We study the defocusing energy-critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random pertubations of the initial data. The random pertubation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space.  In contrast to the previous literature, we do not require the spherical symmetry of the pertubation. 
 
The main novelty lies in a wave packet decomposition of the random linear evolution. Through this decomposition, we can adaptively estimate the interaction between the rough and regular components of the solution. Our argument relies on techniques from restriction theory, such as Bourgain's bush argument and Wolff's induction on scales. 







§ INTRODUCTION

We consider the defocusing cubic nonlinear wave equation in four space dimensions, that is, 

    -∂_tt u + Δ u = u^3      (t,x)∈×^4 ,
    
    u|_t=0  = u_0 ∈Ḣ^s(),   ∂_t u|_t=0 = u_1 ∈Ḣ^s-1() .

If u is a regular solution of (<ref>), then it conserves the energy

    E[u](t):= ∫_|∇ u(t,x)|^2/2 + (∂_t u(t,x))^2/2 + u(t,x)^4/4 .

From the Sobolev embedding Ḣ^1() ↪ L^4(), it follows that the initial data has finite energy if and only if (u_0,u_1)∈Ḣ^1()× L^2(). Thus, we also refer to Ḣ^1()× L^2() as the energy space. In addition to the energy conservation law, (<ref>) obeys the scaling symmetry u(t,x) ↦ u_λ(t,x) = λ u(λ t, λ x). Since the scaling leaves the energy invariant, the equation is called energy critical. Due to the positive sign in front of the potential term u^4, we call (<ref>) defocusing. There also exists analogues of (<ref>) with a power-type nonlinearity in any dimension d ≥ 3. 


The Cauchy problem for (deterministic) initial data in the energy space is well-understood.  We summarize the relevant results in the following theorem. 

Let (u_0, u_1) ∈Ḣ^1()× L^2(). Then, there exists a maximal time interval of existence I and a unique solution u I×→ of (<ref>) such that u ∈ C_t^0 Ḣ_x^1(I ×)  ⋂  L_ t,loc^3 L_x^6(I×) and ∂_t u ∈ C_t^0 L_x^2(I×). Furthermore, we have that 


  * u is global, i.e., I =. 

  * u obeys a global space-time bound of the form

    u _L_t^3 L_x^6(×)≤ C(E[u_0,u_1]) .


  * u scatters to a solution of the linear wave equation. Thus, there exist scattering states (u_0^±, u_1^± ) ∈Ḣ^1()× L^2() s.t.

    lim_t→±∞ (u(t)-W(t)(u_0^±,u_1^±), ∂_t u(t) - ∂_t W(t) (u_0^±,u_1^±)) _Ḣ^1× L^2 = 0 .

Here, W(t)(u_0^±,u_1^±) = cos(t|∇|) u_0^± + (sin(t|∇|)/|∇|)u_1^± denotes the solution to the linear wave equation with initial data (u_0^±,u_1^±). 



Global well-posedness and scattering results such as Theorem <ref> are known for many defocusing dispersive partial differential equations, and hold for the energy-critical nonlinear Schrödinger equation <cit.>, the mass-critical nonlinear Schrödinger equation <cit.>, the mass-critical generalized KdV <cit.>, and the Ḣ^1/2-critical radial nonlinear wave equation <cit.>. 

Since Theorem <ref> provides a complete description of the Cauchy problem with initial data in the energy space, we now seek a similar result for initial data in a rough Sobolev space H_x^s × H_x^s-1, where s ∈ [0,1). However, since this leads to a scaling super-critical problem,  all of the above properties can fail. In fact, <cit.> proved that (<ref>) exhibits norm inflation, which means that arbitrarily small data in H^s × H^s-1 can grow arbitrarily fast. More precisely, we have for all ϵ > 0 that there exists Schwartz initial data (u_0,u_1) and a time 0 < t_ϵ < ϵ  such that 
(u_0,u_1)_H^s× H^s-1 < ϵ and (u(t_ϵ),∂_t u(t_ϵ)) _H^s × H^s-1 > ϵ^-1.  Using finite speed of propagation, one may then also construct solutions whose H^s × H^s-1-norm blows up instantaneously. 




 §.§ The random data Cauchy problem


Many researchers in dispersive partial differential equations have recently examined whether blow-up behaviour, such as the norm-inflation described above, occurs for generic or only exceptional sets of rough initial data. To quantify this, one is quickly lead to random initial data. Indeed, one natural form of rough initial data is (u_0+ f_0^ω, u_1+ f_1^ω), where the functions (u_0,u_1) ∈Ḣ^1× L^2 are regular and deterministic, while the functions (f_0^ω,f_1^ω)∈ H^s× H^s-1 are rough and random. An analogue of Theorem <ref> in this case would imply the stability of the scattering mechanism under a perturbation by noise. 

The literature on random dispersive partial differential equations is vast. We refer the interested reader to the survey <cit.>, and mention the related works <cit.>. 
In the following discussion, we focus on the Wiener randomization <cit.> of a function f ∈ H^s(). Let φ∈ C^∞() be a smooth and symmetric function satisfying φ|_[-3/8,3/8]^d=1, φ|_\ [-5/8,5/8]^d = 0, and ∑_k∈φ(ξ-k) = 1 for all ξ∈. We then define the associated operator P_k by 

    P_k f(ξ) := φ(ξ-k) f(ξ) .

Since the translates {φ(·-k)}_k form a partition of unity, we have that 

    f= ∑_k∈ P_k f  ,

which is called the Wiener decomposition of f. The Wiener randomization is obtained by randomizing the coefficients in (<ref>). Let I ⊆ by an index set such that = I 0.25ex⋃̇ { 0 } 0.25ex⋃̇  (-I). Let { X_k }_k∈ I ∪{ 0 } be a sequence of symmetric, independent, and uniformly sub-gaussian random variables (see Definition <ref>). We set X_-k := X_k for all k ∈ I, and assume that X_0 is real-valued. Then, the Wiener randomization f^W is defined as

    f^W := ∑_k∈ X_k · P_k f .

The reason for introducing the set I is to preserve the real-valuedness of f.  The Wiener randomization f^W is a random linear combination of functions with unit-scale frequency uncertainty, and therefore resembles a random Fourier series. We then examine the random data Cauchy problem


    -∂_tt u + Δ u = u^3      (t,x)∈×^4 
    
    u|_t=0  = u_0 + f_0^W,   ∂_t u|_t=0 = u_1 +f_1^W
     .


We now seek an almost sure version of Theorem <ref> for (<ref>). Before we summarize the recent results, let us sketch the overall strategy, which was developed by Pocovnicu in <cit.>.
 We let F:= cos(t|∇|) f_0^W + (sin(t|∇|)/|∇|) f_1^W be the solution of the linear wave equation with the rough and random initial data. We then define the nonlinear component v by  v := u - F, and obtain the forced nonlinear wave equation

    -∂_tt v + Δ v = (v+F)^3      (t,x)∈×^4 ,
    
    v|_t=0  = u_0 ,   ∂_t v|_t=0 = u_1 .

At the cost of introducing a rough forcing term, we have therefore removed the rough part of the initial data. This transformation is related to the Da Prato-Debussche trick <cit.>. Due to the smoothing effect of the Duhamel integral, we hope to control the nonlinear component v in the energy space. The local well-posedness of (<ref>) follows readily from probabilistic Strichartz estimates (cf. <cit.>) and a contraction mapping argument. Thus our main interest lies in the global well-posedness and the long-time behaviour of the solution. Using the deterministic well-posedness theorem and stability theory, it can be shown (cf. <cit.>) that the solution to (<ref>) exists as long as the energy of v remains bounded. Of course, due to the forcing term in (<ref>), the energy is no longer conserved. In addition, a global bound on the energy of v implies a global bound on the L_t^3 L_x^6-norm, and hence also implies scattering. A short calculation shows that 

    d/ E[v](t) = ∫_ (v^3 - (v+F)^3) ∂_t v  ≈ -  3 ∫_ F v^2 ∂_t v .
 
In the formula above, we have neglected terms that contain more than a single factor of F, since they are simpler to estimate. Therefore, the remaining obstacle lies in the control of the right-hand side of (<ref>). With this overall strategy in mind, we summarize the recent literature. 



In <cit.>, Pocovnicu proved the almost sure global existence of solutions for all s>0. Using a Gronwall-type argument and a probabilistic Strichartz estimate, (<ref>) leads (at top order) to the growth estimate

    E[v](T) ≲ E[v](0) exp(C  F _L_t^1 L_x^∞([0,T]×)) ≲ E[v](0) exp(C_ω T^1/2) .

Since this prevents the finite time blow-up of the energy, this yields an analogue of Theorem <ref>.<ref>. Similar theorems are also known in dimension five <cit.>, dimension three <cit.>, and for the high-dimensional energy-critical nonlinear Schrödinger equation <cit.>. 

The bound (<ref>), however, is not sufficient to obtain global control on the energy of v, and hence does not prove almost sure scattering. Assuming the regularity condition s > 1/2 and that the (deterministic) data (f_0,f_1) is spherically symmetric, Dodson, Lührmann, and Mendelson proved almost sure scattering  in <cit.>. In their argument, the energy increment is estimated by

    | ∫_0^T ∫_ F v^2 ∂_t v  | ≲ |x|^1/2 F _L_t^2 L_x^∞([0,T]×)  |x|^-1/4 v_L_t^4 L_x^4([0,T]×)^2 ∂_t v_L_t^∞ L_x^2(×) .

The first factor is controlled using Khintchine's inequality and a square-function estimate, and heavily relies on the spherical symmetry of f_0 and f_1. The main novelty lies in the control of the second factor, and involves a double bootstrap argument in the energy and a Morawetz term. Under the bootstrap hypothesis, one can then control the second factor in (<ref>) by the square-root of the energy, and this eventually leads to a global bound. 

The method of <cit.> has since been used in several related works. In <cit.>, Dodson, Lührmann, and Mendelson used local energy decay to improve the regularity condition to s > 0. After replacing the cubes in the Wiener randomization by thin annuli, the author proved almost sure scattering for radial data in dimension three <cit.>. The main new ingredient is an interaction flux estimate between the linear and nonlinear components of the solution. Finally, the almost sure scattering for the radial energy-critical nonlinear Schrödinger equation in four dimensions has been obtained in <cit.>. 






 §.§ Main result and ideas


The remaining open question is concerned with almost sure scattering for non-radial data. In order to state the main result of this paper, we first need to introduce a microlocal randomization. While the Wiener randomization is based on a unit-scale decomposition in frequency space, the microlocal randomization is based on a unit-scale decomposition in phase space (see Figure <ref>). 


Let { X_k,l}_k∈ I ∪{ 0 }, l∈ be a sequence of symmetric, independent, and uniformly sub-gaussian random variables. We set X_-k,l := X_k,l for all k ∈ I, and assume that X_0,l is real-valued. For any f ∈ H^s(), we define its microlocal randomization f^ω by 

    f^ω(x):= ∑_k,l∈ℤ^d X_k,l P_k (φ(·-l) f)(x) .


The microlocal randomization is inspired by <cit.>, which used a randomization in physical space. 

 
Let (u_0,u_1)∈Ḣ^1()× L^2(), and let (f_0,f_1)∈ H^s()× H^s-1(), where s > 11/12. Then, there exists a random maximal time interval of existence I and a solution u  I ×→ of (<ref>) such that

    u∈ W(t)(f_0^ω,f_1^ω)+ (C_t^0 Ḣ_x^1(I×)  ⋂  L_t,loc^3L_x^6(I× ) )   and  ∂_t u ∈∂_t W(t) (f_0^ω,f_1^ω) + C_t^0 L^2_x(I×) .

Furthermore, we have that 


  * u is almost surely global, i.e., I =. 

  * u almost surely satisfies the global space-time bound
u _L_t^3 L_x^6(×) < ∞ .

  * u almost surely scatters to a solution of the linear wave equation. Thus, there exist random scattering states (u_0^±, u_1^± ) ∈Ḣ^1()× L^2() s.t.

    lim_t→±∞ (u(t)-W(t)(u_0^±+f_0^ω,u_1^±+f_1^ω), ∂_t u(t) - ∂_t W(t) (u_0^±+f_0^ω,u_1^±+f_1^ω)) _Ḣ^1× L^2 = 0 .



While Theorem <ref> is only proven for the microlocal randomization, the majority of our argument directly applies to the Wiener randomization. 

The main novelty in this paper lies in the application of a wave packet decomposition. To illustrate this idea, fix some k ∈ with k_∞∼ N, and assume that f_k(ξ) = N^-sφ(ξ-k). Then, f_k will essentially be unaffected by both the Wiener and microlocal randomizations, and hence forms an important example. From the method of non-stationary phase, it follows for all times t∈ [0,N] that the evolution exp(± it|∇|) f_k is concentrated in the ball |x± t k/k_2|≲ 1, and has amplitude ∼ N^-s. In space-time, we can therefore view the evolution as a tube, see Figure <ref>. For larger times, the dispersion of the evolution becomes significant, and the physical localization deteriorates. The wave packet perspective also explains the effect of the frequency randomization on the evolution. In Figure <ref>, we display a bush (cf. <cit.>), which is a collection of wave packets intersecting at a single point. If all wave packets in the bush have comparable amplitudes and the data is deterministic, one expects that the L_t^∞ L_x^∞-norm is proportional to the number of wave packets. For random data, however, the phases of the wave packets are all independent, and the central limit theorem predicts that the L_t^∞ L_x^∞-norm should instead be proportional to the square-root of the number of wave packets. 




The examples in Figure <ref> also illustrates an important heuristic: The natural timescale for the randomized evolution at frequency N is T=N. This differs from the natural timescale predicted by the (deterministic) bump-function heuristic, which is T= N^-1. We therefore decompose the positive time-interval as 

    [0,∞) = ( ⋃_n=0^⌊ N^θ⌋ [nN,(n+1)N) )  ⋃  [N^1+θ,∞) ,

where θ >0 is a parameter. Our argument then splits into two separate parts.




On the long-time interval [N^1+θ,∞), we use the additional decay obtained through the physical randomization. The basic idea is that after such a long time, the linear evolution could only be concentrated through constructive interference of a large portion of the initial data, which is highly unlikely due to the physical randomness (see Figure <ref>). To make this rigorous, we prove an L_t^1 L_x^∞([N^1+θ,∞)×)-bound on P_N F, and this is sufficient to control the energy increment. This part of the proof requires the condition s > 1- θ/2. 


The majority of this paper focusses on time intervals such as [0,N). This part of the argument  does not rely on the physical randomness, and therefore also applies to the Wiener randomization. On this interval, we decompose the evolution into a family of wave packets, see Figure <ref>. As can be seen from a single wave packet, we cannot (always) control the evolution in L_t^1 L_x^∞. Instead, we use the following dichotomy: Either F consists of only a few wave packets, in which case its support lies on a few light-cones, or it consists of many wave packets, in which case the L_t^∞ L_x^∞-norm should be small.  



We now present a heuristic and simplified version of the main argument. 
In order to illustrate the ideas, let us first assume that all wave packets belong to a single frequency k∈. After a dyadic decomposition, we may further assume that all wave packets have amplitudes comparable to 2^m. Using the same notation as in Section <ref>, we denote the number of wave packets with this amplitude by # . Due to the L^2-orthogonality of the wave packets, we have that 2^m (#)^1/2≲ N^-s. 

In the case of only a few wave packets, we control the contribution on each tube separately. We have that 

    | ∫_0^N F_N v^2 ∂_t v | ≲ (#) N^1/2 2^m    ( sup_tubes T v_L_t^4 L_x^4(T)^2 ) ∂_t v _L_t^∞ L_x^2([0,N)×)≲ N^1/2 2^m #sup_t∈[0,N) E[v](t) .

The supremum ranges over all tubes of length ∼ N, width ∼ 1, and unit-speed direction inside  [0,N)×. Using a flux estimate and a bootstrap argument, we controll this supremum by the square-root of the energy. 

In the case of many wave packets (with the same direction), we use that their supports are disjoint, and obtain that 

    | ∫_0^N F_N v^2 ∂_t v |   ≲ N  F_N _L_t^∞ L_x^∞([0,N)×) v _L_t^∞ L_x^4([0,N)×)^2 ∂_t v _L_t^∞ L_x^2([0,N)×)≲ N2^m  sup_t∈[0,N) E[v](t) .

By combining both estimates, it follows that

    | ∫_0^N F_N v^2 ∂_t v |    ≲min(N^1/2 2^m #, N2^m) sup_t∈[0,N) E[v](t) 
    ≤ N^3/4 2^m (#)^1/2sup_t∈[0,N) E[v](t) .

We insert the bound 2^m (#)^1/2≲ N^-s, sum over N^θ intervals, and arrive at the condition s> 3/4+θ. In order to match the conditions from the intervals [nN,(n+1)N) and the long-time interval [N^1+θ,∞), we choose θ= 1/6, and obtain the regularity condition s> 11/12. 


In order to remove the restriction to a single frequency, we need to consider both multiple directions and multiple scales.  For this, we rely on techniques from the literature on the Kakeya and restriction conjectures. In order to control multiple directions, we use Bourgain's bush argument <cit.>. The basic idea is to distinguish points which lie in multiple tubes from points which lie only in a few tubes. To this end, we group the wave packets into several bushes and a collection of (almost) non-overlapping wave packets (see Figure <ref>). We then almost argue as for a single frequency, but also use that each bush lies on the surface of a light-cone, which is crucial for the flux estimate. In order to control multiple scales, we rely on Wolff's induction on scales strategy <cit.>. To fix ideas, let us try to bound the energy increment E[v](N) -E[v](0). We have already described the estimates for wave packets of length greater than or equal to N, but the space-time region [0,N] × also contains many shorter wave packets. By induction on scales, we can already close the bootstrap argument at these shorter scales, which greatly reduces the complexity of the proof.  We postpone a more detailed discussion to the Sections <ref> and <ref>.









  
Acknowledgements:  
I want to thank my advisor Terence Tao for his invaluable guidance and support. In particular, he proposed the greedy selection algorithm in Section <ref>. Furthermore, I want to thank Rowan Killip and Monica Visan for several interesting discussions. 
The figures in this paper have been created using TikZ and GeoGebra. 






§ NOTATION AND PRELIMINARIES



For the rest of this paper, the positive integer d denotes the dimension of physical space. In the analysis of the nonlinear evolution, we will eventually specialize to d=4. Furthermore, we also fix positive absolute constants δ, θ, and η. The parameter δ will be used to deal with the spatial tails of the wave packets and certain summability issues. The parameter θ is used in the division of time (see (<ref>)). We will eventually choose θ= 1/6, but prefer to keep θ as a free parameter until the end of the argument. Finally, η describes the size of the frequency truncated data, see Proposition <ref>. 

If A, B are positive quantities, we write A ≲ B if and only if there exist a constant C=C(δ,θ) such that A ≤ CB. Furthermore, most capital letters, such as N, M, and R, will denote dyadic numbers greater than or equal to 1. 

Finally, we define the Fourier transform f of an integrable function f →ℂ by 

    f(ξ) := 1/(2π)^d/2∫_exp(-ixξ) f(x)  .

We now summarize a few basic results from probability theory, harmonic analysis, and dispersive partial differential equations. 




 §.§ Probability theory



We recall a few basic estimates for sub-gaussian random variables. For an accessible introduction, we refer the reader to <cit.>. 


Let (Ω,ℱ,ℙ) be a probability space, and let X  (Ω,ℱ)→ be a random variable. We then define the sub-gaussian norm by

    X _Ψ_2 := sup_p≥ 1(𝔼[|X|^p])^1/p/√(p)

We call a random variable X sub-gaussian if and only if X _Ψ_2 <∞. Furthermore, we call a family of random variables { X_i }_i∈ I uniformly sub-gaussian if and only if sup_i∈ I X_i _Ψ_2 <∞. 

The relationship to the Gaussian distribution may not be obvious from (<ref>). However, it follows from <cit.> that (<ref>) implies  

    ℙ(|X|> x) ≤ 2 exp( - c x^2/ X_Ψ_2^2 )     ∀  x > 0  .

Many concentration inequalities for the sums of independent sub-gaussian random variables can be found in the literature. In the following, we mainly rely on Khintchine's inequality. 

Let  (X_j)_j=1,,J be a finite sequence of independent sub-gaussian random variables with zero mean. Then, it holds for all deterministic sequences (a_j)_j=1,,J, and all p ≥ 1, that 

    ( 𝔼[  |∑_j=1^J a_j X_j|^p ] )^1/p≲√(p)( max_j=1,,J X_j _Ψ_2) ( ∑_j=1^J |a_j|^2 )^1/2

In particular, the sum ∑_j=1^J a_j X_j is sub-gaussian. 

In this paper, Khintchine's inequality will often be combined with Minkowski's integral inequality, which we recall below.

Let (X,μ) and (Y,ν) be two σ-finite measure spaces, and let 1 ≤ p ≤ q ≤∞. Then, we have for all measurable functions 
f  X× Y → that 

    f(x,y) _L^p(X)_L^q(Y)≤ f(x,y) _L^q(Y)_L^p(X) .


The special case p=1 is the standard Minkowski inequality, and it can be found in most real analysis books (see e.g. <cit.>). Since Lemma <ref> is central to many arguments in this paper, we prove the general statement from this special case. 

Since q/p ≥ 1, we have that 

    f(x,y) _L^p(X)_L^q(Y) =  f(x,y)^p _L^1(X)_L^q/p(Y)^1/p≤ f(x,y)^p _L^q/p(Y)_L^1(X)^1/p =  f(x,y) _L^q(Y)_L^p(X) .



We will also need  a crude bound on the maximum of dependent sub-gaussian random variables 

Assume that (X_j)_j=1,,J are (possibly dependent) sub-gaussian random variables. Then, 

    𝔼[ max_j=1,,J |X_j| ] ≲√(log(2+J))max_j=1,,J X_j _Ψ_2 .



Let 1 ≤ p < ∞. Using Hölder's inequality, we obtain that

    𝔼[ max_j=1,,J |X_j| ]  ≤( 𝔼[ max_j=1,,J |X_j|^p ] )^1/p≤( ∑_j=1^J 𝔼[ |X_j|^p ] )^1/p≤ J^1/p√(p)max_j=1,,J X_j _Ψ_2 .

After choosing p= log(2+J), we arrive at (<ref>). 




 §.§ Harmonic analysis


Let N ∈ 2^ℕ_0 and k ∈. As in the introduction, we let φ∈ C^∞_c() be a smooth and symmetric function satisfying φ|_[-3/8,3/8]^d =1, φ|_\ [-5/8,5/8]^d = 0, and ∑_k ∈φ(· - k ) = 1. We also define ψ(ξ) = φ(ξ) - φ(2ξ). 
Then, the re-centered Littlewood-Paley operators are defined as 


    P_N;k f(ξ) = ψ(ξ-k/N) f(ξ)    if N > 1
    φ(ξ-k)  f(ξ)    if N =1  .

With this choice of ψ, it holds that P_N;k P_M;k = 0 if M≥ 4N or N ≥ 4M. 
To simplify the notation, we also set  P_N := P_N;0 and P_k := P_1;k. Furthermore, we define the fattened Littlewood-Paley operators 

    P_N;k:= ∑_2^-10 N≤ M ≤ 2^10 N  P_M;k .



Let N ∈ 2^ℕ_0, k ∈, and 1 ≤ p ≤ q ≤∞. Then, we for all f ∈ L_x^p() that 

    P_N;k f _L_x^q()   ≲ N^d( 1/p - 1/q )  P_N;k f _L_x^p() , 
     |∇| P_N;0 f _L_x^p()   ≲ N  P_N;0 f _L_x^p() .


We emphasize that the constant in  (<ref>) is independent of k ∈, since the phase exp(ikx) does not affect the L_x^p-norms. 



We also record the following standard consequence of Bernstein's inequality and the uncertainty principle.  


Let N ≥ 1, let a,b ∈, and assume that b≥ a+1/N. Then, we have for all f ∈ L_x^2(), all 1 ≤ q < ∞, and all 1 ≤ p ≤∞, that 

    exp(± it|∇|) P_N f _L_t^∞ L_x^p([a,b]×)≲ N^1/qexp(± it|∇|) P_N f _L_t^q L_x^p([a,b]×) .


The argument is essentially taken from <cit.>. 


Pick t_0 ∈ [a,b], and let I be any interval such that t_0 ∈ I ⊆ [a,b]. For all t ∈ I, it holds that 


    exp(± it_0 |∇|) P_N f^ω = exp(± it |∇|) P_N f^ω± i ∫_t^t_0 |∇| exp(± it^' |∇|) P_N f^ω

From Bernstein's inequality, we obtain that

    exp(± it_0 |∇|) P_N f^ω_L_x^p()   ≤exp(± it |∇|) P_N f^ω_L_x^p()  +  |I|^1-1/q |∇| exp(± it |∇|) P_N f^ω_L_t^q L_x^p(I×)
       ≲exp(± it |∇|) P_N f^ω_L_x^p()  +  N |I|^1-1/qexp(± it |∇|) P_N f^ω_L_t^q L_x^p(I×)

Taking the q-th power and averaging over all t ∈ I, we obtain that 

    exp(± it_0 |∇|) P_N f^ω_L_x^p()≲( |I|^-1/q + N |I|^1-1/q) exp(± it |∇|) P_N f^ω_L_t^q L_x^p(I×) .

By choosing |I|= N^-1, and taking the supremum over all t_0 ∈ [a,b], it follows that 

    exp(± it |∇|) P_N f^ω_L_t^∞ L_x^p([a,b]×)≲ N^1/qexp(± it |∇|) P_N f^ω_L_t^q L_x^p([a,b]×)



The following estimate, which appeared in the almost sure scattering problem for the nonlinear Schrödinger equation <cit.>, is useful in combination with Khintchine's inequality. 

Let f ∈ L_x^2() and let φ be as above. Then, it holds that 

    ∑_k∈ |P_k f(x)|^2 ≲ (|φ| * |f|^2)(x) .



In addition to the dyadic decomposition in frequency, we also need a dyadic decomposition in physical space. To avoid confusion, we denote the cut-off function in physical space by χ. More precisely, we set 
χ_1(x):= φ(x) and χ_L(x):= ψ(x/L), where L ≥  2. 

Let 1 ≤ p ≤∞ and M, N,L ≥ 1. We further assume the mismatch conditions max(M/N,N/M) ≥ 8 and L ≥ 8. Then, it holds for all absolute constants D  > 0 that 

    χ_1 P_M χ_L _L_x^p → L_x^p≲_D (ML)^-D  ,
     P_N χ_1 P_M _L_x^p→ L_x^p≲_D (NM)^-D .



The inequality (<ref>) can be found in <cit.>. 
An inequality similar to (<ref>) can be found in <cit.>, and we present a different argument. 

Using duality and (P_N χ_1 P_M)^∗ = P_M χ_1 P_N, we can assume that N ≥ M. From the mismatch condition, it then follows that N ≥ 8 M. Thus, we obtain for all f ∈ L_x^p() that 

    P_N( χ_1 P_M f) _L_x^p =  P_N( (P_≥ N/8χ_1) P_Mf ) _L_x^p≲ P_≥ N/8χ_1 _L_x^∞ P_M f _L_x^p≲ N^-2D f _L_x^p .



The following auxiliary lemma will be helpful in the proof of probabilistic Strichartz estimates. 
 
Let s ∈ and let f ∈ H_x^s(). For any 2 ≤ p ≤∞, we have that 


    P_k( φ_l f) _ℓ_l^2 ℓ_k^2 L_x^p^'(×{ k ∈ k _∞∈ (N/2,N]}×)≲P_N f _L_x^2() + N^-s-10d f _H_x^s()




The error term N^-s-10d f _H_x^s() is a result of the non-compact support of φ_l, but may essentially be ignored. On a heuristic level, each P_k (φ_l f) is supported on a spatial region of volume ∼ 1, and thus (<ref>) should follow from Hölder's inequality. To make this argument rigorous, we use the square-function estimate and the mismatch estimates above. 


 Let P_N be the fattened Littlewood-Paley operator as in (<ref>). We write M  N if either M< 2^-10 N or M > 2^10 N. In the following, we implicitly assume that k _∞∈ (N/2,N]. We then estimate

    P_k( φ_l f) _ℓ_l^2 ℓ_k^2 L_x^p^'≤ P_k( φ_l P_N f ) _ℓ_l^2 ℓ_k^2 L_x^p^' +  ∑_M  N  P_k( φ_l P_M f) _ℓ_l^2 ℓ_k^2 L_x^p^' .

We begin by  controlling the first summand in (<ref>). Using Minkowski's integral inequality and the square-function estimate (Lemma <ref>), we obtain that

    P_k ( φ_l P_N f ) _ℓ_l^2 ℓ_k^2 L_x^p^' 
    = φ_l^' P_k ( φ_l P_N f ) _ℓ_l^2 ℓ_k^2 ℓ_l^'^1 L_x^p^'≲φ_l^' P_k ( φ_l P_N f ) _ℓ_l^2 ℓ_k^2 ℓ_l^'^1 L_x^2≲φ_l^' P_k ( φ_l P_N f ) _ℓ_l^2 ℓ_l^'^1 L_x^2ℓ_k^2
       =φ_l^'( |φ| * |φ_l P_N f|^2 )^1/2_ℓ_l^2 ℓ_l^'^1 L_x^2 .

Using simple support considerations, we have that

    φ_l^'( |φ| * |φ_l P_N f|^2 )^1/2_L_x^2^2 
    ≤φ_l^'( |φ| * |φ_l P_N f|^2 ) _L_x^1≲⟨ l^' - l ⟩^-10d (φ_l P_N f)^2 _L_x^1
       ≲⟨ l^' - l ⟩^-10dφ_l P_N f_L_x^2^2 .

Inserting this back into (<ref>), we obtain that 

    P_k ( φ_l P_N f ) _ℓ_l^2 ℓ_k^2 L_x^p^'≲⟨ l^' - l ⟩^-5dφ_l P_N f_ℓ_l^2 ℓ_l^'^1 L_x^2≲P_N f _L_x^2 .

Thus, this yields the first term in (<ref>). We now control the second summand in (<ref>). First, note that P_k = ∑_2^-5N ≤ N^'≤ 2^5 N P_N^' P_k. Since there exist only ∼ N^d frequencies of magnitude ∼ N, we have that 

    P_k ( φ_l P_M f ) _ℓ_k^2 ℓ_l^2 L_x^p^'   ≲ N^d/2∑_2^-5N≤ N^'≤ 2^5 N P_N^' (φ_l P_M f) _ℓ_l^2 L_x^p^' .

It now suffices to prove for all g ∈ S(), all M  N, and all absolute constants D  > 0 that 

    P_N^'  ( φ_l P_M g) _L_x^p^'≲_D (NM)^-D⟨ x-l ⟩^-D g _L_x^2 .

Using spatial translation invariance, we may set l = 0. Let {χ_L}_L≥ 1 denote the dyadic decomposition in physical space. Using the mismatch estimates (Lemma <ref>), we obtain 

    P_N^' (φ_0 P_M g) _L_x^p^'≤∑_L≥ 1P_N^'  (φ_0 P_M  χ_L g) _L_x^p^'≤∑_L≥ 1 P_N^'φ_0 P_M χ_L _L_x^p^'→ L_x^p^'χ_L g _L_x^p^'
       ≲ (NM)^-D∑_L≥ 1 L^-2Dχ_L g _L_x^p^'≲ (NM)^-D⟨ x ⟩^-D g _L_x^2 .



As a direction consequence of (<ref>), we also obtain the following estimate on the H^s-norm of the microlocal randomization.

Let f ∈ H_x^s() and let f^ω be its microlocal randomization. We further set 

    f_1^ω := ∑_l∈  X_0,l P_0(φ_l f)     and     f_N^ω := ∑_k,l∈
     k_∞∈ (N/2,N] X_k,l P_k(φ_l f) ,   where N ≥ 2 .

Then, we have for all 2 ≤ r < ∞ that 

    f^ω_L_ω^r H_x^s≃ N^s f_N^ω_L_ω^r ℓ_N^2 L_x^2≲√(r) f _H_x^s .



The first equivalence in (<ref>) is a direct consequence of the definition of the H_x^s-norm. Now, we prove the bound in (<ref>). From Minkowski's integral inequality, Khintchine's inequality, and Lemma <ref>, we have for all N ≥ 2 that 

    N^s f_N^ω_L_ω^r  L_x^2   ≤ N^s  ∑_ k _∞∈ (N/2,N] X_k,l P_k(φ_l f ) _L_x^2 L_ω^r
       ≤√(r) N^s  P_k(φ_l f)_L_x^2 ℓ_k,l^2(k_∞∈ (N/2,N]
       ≲√(r)( N^s P_N f _L_x^2() + N^-10d f _H_x^s())

The same argument also applies to N=1. After taking the ℓ_N^2-norm, this completes the proof. 





 §.§ Strichartz estimates

The individual blocks in the microlocal randomization or the Wiener randomization have frequency support inside a unit-sized cube (at a large distance from the origin). Since this rules out the Knapp example, one expects a refined dispersive estimate. The following lemma is due to Klainerman and Tataru <cit.>, and it has first been used in the probabilistic context by <cit.>. 


Let f ∈ L^1(^d), let k ∈ℤ^d satisfy k _∞∈ (N/2,N], and let M ≤ N.  Then it holds for all t ∈ and  2 ≤ p ≤∞ that 

    P_M;k f _L_x^p(ℝ^d)≲M^d (1-2/p)/(1+M^2/N |t|)^(d-1)(1/2-1/p) f _L_x^p^'(^d) .


As stated, the inequality (<ref>) essentially follows from <cit.>. For the sake of completeness, we present the modification below. 
 By interpolation against the energy estimate P_M;k f _L_x^2(ℝ^d)≤ f_L_x^2(^d), it suffices to prove (<ref>) for p=∞. 
The inequality <cit.>, where μ = M/N,  and a scaling argument yield

    P_M;k f _L_x^∞(ℝ^d)≲MN^d-1/(1+N|t|)^d-1/2 f _L_x^1(^d) .

We now distinguish two cases. If |t|≲ N/M^2, then Bernstein's inequality (Lemma <ref>) yields that 

    P_M;k f _L_x^∞(ℝ^d)≲  M^d/2  P_M;k f _L_x^2(ℝ^d) = M^d/2  P_M;k f _L_x^2(ℝ^d)≲  M^d  P_M;k f _L_x^1(ℝ^d) .

If |t|≲ N/M^2, then (<ref>) yields that 

    P_M;k f _L_x^∞(ℝ^d)≲MN^d-1/(N|t|)^d-1/2 f _L_x^1(^d)  = M^d/(M^2/Nt)^d-1/2 f _L_x^1(^d)≲M^d/(1+ M^2/Nt)^d-1/2 f _L_x^1(^d) .



In this paper, we are mainly concerned with the case M= 1. Then, (<ref>) describes the linear evolution on short time intervals more accurately than  (<ref>). 
As a corollary of the refined dispersive estimate, we obtain the following refined Strichartz estimate. 

Let 2 ≤ q,p ≤∞. We call the pair (q,p) wave-admissible if 

    1/q + d-1/2p≤d-1/4    and     (q,p,d) ≠ (2,∞,3) .



Let f ∈ L_x^2(), let k ∈ℤ^d satisfy k _∞∈ (N/2,N], and let M≤ N. Then, we have for all wave-admissible pairs (q,p) that 

    exp(± it|∇|) P_M;k f _L_t^q L_x^p(×)≲ M^d/2-2/q-d/p N^1/q P_M;k f_L_x^2()


The derivation of the refined Strichartz estimate from Lemma <ref> follows from a standard TT^∗-argument, and we therefore omit the proof. For the endpoint (2,2 (d-1)/(d-3)), we also refer to <cit.>. Let us emphasize two special cases: If M= N, we obtain the usual scaling factor N^d/2-1/q-d/p, and if M=1, we obtain the factor N^1/q, which does not depend on p. 




§ PROBABILISTIC STRICHARTZ ESTIMATES


In this section, we derive probabilistic Strichartz estimates (cf. <cit.>) and a probabilistic long-time decay estimate (cf. <cit.>). To keep the exposition self-contained, we include the (short) proofs. Recall from (<ref>) that

    f_1^ω := ∑_l∈  X_0,l P_0(φ_l f)     and     f_N^ω := ∑_k,l∈
     k_∞∈ (N/2,N] X_k,l P_k(φ_l f) ,   where N ≥ 2 .



Let f ∈ H_x^s() and let f^ω be its microlocal randomization. Then, it holds for all N ≥ 1, all wave-admissible exponent pairs (q,p), and all 1 ≤ r < ∞ that 

    exp(± it |∇|) f_N^ω_L_ω^r L_t^q L_x^p(Ω××)≲√(r) N^1/q-s+ f _H_x^s() .


This estimate has previously appeared for the Wiener randomization in <cit.>. 

 In the following, we implicitly assume that k ∈ always satisfies k _∞∈ (N/2,N]. 
First, we assume that 2 ≤ p,q < ∞, and that max(p,q)≤ r < ∞.  Using Minkowski's integral inequality (Lemma <ref>), Khintchine's inequality (Lemma <ref>), and the refined dispersive estimate (Lemma <ref>), we have that 

    exp(± it |∇|) f_N^ω_L_ω^r L_t^q L_x^p≤exp(± it |∇|) f_N^ω_L_t^q L_x^p L_ω^r≲√(r)exp(± it |∇|) P_k(φ_l f) _L_t^q L_x^p ℓ_k,l^2
       ≤√(r)exp(± it |∇|) P_k(φ_l f) _ℓ_k,l^2 L_t^q L_x^p≲√(r) N^1/q P_k(φ_l f) _ℓ_k,l^2 L_x^2≲√(r) N^1/q-s f _H_x^s .

In the last inequality, we have also used Lemma <ref>. The estimate for 1 ≤ r ≤max(p,q) then follows from Hölder's inequality. Thus, it remains to treat the cases q=∞ and/or p=∞. This is a know technical issue, see <cit.> for a discussion. Both cases can be reduced to the previous estimate by using Lemma <ref> and Bernstein's inequality.  




Let f ∈ L_x^2(ℝ^d) and let f^ω be its microlocal randomization. Furthermore, let 1 ≤ q < ∞ and 2 ≤ p ≤∞ be such that 

    1/q + d-1/p < d-1/2 .

Then, we have for all 1 ≤ r < ∞ that

    f_N^ω_L_ω^r L_t^q L_x^p( Ω× [T,∞) ×^d)≲_q,p√(r) N^1/q-s+( 1 + T/N)^1/q+d-1/p-d-1/2 f_H_x^s().



Lemma <ref> has previously been used for a physical space randomization in <cit.>. In contrast to the standard Strichartz estimates, which are time-translation invariant, (<ref>) provides a quantitative decay rate. The motivation behind this estimate is illustrated in Figure <ref>. In this paper, we only require the following special case. 

Let f ∈ L_x^2(ℝ^4) and let f^ω be its microlocal randomization. Then, we have for all θ >0 that 

    f_N^ω_L_ω^r L_t^1 L_x^∞(Ω× [N^1+θ,∞) ×^4)≲√(r) N^1-θ/2+ f_N _L_x^2() .







Due to (<ref>), the L_t^1 L_x^∞-estimate fails logarithmically in three dimensions. 



We essentially follow the argument in <cit.>. Let us first assume that 2≤ q,p < ∞. 

We further assume that r ≥max(q,p), the corresponding estimate for 1≤ r < max(q,p) then follows from Hölder's inequality. Using Minkowski's integral inequality (Lemma <ref>), Khintchine's inequality (Lemma <ref>), and the refined dispersive estimate (Lemma <ref>), we have that 

    f_N^ω_L_ω^r L_t^q L_x^p( Ω× [T,∞) ×^d)[1] 
       ≤ f_N^ω_L_t^q L_x^p L_ω^r(  [T,∞) ×^d ×Ω)[1]
       ≲√(r) P_k(φ_l f) _L_t^q L_x^p ℓ_k,l^2([T,∞)×^d ×ℤ^d+d)[1]
       ≲√(r) P_k(φ_l f) _ℓ_k,l^2 L_t^q L_x^p (ℤ^d+d× [T,∞)×^d )[1]
       ≲√(r)( 1+ |t|/N)^-(d-1) (1/2-1/p) P_k(φ_l f) _L_x^p^'_ℓ_k,l^2 L_t^q(ℤ^d+d× [T,∞))[1] 
       ≲√(r)( 1+ |t|/N)^-(d-1) (1/2-1/p)_L_t^q([T,∞)) P_k(φ_l f) _ℓ_k,l^2 L_x^p^'(ℤ^d+d×^d) .

Using condition (<ref>), we obtain

    ( 1+ |t|/N)^-(d-1) (1/2-1/p)_L_t^q([T,∞))≲ N^1/q( 1 + T/N)^1/q+d-1/p-d-1/2

 Finally, from Lemma <ref> we have that 
 
    P_k(φ_l f) _ℓ_k,l^2 L_x^p^'(ℤ^d+d×^d)≲ N^-s f_H_x^s .

 This finishes the proof in the case 2 ≤ q,p < ∞. Using Bernstein's inequality, we can reduce the case p = ∞ to p< ∞.  Thus, it remains to treat the range 1 ≤ q < 2. Using a dyadic decomposition in time, we have for all T ≥ N that 
 
    f_N^ω_L_ω^r L_t^q L_x^p( Ω× [T,∞) ×^d)[1] 
       ≤∑_m=0^∞ f_N^ω_L_ω^r L_t^q L_x^p( Ω× [2^m T,2^m+1T) ×^d)[1]
       ≤∑_m=0^∞  (2^m T)^1/q-1/2 f_N^ω_L_ω^r L_t^2 L_x^p( Ω× [2^m T,2^m+1T) ×^d)[1]
       ≲∑_m=0^∞ (2^m T)^1/q-1/2  N^1/2-s+( 2^m T/N)^1/2+d-1/p-d-1/2 f_H_x^s[1]
       ≲ N^1/q-s+(T/N)^1/q+d-1/p-d-1/2 f_H_x^s .

 In the second last line, we used condition (<ref>). For T ≤ N, we also have that 
 
    f_N^ω_L_ω^r L_t^q L_x^p( Ω× [T,N) ×^d)≲ N^1/q- 1/2 f_N^ω_L_ω^r L_t^2 L_x^p( Ω× [0,N) ×^d)≲  N^1/q-s+ f_H_x^s .





Let 0≤ s < 1, let (f_0,f_1)∈ H^s()× H^s-1(), and let N_0≥ 1.  We then define

    (f_0,f_1) _Z(N_0)   := ∑_N≥ N_0 N^s+θ/2-1-δcos(t|∇|)  f_0,N + sin(t|∇|)/|∇|  f_1,N_L_t^1 L_x^∞([N^1+θ,∞)×)
        +∑_N≥ N_0 N^s-δcos(t|∇|) f_0,N + sin(t|∇|)/|∇| f_1,N_L_t^∞ L_x^∞([0,∞)×)



From Proposition <ref> and Corollary <ref>, it follows that 

    (f_0^ω,f_1^ω) _L_ω^r Z(1)≲√(r) (f_0,f_1) _H_x^s× H_x^s-1 .





§ WAVE PACKET DECOMPOSITION

In this section, we use a wave packet decomposition to better understand the (random) linear evolution. 

This part of the argument does not rely on the additional randomization in physical space. We therefore phrase all results in a way that applies to both the microlocal and the Wiener randomization, and hope that this 
facilitates future applications. With this in mind, we now rewrite the microlocal randomization in a form that resembles the Wiener randomization. 

Let the random variables { X_k,l}_k,l ∈ be as in Definition <ref>, let {ϵ_k }_k∈ I∪{ 0 } be a family of independent random signs, and set ϵ_-k = ϵ_k for all k ∈ I. We can then define Y_k,l := ϵ_k X_k,l. For a sequence of multi-indices k,l_1,,l_J∈ and any sequence of Borel-measurable sets A_1,,A_J ⊆, we have that 

    ℙ(ϵ_k =1 , Y_k,l_1∈ A_1,,Y_k,l_J∈ A_J) =ℙ(ϵ_k =1 , X_k,l_1∈ A_1,,X_k,l_J∈ A_J)   
    
    = ℙ(ϵ_k =1) ∏_j=1^J ℙ(X_k,l_j∈ A_j)= ℙ(ϵ_k =1) ∏_j=1^J ℙ(Y_k,l_j∈ A_j)  .

In the last equality, we have used that the random variables X_k,l are symmetric. Therefore, for a fixed k ∈, the family {ϵ_k } ⋃ { Y_k,l}_l∈ is independent.  
From this, it then easily follows that the whole family {ϵ_k }_k∈ I ∪{ 0 } ⋃ { Y_k,l}_k ∈ I ∪{ 0 }, l∈ is independent. We then rewrite the microlocal randomization as 

    f^ω = ∑_k,l∈ X_k,l P_k(φ_l f) = ∑_k∈ϵ_k P_k( ∑_l∈ Y_k,lφ_l f ) =∑_k∈ϵ_k f_k,     where   f_k := P_k  (∑_l∈ Y_k,lφ_l f) .

Due to the independence properties discussed above, we can regard the functions { f_k } as deterministic by conditioning on the random variables { Y_k,l}_k,l, and only utilize the randomness through the random signs {ϵ_k }_k. Note that (<ref>) closely resembles the Wiener randomization.  

To motivate the wave packet decomposition below, we now rewrite the linear evolution with initial data (f_0^ω,f_1^ω). Using the notation from (<ref>), we first introduce the half-wave operators by writing

    cos(t|∇|) f_0^ω + sin(t|∇|)/|∇| f_1^ω
       = ∑_k∈ϵ_k (  cos(t|∇|) f_0;k^ω + sin(t|∇|)/|∇| f_1;k^ω) 
       = ∑_k∈ϵ_k  [ exp(it|∇|) ( f_0;k+ i |∇|^-1 f_1;k/2) + exp(-it|∇|) ( f_0;k- i |∇|^-1 f_1;k/2) ] 
       =: ∑_k∈ϵ_k [ exp(it|∇|) f_k^+ + exp(-it|∇|) f_k^- ] .

As in (<ref>), we also decompose dyadically in frequency space, and write 

    F_N^±:=  ∑_ k_∞∈ (N/2,N]ϵ_k exp(± it|∇|) f_k^±    and     F_N := F_N^+ + F_N^- .

Let k ∈ℤ^d with k_∞∈ (N/2,N], and let l ∈ℤ^d. We define the tubes T_k,l^± by

    T_k,l^+ := { (t,x)∈ [0,N] ×^d  x- ( l ∓ t · k/k_2 ) _2 ≤ 1 } .

Here, the superscripts ± are chosen so that the tubes correspond to the operators . The dimensions and the shape of the tubes are illustrated in the introduction, see Figure <ref>. Motivated by the Doppler effect, the tubes T_k,l^+ are sometimes called red tubes, and the tubes T_k,l^- are sometimes called blue tubes. 


Let k ∈ℤ^d with k_∞∈ (N/2,N]. Let f_k ∈ be a function such that f_k⊆ k + [-1,1]^d. Then, there exists a decomposition 

    f_k = ∑_l∈ℤ^d f_k,l

such that


  *   f_k,l⊆ k+[-4,4]^d for all l ∈ℤ^𝕕, 

  *  the family { f_k,l}_l∈ satisfies the almost-orthogonality condition
    ∑_l∈ℤ^d f_k,l_^2 ≲ f _^2 ,


  *  and for any D ≥ 1, any l ∈ℤ^d, and all (t,x)∈ [0,N]×^d, it holds that 

    | exp(± it|∇|) f_k,l(x) |≲_D (1+ ((t,x), T_k,l^±) )^-D f_k _ .




Wave packet decomposition as in Proposition <ref> have been used extensively in the literature, see e.g. <cit.> and the survey <cit.>. We present the details below, but encourage the expert reader to skip ahead to the end of the proof. 


We define the fattened projection

    P_k := ∑_k^' -k_∞≤ 2 P_k^' .

Then, it holds that 

    f= P_k f = ∑_l∈ℤ^dP_k( φ_l f ) =: ∑_l∈ℤ^d f_k,l .

The frequency support condition <ref> directly follows from the definition of P_k. Furthermore, the almost orthogonality <ref> follows from

    ∑_l∈ f_k,l_^2 ≲∑_l∈φ_l f_k_^2 ≲ f_k _^2 .

Thus, it remains to prove the decay estimate <ref>. We only treat the operator exp(it|∇|), since the proof for exp(-it|∇|) is similar. If N ≲ 1, the estimate is trivial.  Thus, we may assume that N ≫ 1. The argument is based on the method of non-stationary phase. For all t∈ [0,N] and x ∈, we have that 

    exp(it|∇|) f_k,l(x) 
       = 1/(2π)^d/2∫_exp(ixξ+ i t |ξ|) ψ(ξ-k) (φ_l * f)(ξ) 
       = 1/(2π)^d∫_∫_∫_exp(ixξ+ i t |ξ|) ψ(ξ-k) exp(-iy(ξ-η)) φ(y-l) f(η) 
       = 1/(2π)^dexp(ixk+it|k|) ∫_ K(η;t,x) exp(-il(k-η)) f(η)  ,

where the kernel K(η;t,x) is given by 

    K(η;t,x):= ∫_∫_exp(i(x-l) ξ+ i t (|ξ+k|-|k|))  exp(-iy(ξ+k-η)) ψ(ξ) φ(y)  .

Since φ⊆ [-1,1]^d, the function a(ξ;y,η):= exp(-iy(ξ+k-η)) ψ(ξ) φ(y) has uniformly bounded derivatives in ξ, i.e, we have for all α∈ℕ_0^d that |∂_ξ^α a(ξ;y,η)|≲_α 1. Using the support conditions in the variables y and η, it thus suffices to prove for all a ∈ C^∞_c([-2,2]^d) that

    | ∫_exp(i(x-l) ξ+ i t (|ξ+k|-|k|))  a(ξ) | ≲_M ( 1+ | x - l + t · k/ k_2 )^-M .

Due to the compact support of a(ξ), we restrict to |ξ|≤ 2. The bound for |x-l + t · k/k_2 |≲ 1  is trivial. Thus, we may assume that |x-l+ t · k/ k_2 |≫ 1. We define the phase function 

    Φ_k(ξ)= Φ_k(ξ;t,x,l)= i(x-l) ξ+ i t (|ξ+k|-|k|) .

Then, we have that 

    ∇_ξΦ_k(ξ)  
       =∇_ξ( (x-l) ξ + t ( |ξ+k| - |k|) ) 
       = x-l + t ξ+k/|ξ+k|
       = x-l + t k/|k| + t ( |k|-|ξ-k|/| ξ+k| |k|   k + ξ/|ξ+k|)  .

From the assumption |t|≤ N, it follows that ∇_ξΦ_k = x-l + t k/|k| + 𝒪(1). We also write 
    ∇_ξΦ_k (ξ)=  x-l + t k/|k| + t Ψ_k (ξ/|k|) k/|k| ,

 where 

    Ψ_k(ν):= 1-|ν-k/|k||/|ν+k/|k|| + ν/|k/|k|+ν| .

From rotation invariance, it follows easily that |∇^αΨ_k(ν)| ≲_α 1 for all |ν|≤ 1/10, uniformly in k. This leads to 
    |∇^αΦ_k(ξ)|≲_α|t|/|k|^|α|-1≲ 1     for all  |α|≥ 2 .

We then rewrite the integral in (<ref>) as 

    ∫_exp(i(x-l) ξ+ i t (|ξ+k|-|k|))  a(ξ) 
       = ∫_( ( (-i) ∇_ξΦ_k(ξ) ∇_ξ/|∇Φ_k(ξ)|^2)^Mexp(iΦ_k(ξ)) ) · a(ξ)
       = ∫_exp(iΦ_k(ξ))  ( ∇_ξ·i ∇_ξΦ_k(ξ)/|∇Φ_k(ξ)|^2)^M a(ξ)  .

The inequality (<ref>) then follows from the bounds on the phase function above. 



The wave packet decomposition in Proposition <ref> is valid on the time interval [0,N], and the physical localization deteriorates for larger times. When analyzing the linear evolution on an interval of the form [t_0,t_0+N), with t_0∈ Nℕ_0, we therefore use the wave packet decomposition of exp(± it_0 |∇|) f_k. To state the result, we set

    T_k,l;t_0^± := { (t,x) ∈ [t_0,t_0+N]× x- ( l ∓ (t-t_0) · k/k_2 ) _2 ≤ 1 } .



Let k ∈ℤ^d with k_∞∈ (N/2,N], and let t_0 ∈ N ℕ_0. Let f_k ∈ be a function satisfying f_k⊆ k + [-1,1]^d. Then, there exists a decomposition 

    exp( ± i t_0 |∇|) f_k = ∑_l∈ℤ^d f_k,l;t_0^±

such that


  * f_k,l;t_0^±⊆ k+[-2,2]^d for all l ∈ℤ^𝕕, 

  * the family { f_k,l;t_0^±}_l∈ satisfies the almost-orthogonality condition
    ∑_l∈ℤ^d f_k,l;t_0^±_^2 ≲ f _^2 ,


  * and for any D ≥ 1, any l ∈ℤ^d, and all (t,x)∈ [t_0,t_0+N]×^d, it holds that 

    | exp(± i(t-t_0)|∇|) f_k,l;t_0^±(x) |≲_D (1+ ((t,x), T_k,l;t_0^±) )^-D f_k _ .




We apply Proposition <ref> to exp(± it_0 |∇|) f_k. 




As discussed in the introduction, we now group the wave packets into bushes and a (nearly) non-overlapping collection (see Figure <ref>). This argument is inspired by Bourgain's bush argument from <cit.>, and we also refer the reader to <cit.>. 



Before we state main proposition, we define the truncated and fattened ℓ^∞-cone 

    := { (t,x) ∈ [t_0,t_0+N] ×x-x_0_∞≤ 16N-|t-t_0| } .

The significance of  will be explained in Section <ref> and Section <ref>. For now, we encourage the reader to treat  as space-time cube of scale N. 


Let { f_k^±}_k⊆ L_x^2() be a family of functions, where k _∞∈ (N/2,N], and f^±_k⊆ k + [-1,1]^d. Let t_0 ∈ Nℕ_0, let x_0 ∈ N, and let the wave packets { f_k,l;t_0^±} be as in . Furthermore, let  be a collection of disjoint space-time cubes with sidelength ∼ N^δ covering . 
We group the wave packets according to their amplitude by setting 

    =  := { (k,l) ∈× f_k,l;t_0^±_L_x^2()∈ [2^m,2^m+1],  l - x_0 _∞≤ 3N} .

Then, there exists a family of bushes {}_j = {}_j, where j=1,, J^N,±_m,t_0,x_0, and a nearly 
non-overlapping set =, depending only on the set , so that the following holds:


  *  The sets form a partition of , i.e.,  
    =  ∙⋃ ( ∙⋃_j=1,, J)


  *  We have the bound on the number of wave packets 
    ∑_x_0∈ N∑_m∈ℤ 2^2m#≲∑_k f_k_L_x^2()^2 .


  *  Each bush  contains at least μ=μ(N,m) :=N^-1/2# wave packets. 

  * For each bush , all corresponding wave packets intersect in the same region of space-time. More precisely, there exists a cube Q∈ s.t. 

    T_k,l;t_0^± ⋂  2Q ≠∅    ∀ (k,l) ∈ .


  *  At μ= N^-1/2# wave packets in  overlap, i.e.,  we have for all cubes Q ∈ that 

    #{ (k,l) ∈ T_k,l;t_0^± ⋂  2Q ≠∅} < P .
 



The choice of the number of packets/multiplicity μ=N^-1/2# will be justified in the proof of Proposition <ref>, see (<ref>). The parameter μ corresponds to the multiplicity parameter in Bourgain's bush argument, see <cit.>. 


We will later apply this proposition to a set of random functions {ϵ_k f_k^±}_k. From (<ref>), it follows that the sets , and hence also  and , do not depend on the random signs {ϵ_k }. 



Let us first prove the inequality in <ref>. From Corollary <ref>, it follows that

    ∑_x_0∈ N∑_m∈ℤ 2^2m#≲∑_ k,l∈
     k_∞∈ (N/2,N] f_k,l;t_0^±_L_x^2^2 ≲∑_k∈
     k_∞∈(N/2,N] f_k^±_L_x^2^2 .


We now construct the sets  and . 
To simplify the expressions, we drop the super- and subscripts ±, N, t_0, and x_0 from our notation. The basic idea is to form the bushes through a greedy selection algorithm. For any Q ∈, we define

    := { (k,l)∈ T_k,l ⋂  Q ≠∅} .

We further set 𝒯_m^(1)(Q) :=. We then choose a cube Q_1 ∈ such that 

    #𝒯_m^(1)(Q_1) = max_Q∈#𝒯_m^(1)(Q) ,

and define the first bush as ℬ_1,m :=  𝒯_m^(1)(Q_1). By setting 

    𝒯_m^(2)(Q) = 𝒯_m^(1)(Q)\ℬ_1,m ,

we remove all of the wave packets in the first bush from the collection. We then iteratively define := 𝒯^(j)_m(Q_j), where 

    #𝒯_m^(j)(Q_j) = max_Q∈#𝒯_m^(j)(Q) ,

and the collections 𝒯_m^(j)(Q) are defined as 𝒯_m^(j-1)(Q)\. Once 𝒯_m^(j+1)(Q_j+1) < μ, we no longer create a new bush, and instead stop the algorithm. Since  contains at most ∼ N^8 wave packets, the greedy selection algorithm terminates after finitely many steps. From the construction, we see that the sets ⊆×  are disjoint (even though the corresponding tubes may still overlap). Finally, we define the collection  by 
    := \⋃_j=1^J  .

The properties <ref>, <ref>, <ref>, and <ref> then follow directly from the construction. 


We now prove a probabilistic estimate for the wave packets with random coefficients. 


Let { f_k^±}_k⊆ L_x^2() be a deterministic family of functions, where k ∈ satisfies k _∞∈ (N/2,N], and assume that ∑_k f_k^±_L_x^2()^2 ≲ 1. Let θ > 0 be a parameter, and let C_d > 0 be any large absolute constant. Then, we have for all m ∈ℤ satisfying - C_d log(N) ≤ m ≤ C_d log(N) that 

    𝔼[ sup_ t_0=0,⌊ N^θ⌋ N 
     x_0∈ N sup_j=1,, J^N,±_m,t_n,x_0∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_t,x^∞(×)/2^m (#)^1/2] 
    ≲ N^δ d

and

    𝔼[ sup_t_0=0,⌊ N^θ⌋ N 
     x_0∈ N∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_t^∞ L_x^∞([t_0,t_0+N]  ×)/2^m μ^1/2] 
    ≲ N^δ d .

Here, μ= N^-1/2# is as in Proposition <ref>. To be perfectly precise, we use the convention 0/0:= 0 in (<ref>).

The expressions in (<ref>) and (<ref>) may seem complicated. To make sense of them, recall that the square function heuristic predicts that ∑_k ϵ_k a_k is roughly of size ∼ (∑_k a_k^2 )^1/2. Then, Proposition <ref> simply states that the square function heuristic can be justified for all relevant amplitudes, for all relevant times, all positions, all families of bushes, and all non-overlapping collections.  

For instance, let us heuristically motivate (<ref>). By the definition of , any fixed point in the space-time region [t_0,t_0+N] × is contained in the (moral) support of at most μ wave packets. Since each of the wave packets has amplitude ∼ 2^m, and they all correspond to different frequencies k ∈, the square-function heuristic predicts a contribution of size ∼ 2^m μ^1/2. 

In this proof, we make extensive use of Lemma <ref>. First, we prove that the suprema in (<ref>) and (<ref>) are over at most N^𝒪(C_d)-many terms. From (<ref>), it follows for all m≥ -C_d log(N) that 

    ∑_x_0∈ N#≲ 2^-2m∑_k,l f_k,l^±_^2 
    ≲ 2^-2m∑_k f_k^±_^2 
    ≲ N^2C_d .

Thus, this bounds the number of all wave packets with amplitude ∼ 2^m. 
Since each bush  contains at least one wave packet, the supremum in (<ref>) is over at most 𝒪(N^2C_d)  non-zero terms. The same applies to the non-overlapping families  in (<ref>). From Lemma <ref>, it then suffices to obtain uniform sub-gaussian bounds on each individual term in  (<ref>) and (<ref>). 

We start with the contribution of the bushes. To simplify the notation, we write =. From Bernstein's inequality and Lemma <ref>, we have for all 2 ≤ p < ∞ that 

    ∑_(k,l)∈ϵ_k exp(± i(t-t_0)|∇|) f_k,l;t_0^+ _L_t^∞ L_x^∞(×)
       ≤ N^d+1/p∑_(k,l)∈ϵ_k exp(± i(t-t_0)|∇|) f_k,l;t_0^+ _L_t^p L_x^p(×) .

Before we utilize the randomness, we observe that for each k ∈ at most 𝒪(N^δ d) tubes T_k,l;t_0^± can intersect a space-time cube of sidelength ∼ N^δ. As a result, it follows from (<ref>) that 

    #{ l∈ (k,l)∈}≲ N^δ d .
 
For all p≤ r < ∞, we then obtain from Minkowski's integral inequality, Khintchine's inequality, and the refined Strichartz estimate (Corollary <ref>) that

    ∑_(k,l)∈ϵ_k exp(± i(t-t_0)|∇|) f_k,l;t_0^+ _L_ω^rL_t^∞ L_x^∞(Ω××)
       ≲ N^d+1/p∑_(k,l)∈ϵ_k exp(± i(t-t_0)|∇|) f_k,l;t_0^+ _L_t^p L_x^pL_ω^r(××Ω)
       ≲√(r) N^d+1/p( ∑_k∈(∑_l (k,l)∈exp(± i(t-t_0)|∇|) f_k,l;t_0^+ )^2 )^1/2_L_t^p L_x^p(×)
       ≲√(r) N^d+1/p+ δ d/2exp(± i(t-t_0)|∇|) f_k,l;t_0^+  _L_t^p L_x^p ℓ_k,l^2(××)
       ≲√(r) N^d+1/p+ δ d/2exp(± i(t-t_0)|∇|) f_k,l;t_0^+  _ℓ_k,l^2 L_t^p L_x^p (××)
       ≲√(r) N^d+2/p+ δ d/2 f_k,l;t_0^+ _ℓ_k,l^2 L_x^2(×)
       ≲√(r) N^d+2/p + δ d/2 2^m (#)^1/2 .

By taking p≥ 2 to be sufficiently large, we then obtain the desired sub-gaussian bound. This completes the proof of (<ref>). 

We now control the contribution of a single non-overlapping family =. For the technical aspects of this part, recall that the collection  from Proposition <ref> covers , but due the definition of  in (<ref>), all the tubes T_k,l;t_0^± with indices in  are contained in the region x- x_0 _∞≤ 6N. This gives us sufficient room for the following argument. 

We let 2 ≤ p < ∞. As before, it follows from Bernstein's inequality and Lemma <ref> that 

    ∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_t^∞ L_x^∞([t_0,t_0+N] ×)
       ≲ N^d+1/p∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_t^p L_x^p([t_0,t_0+N]  ×)

For all p ≤ r <∞, we then obtain from Minkowski's integral inequality and Khintchine's inequality that

    ∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_ω^r L_t^∞ L_x^∞(Ω× [t_0,t_0+N]  ×)[1] 
       ≲ N^d+1/p∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_t^p L_x^p L_ω^r([t_0,t_0+N]  ××Ω)[1] 
       ≲√(r) N^d+1/p( ∑_k∈( ∑_l (k,l)∈exp(± i (t-t_0)|∇|) f_k,l;t_0^±)^2 )^1/2_L_t^p L_x^p([t_0,t_0+N]  ×)[1] 
       ≲√(r) N^d+1/p( ∑_k∈( ∑_l (k,l)∈exp(± i (t-t_0)|∇|) f_k,l;t_0^±)^2 )^1/2_L_t^p L_x^p()
          + √(r) N^d+1/p( ∑_k∈( ∑_l (k,l)∈exp(± i (t-t_0)|∇|) f_k,l;t_0^±)^2 )^1/2_L_t^p L_x^p(([t_0,t_0+N]  ×)\)

Since μ = N^-1/2#≥ N^-1/2, the bound on ([t_0,t_0+N]  ×)\ easily follows from the decay estimate (<ref>).  Thus, we now control the contribution on . From Hölder's inequality, we have that 

    ( ∑_k∈( ∑_l (k,l)∈exp(± i (t-t_0)|∇|) f_k,l;t_0^±)^2 )^1/2_L_t^p L_x^p()
       ≲ N^d+1/p( ∑_k∈( ∑_l (k,l)∈exp(± i (t-t_0)|∇|) f_k,l;t_0^±)^2 )^1/2_L_t^∞ L_x^∞()

Now pick any cube Q ∈. In analogy to (<ref>), we define the collection of “remaining” tubes by 

    𝒯_m^r(Q):= { (k,l)∈ T_k,l;t_0^± ⋂  2Q ≠∅} .

From Proposition <ref>, it follows that #𝒯_m^r(Q)  ≤μ. As above, we have for each frequency k ∈ the bound #{ l ∈ (k,l) ∈𝒯_m^r(Q) }≲ N^δ d. Using the decay estimate (<ref>) to treat distant wave packets, we obtain

    [1]
       ( ∑_k∈( ∑_l (k,l)∈exp(± i (t-t_0)|∇|) f_k,l;t_0^±)^2 )^1/2_L_t^∞ L_x^∞(Q)
       ≲( ∑_k∈( ∑_l (k,l)∈𝒯_m^r(Q) exp(± i (t-t_0)|∇|) f_k,l;t_0^±)^2 )^1/2_L_t^∞ L_x^∞(Q)[1] 
          +( ∑_k∈( ∑_l (k,l)∈\𝒯_m^r(Q) exp(± i (t-t_0)|∇|) f_k,l;t_0^±)^2 )^1/2_L_t^∞ L_x^∞(Q)[1] 
       ≲ N^δ d/2exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_t^∞ L_x^∞ℓ_k,l^2(Q×𝒯_m^r(Q)) + N^-10C_d[1]
       ≲ N^δ d/2 2^m (#𝒯_m^r(Q))^1/2 + N^-10C_d[1]
       ≲ N^δ d/2 2^m μ^1/2 .

After taking the supremum over all cubes Q ∈, we finally arrive at

    ∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_ω^r L_t^∞ L_x^∞(Ω× [t_0,t_0+N]  ×)≲√(r) N^2 d+2/p+ δ d/2  2^m μ^1/2 .
 
By choosing p ≥ 2 sufficiently large, we arrive at the desired sub-gaussian bound.



 
Let (f_0,f_1)∈ H^s() × H^s-1() and let f_k^± be defined as in (<ref>) and (<ref>).
For any N_0 ≥ 1, we then define the wave packet “norm” of the random data ( f_0^ω,f_1^ω) by

    (f_0^ω,f_1^ω) _(N_0) := 
       ∑_N≥ N_0
     |m|≤ C_d log(N) N^-2δ dsup_  t_0=0,⌊ N^θ⌋ N 
     x_0∈ Nsup_j=1,, J^N,±_m,t_0,x_0∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_t^∞ L_x^∞(×)/2^m (#)^1/2
        + ∑_N≥ N_0
     |m|≤ C_d log(N) N^-2δ dsup_  t_0=0,⌊ N^θ⌋ N 
     x_0∈ N∑_(k,l)∈ϵ_k exp(± i (t-t_0)|∇|) f_k,l;t_0^±_L_t^∞ L_x^∞([t_0,t_0+N]  ×)/2^m μ^1/2

While the quantity (f_0^ω,f_1^ω) _(N_0) measures the size of the wave packets (over their expected size), it is certainly far from being an actual norm. 



Let (f_0,f_1)∈ H^s() × H^s-1() and let f_k^± be defined as in (<ref>) and (<ref>). Furthermore, we assume that N_0 = N_0( { Y_k,l}) satisfies

    ∑_k _∞≥ N_0/2(  f_0;k_L_x^2()^2 +  f_1;k_Ḣ_x^-1()^2  )≲ 1  .

Then, it holds that 

    𝔼_ϵ (f_0^ω,f_1^ω) _(N_0) < ∞ ,

where 𝔼_ϵ denotes the expectation over the random signs {ϵ_k }. 




This follows directly from Proposition <ref> and Definition <ref>. 


For the bootstrap argument in Section <ref>, it will be convenient to create a small forcing term by truncating to high frequencies. If N_ = N_(ω) is a (possibly random) frequency parameter, we set 

    F_ := ∑_N≥ N_( cos(t|∇|) f_0,N^ω + sin(t|∇|)/|∇| f_1,N^ω) .



Let 0 <η≤ 1 be an absolute constant and let s > 1/3. Let (f_0,f_1) ∈ H_x^s()× H_x^s-1(), let (f_0^ω,f_1^ω) be their microlocal randomizations, and let { f_k^±} be as in (<ref>) and (<ref>). Then, there exists a random frequency parameter N_≥η^-1 such that 

    (P_≥ N_/4 f_0^ω ,  P_≥ N_/4 f_1^ω) _H^s × H^s-1   ≤η , 
     (f_0^ω,f_1^ω) _Z(N_)   ≤η , 
    (f_0^ω,f_1^ω) _(N_)   ≤η , 
     F__L_t^3L_x^6(×)   ≤η .





We only need to combine the previous estimates. 
From Lemma <ref>, it follows that 

    ∑_N≥ 2∑_k_∞∈ (N/2,N]( N^2s f_0;k_L_x^2^2 + N^2(s-1) f_1;k_L_x^2^2 ) < ∞    a.s.

From dominated convergence, it then follows that there exists some random frequency N_,1, depending only on the random variables { Y_k,l},  satisfying

    ∑_N≥ N_,1∑_k_∞∈ (N/2,N]( N^2s f_0;k_L_x^2^2 + N^2(s-1) f_1;k_L_x^2^2 ) ≤η    a.s.

From Corollary <ref>, it then follows that 

    𝔼_ϵ (f_0^ω,f_1^ω) _(N_,1) < ∞ .

From dominated convergence, it follows that there exists a random frequency N_,2, depending on the random variables {ϵ_k } and { Y_k,l}, which satisfies

    (f_0^ω,f_1^ω) _(N_,2)≤η    a.s.

By similar arguments, it also follows from  Proposition <ref> and Corollary <ref> that there exists random frequencies N_,3 and N_,4 such that 

    (f_0^ω,f_1^ω) _Z(N_,3)≤η  and  ∑_N≥ N_,4cos(t|∇|) f_0,N^ω + sin(t|∇|)/|∇| f_1,N^ω_L_t^3L_x^6(×)≤η

For the second inequality, we have used the condition s > 1/3.

By choosing N_ := max( 4 N_,1,N_,2 ,N_,3 ,N_,4,η^-1), we arrive at the desired conclusion. 





§ NONLINEAR EVOLUTION: LOCAL WELL-POSEDNESS, STABILITY THEORY, AND FLUX ESTIMATES

In this section, we first apply to Da Prato-Debussche trick <cit.> to the nonlinear wave equation with random initial data. Then, we recall certain properties of the (forced) energy critical nonlinear wave equation. In our exposition of the local well-posedness and stability theory, we mainly rely on <cit.>. The flux estimate already played a major role in the author's work on almost sure scattering for the radial energy critical NLW <cit.>, and we loosely follow parts of <cit.>. 


Let N_ be as in Proposition <ref>, and let F = F_ be as in (<ref>). We then decompose the solution u of (<ref>) by setting v:= u - F. Then, the nonlinear component v solves the forced nonlinear wave equation

    -∂_tt v+ Δ v = (v+F)^3      (t,x)∈×^4 ,
    
    v|_t=0  = v_0 ∈Ḣ^1,   ∂_t v|_t=0 = v_1 ∈ L^2 ,

where v_0 := u_0 + f^ω_0,<N_ and v_1 := u_1 + f^ω_1,<N_. The randomness in the initial data (v_0,v_1) is not important, and we treat it as arbitrary data in the energy space. For the rest of this section, we treat F as an arbitrary forcing term in L_t^3 L_x^6(×), since the finer properties of F will only be relevant in Section <ref>.



 §.§ Local well-posedness and stability theory


In this section, we recall the local well-posedness of (<ref>). Using stability theory, we recall the reduction of Theorem <ref> to an a-priori energy bound. These results are well-known in the literature, see e.g. <cit.>. 


Let (v_0,v_1) ∈Ḣ^1()× L^2() and F ∈ L_t^3 L_x^6([0,∞)×). Then, there exists a time T > 0 and a unique solution v  [0,T) ×→ satisfying

    v ∈ C_t^0 Ḣ^1_x([0,T)×)  ⋂  L_t^3 L_x^6([0,T)×)     and    ∂_t v ∈ C_t^0 L_x^2([0,T)×) .



Using stability theory, <cit.> proved the following proposition. 


Let (v_0,v_1) ∈Ḣ^1()× L^2() and F ∈ L_t^3 L_x^6(×). Let v be a solution of (<ref>), and let T_+ > 0 be its maximal time of existence. Furthermore, assume the a-priori energy bound 
    sup_t∈ [0,T_+) E[v](t) < ∞ .

Then, v is a global solution and satisfies the global space-time bound v_L_t^3 L_x^6([0,∞)×) < ∞. As a result, there exist a scattering states (v_0^+,v_1^+)∈Ḣ^1()× L^2() such that 

    lim_t→ +∞ (v(t)-W(t)(v_0^+,v_1^+), ∂_t v(t) - ∂_t W(t) (v_0^+,v_1^+)) _Ḣ^1× L^2 = 0 .



Using Lemma <ref> and Proposition <ref>, we have reduced the proof of Theorem <ref> to an a-priori bound on the energy of v.



 §.§ Flux estimates

As before, we let v  I ×→ be a solution to the forced equation (<ref>). 
Recall that the (symmetric) energy-momentum tensor for the energy critical nonlinear wave equation is given by 

    T^00= T^00[v]    := 1/2( (∂_t v)^2+ |∇ v|^2) + 1/4 v^4 , 
    
    T^j0 = T^j0[v]    := - ∂_t v ·∂_x_j v , 
    
    T^jk = T^jk[v]    := ∂_x_j v ∂_x_k v - δ_jk/4 (-∂_tt+ Δ)(v^2) + δ_jk/2 v^4 .

The component T^00 is the energy density, the component T^j0 is the j-th momentum/energy flux, and the components T^jk are called the momentum flux. If v solves the energy critical nonlinear wave equation (<ref>), then the energy-momentum tensor is divergence free. This fails for solutions to the forced equation (<ref>); however, one can still expect that the error terms have lower order. Setting 𝒩 = (v+F)^3 - v^3, a short computation shows that 

    ∂_t T^00 + ∂_x_k T^0k    = - 𝒩∂_t v   ,  
    ∂_t T^j0 + ∂_x_k T^jk   = 𝒩∂_x_j v - 1/2∂_x_j ( 𝒩 v )  .

As in earlier work on almost sure scattering for radial data <cit.>, the main goal of this paper is to bound the energy of v. In terms of  the energy-momentum tensor, the (total) energy can be written as 

    E[v](t):= ∫_ T^00(t,x)  .

For future use, we record the following consequence of (<ref>). 

Let v  I ×→ be a solution of (<ref>), and let a,b ∈ I with a ≤ b. Then, we have that 

    E(b)-E(a)  = - ∫_a^b ∫_𝒩∂_t v ≤ 6 ∫_a^b ∫_ |F| |v|^2 |∂_t v| + 3 ∫_a^b ∫_ |F|^3 |∂_t v| .


We will later see that the second summand on the right-hand side of (<ref>) can be bounded directly using Hölder's inequality and probabilistic Strichartz estimates. In contrast, no such estimate is available for the first summand, and we need the wave packet decomposition to control this term. 

Once we employ the wave packet decomposition, it will be natural to study the energy  on a time and length scale ∼ N ≥ 1. We fix t_0 ∈ N ℕ_0 and x_0 ∈ N, and define the local energy

    E_t_0,x_0^N[v](t) := ∫_x-x_0 _∞≤ 2N-|t-t_0| T^00(t,x),     where   t ∈ [t_0,t_0+N] .

Thus, this definition is adapted to the truncated ℓ^∞-cone , which is  given by

    K_t_0,x_0^N := { (t,x) ∈ [t_0,t_0+N] ×x-x_0_∞≤ 2N-|t-t_0| } .

It might be more appropriate to call K_t_0,x_0^N  a pyramid (see Figure <ref>); unfortunately, the letter P is already heavily used in our notation, so that we decided to use the letter K. Our reason for using the ℓ^∞-norm, instead of the more common ℓ^2-norm, lies in the induction on scales argument (Proposition <ref>). Then, it will be an advantage to write  as the union of finitely overlapping smaller cones K_τ_0,y_0^M, which are contained in . Using finite speed of propagation and the inequality ·_ℓ^∞()≤·_ℓ^2(), one can still meaningfully restrict the nonlinear wave equation to . 





Let v be a solution to the forced equation (<ref>), let N ≥ 1,  let 
t_0 ∈ N ℕ_0, and let x_0 ∈ N. Then, we have that 

    sup_t ∈ [t_0,t_0+N] E_t_0,x_0^N[v](t) ≤ E_t_0,x_0^N[v](t_0) + 6 ∫_  |F| |v|^2 |∂_t v|  + 3 ∫_  |F|^3 |∂_t v|  .





Using (<ref>), we have that 

    d/[v](t)    = - ∫_ x - x_0 _∞
    = 2N- |t-t_0| T^00(x) + ∫_ x - x_0 _∞
    ≤ 2N- |t-t_0|∂_t T^00
       =   ∫_ x - x_0 _∞
    = 2N- |t-t_0|  (-T^00+T^0jν_j ) (x) - ∫_ x - x_0 _∞
    ≤2N- |t-t_0|𝒩∂_t v

Here, ν is the outward unit normal to the cube. From Cauchy-Schwarz, it follows that |T^0jν_j |≤ |∂_t v| ∇ v _2 ≤ T^00. After integrating in time, this completes the proof.

To simplify the notation, we now write 

    [v] := sup_t ∈ [t_0,t_0+N] E_t_0,x_0^N[v](t)  .


In the following, we want to deduce a flux estimate for the solution v of the forced NLW (<ref>). Here, we encounter a minor technical problem. Let (t^',x^') ∈  be a point in the truncated ℓ^∞-cone. We then want to control the potential |v|^4 on the truncated light-cone 
    C^N_t^',x^':= { (t,x)∈ [t_0,t_0+N]× |t-t^' | = x-x^'_ 2} .

Unfortunately, C^N_t^',x^' may not be contained in , and hence we cannot expect to bound this solely by [v]. Since the flux estimate is derived through a monotonicity formula for the local energy, this issue persists even if we are only interested in the portion of C^N_t^',x^' intersecting . To solve this problem, while still keeping the same energy increment as in (<ref>), we introduce the notion of a locally forced solution.


Let t_0 ∈ Nℕ_0 and x_0 ∈ N. We call w ×→ a -locally forced solution if it solves

    -∂_tt w + Δ w = (1_ F+w)^3      (t,x)∈×^4
    
    w|_t=t_0  =w_0 ∈Ḣ^1(),   ∂_t w|_t=t_0 = w_1∈ L^2()   .

We also require that the functions  (w_0,w_1) agree with (v(t_0),∂_t v(t_0)) on the cube x-x_0_∞≤ 2N.  



From finite speed of propagation, it follows that w|_ = v|_. 

For the same reasons as described in the last paragraph, we also use the energy on a slightly larger region. To this end, we define

    E_t_0,x_0^N[w](t) := ∫_x-x_0 _∞≤ 16N-|t-t_0| T^00[w](t,x) ,     where   t ∈ [t_0,t_0+N] .

Thus, this definition is adapted to the fattened cone , which is defined in (<ref>). 
We also set 

    [w]:= sup_t∈ [t_0,t_0+N]E_t_0,x_0^N[w](t)  .



Let t_0 ∈ Nℕ_0, let x_0 ∈ N, and let w be  a -locally forced solution. Then, we have that 

    sup_ t^'∈ [t_0,t_0+N] 
     x^'-x_0 _ℓ^∞≤ 8N ∫_x-x^'_2= |t-t^'| 
     t∈[t_0,t_0+N]w^4/4 (t,x) ≤ 4  + 12 ∫_ |F| (|w|+|F|)^2 |∂_t w|  .

 

We emphasize that, even though the energy  is measured on a truncated ℓ^∞-cone, the flux is still controlled on a light cone. 
The estimate (<ref>), however, only controls w on a lower-dimensional surface in space-time, and thus cannot directly be used to bound the energy increment. In our main argument, we rely on the following averaged version. 

 Let N≥ N_, let t_0 ∈ Nℕ_0, let x_0 ∈ N, and let w be  a -locally forced solution. Then, we have that 

    := 
    sup_ t^'∈ [t_0,t_0+N]
    x^' -x_0_∞≤ 4N ∫_|x-x^'_2- |t-t^'||≤ N^10δ
     t∈[t_0,t_0+N]w^4/4 
       ≲ N^40δ(  +  ∫_ |F| (|w|+|F|)^2 |∂_t w| ) .



The appeareance of N^10δ is for technical reasons only, and the reader is encouraged to mentally replaced it by 1. This will later help us to deal with the spatial tails of the wave packets. 

 
For the duration of this proof, we define

    e(t):= ∫_ x - x^'_2 ≤ |t-t^'| T^00(t,x)

From finite speed of propagation, we expect e(t) to be (nearly) non-increasing on [t_0,t^'] and non-decreasing on [t^',t_0+N]. From the assumptions above, it follows that x - x^'_2 ≤ |t-t^'| implies

    x-x_0 _∞≤ x-x^'_∞ +  x^' - x_0 _∞≤ x-x^'_2 + 8N ≤ |t-t^'| + 8N ≤ 10N - |t-t_0| .

Thus, we obtain that e(t) ≤ for all t ∈ [t_0,t_0+N].  Using (<ref>), we obtain for all t ∈ [t^',t_0+N] that 

    d/ e(t)    = ∫_ x - x^'_2 ≤ |t-t^'|∂_t T^00(t,x)  + ∫_ x - x^'_2 = |t-t^'| T^00(t,x) (x)
       = - ∫_ x - x^'_2 ≤ |t-t^'|𝒩∂_t v  - ∫_ x - x^'_2 ≤ |t-t^'|∂_x_k T^0k + ∫_ x - x^'_2 = |t-t^'| T^00(t,x) (x) 
       = - ∫_ x - x^'_2 ≤ |t-t^'|𝒩∂_t v  + ∫_ x - x^'_2 = |t-t^'| (T^00(t,x) + T^0kν_k) (x) 
       ≥ -6 ∫_ x - x^'_2 ≤ |t-t^'| |F| ( |F|+|v|)^2 |∂_t v| + ∫_ x - x^'_2 = |t-t^'|v^4/4(x) .

Integrating this inequality in time, we obtain the result on [t^',t_0+N]. The bound on [t_0,t^'] is similar.





 Since N≥ N_≥η^-1, we have that N^10δ≪ N. 
We then simply integrate (<ref>) over a spatial ball of radius ∼ N^10δ around x^'. 







§ THE ENERGY INCREMENT AND INDUCTION ON SCALES


We are now ready to finally bound the energy increment of the nonlinear component v. The argument roughly splits into two parts: A single scale analysis and induction on scales. 

For technical reasons, we define a flux-term involving a thinner neighborhood of the cone. More precisely, we let 

    [w] :=sup_(t^',x^')∈ [t_0,t_0+N]×
    x^' -x_0_∞≤ 4N∫_|x-x^'_2- |t-t^'||≤ N^5δ
     t∈[t_0,t_0+N]w^4/4

Recall that the light-cone in [w], as defined in (<ref>), has width N^10δ. 




 
Let N ≥ 1. Let t_0 ∈ N ℕ_0,  where 0 ≤ t_0 /N ≤⌊ N^θ⌋, and let x_0 ∈ N. Furthermore, let w_1,w_2 ∈ L_t^∞ L_x^4(×) and w_3 ∈ L_t^∞ L_x^2(×). Then, 

    ∫_ |F_M| |w_1| |w_2| |w_3| 
       ≲η N^3/4-s+8δ  (  w_1 _L_t^∞ L_x^4(×)^4 + [w_1] )^1/4  (  w_2 _L_t^∞ L_x^4(×)^4 + [w_2] )^1/4 w_3 _L_t^∞ L_x^2(×) .




We have two separate reasons for introducing the auxiliary functions w_1, w_2, and w_3. First, it emphasizes that the proof does not depend on the evolution equation of the nonlinear component. Second, it allows us to pass to smaller spatial scales than N with minimal notational effort, see Corollary <ref>.




If 1 ≤ N < N_, there is nothing to show. Thus, we may assume that N ≥ N_. 

Step 1: Wave packet decomposition. Recall from (<ref>) and (<ref>) that 

    F_N^ω = ∑_kϵ_k exp(it|∇|) f_k^+ +  ∑_kϵ_k exp(-it|∇|) f_k^- .

We only control the contribution of ∑_kϵ_k exp(it|∇|)  f_k^+, the other estimate is nearly identical. Then, we may also drop the superscript + from our notation. We now apply 
Proposition <ref> and Proposition <ref> to the family {ϵ_k f_k }_k, and let the sets , , and  be as in Proposition <ref>. 
As before, we implicitly restrict to k _∞∈ (N/2,N]. 
We also write f_k,l = f_k,l;t_0.


Step 2: Distant wave packets and extreme amplitudes. On a heuristic level, the wave packets whose tubes T_k,l do not intersect  should not contribute to the integral. We now make this precise using the decay estimate (<ref>). Indeed,

    ∫_| ∑_ (k,l)∈×:
     l - x_0 _∞ > 3Nϵ_k exp(i(t-t_0)|∇|) f_k,l| |w_1| |w_2| |w_3| 
       ≲ N ∑_ (k,l)∈×:
     l - x_0 _∞ > 3Nϵ_k exp(i(t-t_0)|∇|) f_k,l_L_t^∞ L_x^∞() w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×)
       ≲ N  ( ∑_ (k,l)∈×:
     l - x_0 _∞ > 3N N^-100 (1+ x_0 - l_2)^-100 f_k_L_x^2() ) w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×)
       ≲ N^-99+2 ( ∑_k∈ f_k_L_x^2()^2 )^1/2 w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×)
       ≲η N^3/4-s w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×) .

Thus, this contribution is acceptable. It remains to control the wave packets with indices in ⋃_m∈ℤ. We now use crude estimates to reduce to ∼log(N) amplitude scales. Let m ≤ -20 log(N). Since #≲ N^8, we have that 

    ∫_| ∑_(k,l)∈ϵ_k exp(i(t-t_0)|∇|) f_k,l| |w_1| |w_2| |w_3|[1] 
       ≲ N  ( ∑_(k,l)∈exp(i(t-t_0)|∇|) f_k,l_L_t^∞ L_x^∞(×))   w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×)[1]  
       ≲ N2^m # w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×)[1] 
       ≲ 2^m N^9   w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×) .

Summing this inequality over all m ≤ -20log(N), we obtain that 

    ∫_| ∑_m≤ -20log(N)∑_(k,l)∈ϵ_k exp(i(t-t_0)|∇|) f_k,l| |w_1| |w_2| |w_3|[1]
       ≲ N^-11 w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×)[1]
       ≲η N^3/4-s w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3 _L_t^∞ L_x^2(×) .

Finally, if ≠∅, then #≥ 1. This implies that 

    2^m ≤ 2^m (#)^1/2≲( ∑_k,l f_k,l_L^2()^2 )^1/2≲η N^-s .

For a sufficiently small absolute constant η > 0, this implies that m ≤ 0. This completes the crude part of the argument. 
Step 3: Bushes.
First, we define the fattened tubes by 

    T_k,l  := { (t,x)∈ [t_0,t_0+N] ×^d  | x- ( l - t · k/k_2 ) |≤ N^2δ} .

Furthermore, we define the collection of fattened tubes corresponding to a bush by 

    T() := ⋃_(k,l)∈ T_k,l .

With these definitions in hand, we now write 

    ∫_| ∑_j=1^J∑_(k,l)∈ϵ_k  exp(i(t-t_0)|∇|)f_k,l| |w_1| |w_2| |w_3| 
       ≤∑_j=1^J∫_| ∑_(k,l)∈ϵ_k  exp(i(t-t_0)|∇|) f_k,l|  |w_1| |w_2| |w_3|
       +∑_j=1^J∫_\| ∑_(k,l)∈ϵ_k exp(i(t-t_0)|∇|) f_k,l| |w_1| |w_2| |w_3|

Using that all tubes in  pass through the same space-time cube of size ∼ N^δ, we obtain from Proposition <ref> that 

    ∑_j=1^J∫_| ∑_(k,l)∈ϵ_k  exp(i(t-t_0)|∇|) f_k,l|  |w_1| |w_2| |w_3|
       ≲∑_j=1^J  N^1/2∑_(k,l)∈ϵ_k  exp(i(t-t_0)|∇|) f_k,l_L_t,x^∞(×) w_1_L_t,x^4() w_2 _L_t,x^4() w_3 _L_t^∞ L_x^2(×)[2]
       ≲η N^1/2+8δ 2^m  ( ∑_j=1^J (#)^1/2)  [w_1]^1/4[w_2]^1/4 w_3 _L_t^∞ L_x^2(×)[2]
       ≲η N^1/2+8δ2^m( ∑_j=1^J ) μ^-1/2[w_1]^1/4[w_2]^1/4 w_3 _L_t^∞ L_x^2(×)
       ≲η N^1/2+8δ2^m  (#) μ^-1/2[w_1]^1/4[w_2]^1/4 w_3 _L_t^∞ L_x^2(×) .

Here, μ denotes the minimum number of packets inside a single bush, see Proposition <ref>. 
Using the decay estimate (<ref>), we control the contributions outside the  by 

    ∑_j=1^J∫_\| ∑_(k,l)∈ϵ_k  exp(i(t-t_0)|∇|) f_k,l| |w_1| |w_2| |w_3|
       ≲ N ∑_j=1^J∑_(k,l)∈ϵ_k exp(i(t-t_0)|∇|) f_k,l_L_t,x^∞(\) w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3_L_t^∞ L_x^2(×)
       ≲ N^1 N^-100 2^m ( ∑_j=1^J#) w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3_L_t^∞ L_x^2(×)
       ≲η N^3/4-s w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3_L_t^∞ L_x^2(×) .

In the last line, we have used that 2^m ( #)^1/2≲η and #≲ N^8. 


Step 4: Disjoint wave packets. We now control the contribution of the almost disjoint family . If μ< 1, then  is empty, and there is nothing to prove.  If μ≥ 1, it follows from Proposition <ref> that

    ∫_| ∑_(k,l)∈ϵ_k exp(i(t-t_0)|∇|) f_k,l| |w_1| |w_2| |w_3| 
       ≲ N  ∑_(k,l)∈ϵ_k  exp(i(t-t_0)|∇|) f_k,l_L_t,x^∞(K^N_x_0) w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3_L_t^∞ L_x^2(×)
       ≲η N^1+8δ 2^m  μ^1/2 w_1 _L_t^∞ L_x^4(×) w_2 _L_t^∞ L_x^4(×) w_3_L_t^∞ L_x^2(×).




Step 5: Finishing the proof.

In total, we have shown that 

    ∫_ |F_M| |w_1| |w_2| |w_3| 
       ≲η N^8δ( N^3/4-s + N 2^m  μ^1/2  + N^1/22^m  (#) μ^-1/2)  (  w_1 _L_t^∞ L_x^4(×)^4 + [w_1] )^1/4
         · (  w_2 _L_t^∞ L_x^4(×)^4 + [w_2] )^1/4 w_3 _L_t^∞ L_x^2(×)  .

Due to our choice μ:= N^-1/2# in Proposition <ref>, this completes the proof. 


 
Let N ≥ 1. Let t_0 ∈ N ℕ_0, with 0≤ t_0/N ≤⌊ N^θ⌋, and let x_0 ∈ N. 
Let w be a -locally forced solution. Then, we have for all M ≥ N that 

    ∫_ |F_M| |w|^2 |∂_t w |≲η M^3/4-s+8δ[w]^1/2( [w] + [w] )^1/2 .


We refer to Corollary <ref> as a coarse scale estimate since the wave packets in F_M are atleast as long as the length of  (in time).  


As before, we may take M≥ N_. We distinguish two different cases. If M ≥ N^4/3, we obtain from Proposition <ref> that

    ∫_ |F_M| |w|^2 |∂_t w| ≤ N  F_M _L_t^∞ L_x^∞(×) w _L_t^∞ L_x^4()^2 ∂_t w _L_t^∞ L_x^2()≲η M^3/4-s+δ[w] .

Next, we let M ≤ N^4/3. Then, there exist τ_0 ∈ Mℕ_0 and y_0 ∈ M s.t. ⊆ K_τ_0,y_0^M. Furthermore, since t_0/N≤⌊ N^θ⌋, it holds that  τ_0/M ≤⌊ M^θ⌋. Set w_1 = w_2  := 1_ w and w_3 := 1_ w. Since M ≤ N^4/3, we have that 

    ℱ_τ_0,y_0^M[w_1] = ℱ_τ_0,y_0^M[w_2] ≤ℱ_t_0,x_0^N[w]

Using the single-scale energy increment (Proposition <ref>), we obtain that

    ∫_ |F_M| |w|^2 |∂_t w|
       = ∫_K_τ_0,y_0^M |F_M| |w_1| |w_2| |w_3|
       ≲η M^3/4-s+δ (  w_1 _L_t^∞ L_x^4(×)^4 + [w_1] )^1/4  (  w_2 _L_t^∞ L_x^4(×)^4 + [w_2] )^1/4 w_3 _L_t^∞ L_x^2(×)
       ≲η M^3/4-s+δ[w]^1/2( [w] + [w] )^1/2 .



Due to Proposition <ref> and Corollary <ref>, we understand the energy increment at a single scale. Unfortunately, the cone  may contain many wave packets on smaller scales. Similar problems are often encountered in restriction theory, and can sometimes be solved using Wolff's induction on scales strategy <cit.>. The following argument can be seen as a (simple) implementation of this idea.


Let s> max(1-θ/2,3/4+θ). Let R ≥ 1 be a dyadic integer, and let F be as in Proposition <ref>. Let t_0 ∈ Rℕ_0, with t_0/R ≤⌊ R^θ⌋,  x_0 ∈ R, let w be a -locally forced solution. For a large absolute constant C_1 ≥ 1, we have that

    [w]  ≤ 2 E_|x-x_0|≤ 16R[w](t_0) + C_1 F _L_t^3 L_x^6()^6

and 

    [w] ≤ C_1 R^50δ(  E_|x-x_0|≤ 16R[w](t_0) +   F _L_t^3 L_x^6()^6 ) .






We use induction on the dyadic integers R ≥ 1. 


Step 1: Base case R=1. We have that 

    [w]    ≤ E_|x-x_0|≤ 16[w](t_0) + C ∫_ |F| |w|^2 |∂_t w| + C ∫_ |F|^3 |∂_t w |
       ≤ E_|x-x_0|≤ 16[w](t_0) + C  F _L_t^∞ L_x^∞(×) w _L_t^∞ L_x^4()^2 ∂_t w_L_t^∞ L_x^2()
          + C  F _L_t^3 L_x^6()^3 ∂_t w _L_t^∞ L_x^2()
       ≤  E_|x-x_0|≤ 16[w](t_0) + 1/4[w] + C  F _L_t^3 L_x^6()^3 [w]^1/2
       ≤ E_|x-x_0|≤ 16[w](t_0)  +  1/2[w] + C  F _L_t^3 L_x^6()^6 .

 Insert this bound into Lemma <ref>, we obtain that 

    [w]    ≲[w] +  ∫_ |F| |w|^2 |∂_t w| + C ∫_ |F|^3 |∂_t w |
       ≲[w] 
       ≲ E_|x-x_0|≤ 16[w](t_0) +  F _L_t^3 L_x^6()^6 .

By choosing C_1 sufficiently large, we obtain (<ref>) and (<ref>). This already determines our choice of C_1, which we now regard as a fixed constant. Let R≥ 2 be an arbitrary dyadic integer. Using the induction hypothesis, we can rely on the inequalities (<ref>) and (<ref>) for all scales N ≤ R/2. 


Step 2: Splitting the energy increment. 
From Lemma <ref>, we have that 

    [w]≤ E_|x-x_0|≤ 16R[w](t_0) + C ∫_ |F| |w|^2 |∂_t w|  + C ∫_ |F|^3 |∂_t w |  .

The main term is the second summand in (<ref>). We use a Littlewood-Paley type decomposition of the linear evolution and write 

    ∫_ |F| |w|^2 |∂_t w| 
       ≤∑_N≥ R∫_ |F_N| |w|^2 |∂_t w|  + ∑_N≤ R/2∫_ |F_N| |w|^2 |∂_t w|  .

Step 3: High frequencies.
The high frequencies can be controlled using the single-scale estimate from Proposition <ref>. Indeed, we have that 

    ∑_N≥ R∫_ |F_N| |w|^2 |∂_t w| 
       ≲η∑_N≥ R N^3/4-s+8δ[w]^1/2( [w] + [w] )^1/2
       ≲η R^3/4-s+8δ[w]^1/2( [w] + [w] )^1/2 .

Step 4: Low frequencies. 
For τ_n = Nn ∈ Nℕ_0 and y_j = Nj ∈ N , we write

    ∫_ |F_N| |w|^2 |∂_t w| 
       = ∑_n=0^⌊ N^θ⌋( ∫_([τ_n,τ_n+N]×)  ⋂|F_N| |w|^2 |∂_t w| )
    +  ∫_([N^1+θ,∞)×)  ⋂|F_N| |w|^2 |∂_t w| 
       = ∑_n=0^⌊ N^θ⌋∑_j∈
    ⊆∫_ |F_N| |w|^2 |∂_t w|  +  ∫_([N^1+θ,∞)×)  ⋂|F_N| |w|^2 |∂_t w|  .

In the last line, we have used that 
    = ⋃_(n,j)∈ℕ_0×
    ⊆ .

We first control the contributions on the time intervals [τ_n,τ_n+N]. To this end, 
we define w^(N,n,j) as the -locally forced solution with data 

    w^(N,n,j)(τ_n)= w(τ_n)     and    ∂_t w^(N,n,j)(τ_n) = ∂_t w(τ_n) .

Using finite speed of propagation, w and w^(N,n,j) coincide on . 
Applying Proposition <ref> and the induction hypothesis to w^(N,n,j), it follows that 


    ∫_ |F_N| |w|^2 |∂_t w| [3] 
       = ∫_ |F_N| |w^(N,n,j)|^2 |∂_t w^(N,n,j)| [3]
       ≲η N^3/4-s+8δ[w^(N,n,j)]^1/2( [w^(N,n,j)]^1/2  + [w^(N,n,j)] )^1/2[3] 
       ≲η N^3/4-s+8δ(2 E_|x-y_j|≤ 16N[w^(N,n,j)](τ_n) + C_1  F _L_t^3 L_x^6()^6 )^1/2
          ·  N^50δ( C_1  E_|x-y_j|≤ 16N[w^(N,n,j)](τ_n)  + C_1  F _L_t^3L_x^6()^6 )^1/2
       ≲η N^3/4-s+ 60 δ C_1  (   E_|x-y_j|≤ 16N[w^(N,n,j)](τ_n)  +   F _L_t^3L_x^6()^6 ) 
       ≲η N^3/4-s+ 60 δ C_1  (   E_|x-y_j|≤ 16N[w](τ_n)  +   F _L_t^3L_x^6()^6 )  .

As a consequence, we obtain that 

    ∑_n=0^⌊ N^θ⌋∑_j∈
    ⊆∫_ |F_N| |w|^2 |∂_t w| 
       ≲η N^3/4-s+ 60δ C_1∑_n=0^⌊ N^θ⌋∑_j∈
    ⊆(  E_|x-y_j|≤ 16N[w](τ_n)  +  F _L_t^3L_x^6()^6 ) 
       ≲η N^3/4-s + 60 δ C_1 ∑_n=max(0,t_0/N)^⌊ N^θ⌋(  E_|x-x_0|≤ 16R- |t-t_0|[w](τ_n) +   F _L_t^3L_x^6(([τ_n,τ_n+N]×) ⋂ )^6 ) 
       ≲η N^3/4+ θ -s + 60 δ  C_1  + η N^3/4-s+ 60 δ C_1  F _L_t^3 L_x^6()^6 .

Using the long-time decay estimate, we can control the contribution on the interval [N^1+θ,∞) by 

    ∫_([N^1+θ,∞)×)  ⋂|F_N| |w|^2 |∂_t w| 
       ≲ F_N _L_t^1 L_x^∞([N^1+θ,∞)×) w _L_t^∞ L_x^4()^2 ∂_t w _L_t^∞ L_x^2()
       ≲η N^1-θ/2-s+δ .

Combining (<ref>) and (<ref>), it follows that 

    ∑_N≤ R/2∫_ |F_N| |w|^2 |∂_t w| 
       ≲η C_1 ∑_N≤ R/2( N^3/4+θ-s+ 5 δ + N^1-θ/2- s )   +  η(∑_N≤ R/2  N^3/4-s+ 5 δ) C_1   F _L_t^3 L_x^6()^6 
       ≲η C_1  + η C_1   F _L_t^3 L_x^6()^6  .

Here, we have used that s > max( 1 -θ/2, 3/4+ θ), and that δ = δ(s,θ) > 0 is sufficiently small. 


Step 5: Finishing the proof. 
At this point, we have proven all the necessary estimates on w. It only remains to put them together, and use a “kick back” argument. 
From the energy increment (<ref>), the high frequency estimate (<ref>), the low-frequency estimate (<ref>), and Hölder's inequality, it follows that 

    [w] ≤   E_|x-x_0|≤ 16R[w](t_0) + η C_1 [w]  + η R^3/4-s+8δ[w] + η C_1   F _L_t^3 L_x^6()^6

Inserting the same bound for the energy increment into (<ref>), we also have that 

    [w] ≲ R^40δ( [w] +  η R^3/4-s+8δ[w] + η C_1   F _L_t^3 L_x^6()^6 )

If the absolute constant η = η(C_1,δ)> 0 is chosen sufficiently small, then (<ref>) implies that 

    [w] ≲ R^40δ( [w] + η C_1  F _L_t^3 L_x^6()^6 ) .

Inserting this into (<ref>), we obtain (<ref>). Finally, (<ref>) and (<ref>) imply (<ref>). This completes the proof of the induction step. 




Using Proposition <ref>, we now provide a short proof of the main result.

Assume that the statements in Proposition <ref> hold for ω∈Ω. From Lemma <ref>, it follows that there exists a local solution to (<ref>). From Proposition <ref>, it follows for all R ≥ 1 that 

    sup_t∈[0,R]∫_|x|≤ 2R-t T^00[v](t,x) ≤ 2E[v_0,v_1]+ C_1  .

By letting R→∞, we obtain the a-priori energy bound

    sup_t∈ [0,∞) E[v](t) ≤ 2E[v_0,v_1] + C_1 .

From Proposition <ref>, this implies the global space-time bound v_L_t^3 L_x^6([0,∞) ×) < ∞ and the existence of scattering states (v_0^+ ,v_1^+ ) ∈Ḣ^1()× L^2(). Since u = F + v, we obtain the global space-time bound u _L_t^3 L_x^6([0,∞)×) < ∞ and the scattering states (u_0^+ , u_1^+ ) = (v_0^+- f^ω_0,<N_, v_1^+ - f^ω_1,<N_ ). This completes the proof for positive times. By time-reflection symmetry, we obtain the same result for negative times.





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