Short-time trilinear estimates

Case B:In casej1 =j₁^∗, that is a high frequency carrying a high modulation we estimate the low frequencyg3 by Cauchy-Schwarz to find:

2^j3/22^l/2kg₃k_L2 τ`²_n sup

n₃,τ₃

Z dτ₁

Z

dτ₄X

n1

|g₁(τ₁, n₁)|

X

n₂

|g₂(τ₂, n₂)||g₄(h₄,−n₁−n₂−n₃)|

(5.70)

We consider the setE42={n2∈Z|h4=O(2^j4)}. Since∂n₂h4= 3(n1+n3)(n2−n4) we find |∂n₂h4|&2^2k∗1 and further |E42|.(1 + 2^j4−2k∗1)^1/2. We find after several applications of the Cauchy-Schwarz inequality:

(5.70).(1 + 2^j4−2k∗1)^1/22^j3/22^l/2kg3kL²_τ`²_n sup

n3,τ3

Z dτ1

Z dτ2

X

n1

|g1(τ₁, n₁)| X

n2

|g2(τ₂, n₂)|²|g4(h₄,−n1−n₂−n₃)|²

!^1/2

.2^j3/22^l/2(1 + 2^j4−2k∗1)^1/2kg₃k_L2 τ`²_n sup

n₃,τ₃

Z

dτ₂X

n1

Z

dτ₁|g₁(τ₁, n₁)|^{2 1/2}

× X

n₂

|g2(τ2, n2)|²kg4(h4,−n1−n2−n3)k²_L2 τ1

!^1/2

.2^j3/22^l/2(1 + 2^j4−2k∗1)^1/2kg₃k_L2 τ3`²_n

3 sup

n₃,τ₃

Z

dτ₂kg₁k_L2 τ`²_n

× X

n₂

|g2(τ2, n2)|²X

n₁

kg4(h4,−n1−n2−n3)k²_L2 τ

!1/2

.2^j3/22^l/2(1 + 2^j4−2k∗1)^1/22^j2/2

4

Y

i=1

kgik_L2 τ`²_n.

Clearly, an adapted computation shows the claim ifj₁^∗=j4.

The estimate (5.68) follows from the same considerations as in Case A.

The estimate for High×High×Low → Low-interaction is related, but the minimal size of the support of the modulation variable is different. This is taken into account in the following sections.

In fact, the resonant interaction can be perceived as a special case ofHigh×High×

High → High-interaction, see below. Hence, we only estimate the non-resonant part.

The trilinear estimate

kN(u1, u₂, u₃)kN^s,α(T).T^θku1kF^s,α(T)ku2kF^s,α(T)ku3kF^s,α(T) (5.72) then follows from splitting up the frequency support of the functions and Lemma 2.5.3. Note that it will be enough to estimate one function in (5.71) at a modulation slightly below 1/2 to derive (5.72).

Below, we only derive (5.71) for F_k^α

i-spaces in detail. The systematic modifi-cation to find (5.71) to hold with one modulation regularity strictly less than 1/2 follows by accepting a slight loss in the highest modulation in modulation localized estimates.

We start withHigh×Low×Low→High-interaction:

Lemma 5.2.8. Let k4≥20andk1≤k2≤k3−5 and suppose thatPk_iui=ui for i∈ {1,2,3}. Then, we find the estimate (5.71) to hold withD(α, k) = 2^{−(α/2−ε)k4} for anyε >0.

Proof. Letγ:R→[0,1] be a smooth function with supp(γ)⊆[−1,1] and X

m∈Z

γ³(x−m)≡1.

We find the left hand-side in (5.71) to be dominated by C2^k4 X

m∈Z

sup

t_k4∈R

k(τ−n³+i2^[αk4])⁻¹1I_k4(n) (Ft,x[η0(2^[αk4](t−tk₄))γ(2^[αk∗1](t−tk₄)−m)u1]

∗ Ft,x[γ(2^[αk∗1](t−tk4)−m)u2]∗ Ft,x[γ(2^[αk∗1](t−tk4)−m)u3])^˜kX_k4. We observe that #{m ∈ Z|η0(2^[αk4](· −tk))γ(2^[αk1∗](· −tk)−m) 6= 0} = O(1).

Consequently, it is enough to prove C2^k4 sup

t_k4∈R

k(τ−n³+i2^[αk4])⁻¹1I_k4(n)(Ft,x[η0(2^[αk4](t−tk))γ(2^{[αk∗1](t−tk)−m)}u1]

∗ F_t,x[γ(2^[αk∗1](t−t_k)−m)u₂]∗ F_t,x[γ(2^[αk∗1](t−t_k))u₃])^˜k_Xk

4

.2^{−(α/2−ε)k4}ku₁k_F^α

k1

ku₂k_F^α

k2

ku₃k_F^α

k3

.

We writef_ki =F_t,x[η₀(2^[αk4](t−t_k)γ(2^[αk∗1](t−t_k)−m)u_i] and with additional localization in modulation we use the notation

fk_i,j_i =

(η_≤ji(τ−n³)fk_i, ji= [αk₁^∗], η_ji(τ−n³)f_ki, j_i>[αk₁^∗].

By means of the definition ofF_k^α

i and (2.48), it is further enough to prove X

j₄≥[αk₄]

X

j₁,j₂,j₃≥[αk₁^∗]

2^−j4/2k1D_k4,(≤)j4(f_k1,j1∗f_k2,j2∗f_k3,j3)^˜kL²_τ`²_n

.2^{−(α/2−ε)k1}

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfki,jikL²_τ`²_n.

(5.73)

We see that (5.73) follows from (5.68). In fact, the resonance function, giving a lower bound for the minimal modulation in (5.73), is given by

Ω = (k1+k2+k3)³−k₁³−k³₂−k³₃= 3(k1+k2)(k1+k3)(k2+k3).

Thus, 2^2k∗1 .|Ω|.2^2k∗1+k∗3. To derive effective estimates, we localize|Ω| ∼2^2k4+l. This is equivalent to prescribe|k2+k₃| ∼2^l, and the contribution to (5.73) will be denoted by

kP_Ω^l1_Dk

4,(≤)j4(f_k1,j1∗f_k2,j2∗f_k3,j3)^˜k_L2 τ`²_n. We writeDk₄,j₄ forDk₄,(≤)j4 in the following.

In the above display we split the frequency support offk₂,j₂ up into intervals of length 2^l, that isfk2,j2 =P

If_k^I_2,j2 and, due to the localization of Ω, this also gives a decomposition off_k3,j3 so that the above display is dominated by

X

I

kP_Ω^l1D_k4,j4(fk₁,j₁∗f_k^I_2,j2∗f_k^I_3,j3)^˜k_L2 τ`²_n.

Further, we split after decomposition in 0≤l≤k^∗₃the sum overj₄intoj₄≤2k^∗₁+l andj4≥2k^∗₁+l. For fixedl we find from (5.68)

2^k4 X

[αk4]≤j4≤2k^∗₁+l, j_i≥[αk^∗₁], i=1,2,3

2^−j4/2X

I

kP_Ω^l1D_k4,j4(fk₁,j₁∗f_k^I_2,j2∗f_k^I_3,j3)^˜k_L2 τ`²_n

.2^k4 X

[αk₄]≤j4≤2k^∗₁+l

2^−j4/22^−j1∗/22^l/22^−[αk1]/22^j4/2

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ikL²

.k₁^∗2^−[αk1]/2

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ik_L2 τ`²_n, where f_k^I

i,j_i for i = 2,3 were reassembled by means of almost orthogonality and Cauchy-Schwarz inequality.

For the second partj4≥2k₁^∗+l, we take 2^−j1∗/2≤2^−j4/2 to find similarly 2^k4 X

j₄≥2k₁^∗+l

X

j1,j2,j3≥[αk^∗₁]

2^−j4/2X

I

kP_Ω^l1D_k4,j4(fk₁,j₁∗f_k^I_2,j2∗f_k^I_3,j3)^˜k_L2 τ`²_n

.2^k4 X

j₄≥2k^∗₁+l

2^−j4/22^l/22^−[αk1]/2

3

Y

i=1

2^ji/2kfk_i,j_ik_L2

.2^−[αk1]/2

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ikL²_τ`²_n.

An estimate with one modulation strictly less than 1/2 follows from slight loss in the highest modulation. We omit the details.

We turn toHigh×High×Low→High-interaction.

Lemma 5.2.9. Let k₄ ≥ 20, k₁ ≤ k₂ ≤ k₃, k₁ ≤ k₂−15, |k2−k₄| ≤ 10 and suppose thatP_kiu_i=u_i fori∈ {1,2,3}. Then, we find estimate(5.71)to hold with D(α, k) = 2^{−(1/2−ε)k4} for anyε >0.

Proof. With the reduction steps and notation from above, we have to prove 2^k4 X

j₄≥[αk4]

2^−j4/2 X

j₁,j₂,j₃≥[αk₁^∗]

k1D_k4,j4(fk₁,j₁∗fk₂,j₂∗fk₃,j₃)^˜k_L2 τ`²_n

.^ε2^{−(1/2−ε)k4}

3

Y

i=1

X

ji≥[αk^∗₁]

2^ji/2kfk_i,j_ik_L2 τ`²_n.

(5.74)

We use (5.63) to find

k1D_k4,j4(fk₁,j₁∗fk₂,j₂∗fk₃,j₃)^˜k_L2

τ`²_n.2^−j∗1/22^εk∗12^j4/2

3

Y

i=1

2^ji/2kfk_i,j_ik_L2

τ`²_n. (5.75) We find from the resonance relation that j₁^∗ ≥ 3k₁^∗−10. Now the estimate follows in a similar spirit to the computation above: Splitting up the sum overj4

into [αk4]≤j4≤3k₁^∗ andj4≥3k^∗₁, we find 2^k4 X

[αk4]≤j4≤3k^∗₁

2^−j4/2 X

j1,j2,j3≥[αk^∗₁]

2^εk∗12^−3k∗1/22^j4/2

3

Y

i=1

2^ji/2kfk_i,j_ik_L2 τ`²_n

.^ε3k^∗₁2^{−k1∗/2+(ε/2)k∗1}

3

Y

i=1

X

ji≥[αk^∗₁]

2^ji/2kfk_i,j_ik_L2 τ`²_n

.ε2^{−(1/2−ε)k4}

3

Y

i=1

X

ji≥[αk^∗₁]

2^ji/2kf_ki,jik_L2 τ`²_n. For the remaining part we argue like above

X

j4≥3k^∗₁

2^−j4/22^(ε/2)k∗1

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ikL²_τ`²_n

.2^−k4/2+εk4

3

Y

i=1

X

j_i≥[αk₁^∗]

2^ji/2kfk_i,j_ik_L2 τ`²_n,

and the claim follows. The variant with one function in a strictly less modulation regularity than 1/2 follows from the same considerations like in the previous lemma.

We turn toHigh×High×High→High-interaction. In this case, we do not use a multilinear argument but only the bilinear estimate from Lemma 5.2.6. In the special caseα= 1, this becomes the analysis from [Mol12]. Additionally, the computation reveals that the interaction under consideration can be estimated for negative Sobolev regularities forα >1.

Lemma 5.2.10. Let k₄≥20and |k_i−k₄| ≤20for any i∈ {1,2,3} and suppose that P_kiu_i = u_i for i ∈ {1,2,3}. Then, we find (5.71) to hold with D(α, k) = 2−(α/2−1/2)k4 wheneverα≥1.

Proof. The usual reduction steps lead us to the remaining estimate:

X

j₄≥[αk4]

2^−j4/22^k4 X

j₁,j₂,j₃≥[αk^∗₁]

k1D_k4,j4(fk₁,j₁∗fk₂,j₂∗fk₃,j₃)^˜kL²_τ`²_n

.2−(α/2−1/2)k4

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ik_L2 τ`²_n.

(5.76)

We use duality to write

k1Dk4,j4(f_k1,j1∗f_k2,j2∗f_k3,j3)kL²_τ`²_n= sup

ku4k_L2 t,x=1

Z Z

u₁u₂u₃u₄dxdt.

whereu_i =F_t,x⁻¹[f_ki,ji] fori= 1,2,3.

After splitting the expression according toP_±u_i, whereP_±projects to only pos-itive, respectively negative frequencies, it is easy to see that two bilinear estimates are applicable.

Indeed, the same sign must appear twice. A pair of this kind is amenable to (5.67) as the output frequency must be of order 2^k∗1, and the two remaining factors are also amenable to a bilinear estimate.

Say we can apply bilinear estimates tou4u1andu2u3. This gives k1D_k4,j4(fk₁,j₁∗fk₂,j₂∗fk₃,j₃)kL²_τ`²_n

.2^j1/22^(j4−k4)/22^j2/22^(j3−k4)/4

3

Y

i=1

kfk_i,j_ikL²_τ`²_n

.2^−k4/22^j4/42^−αk4/4

3

Y

i=1

2^j1/2kfk_i,j_ikL²_τ`²_n. The claim follows after summation overj₄.

The resonant interaction is a special case of High×High×High → High-interaction, but we mention that in this case the same estimate like above can be proved by elementary means.

Next, we deal withHigh×High×Low→Low-interaction:

Lemma 5.2.11. Let k1≥20,k1≤k2≤k3,k1≤k2−5,k4≤k2−5 and suppose that Pk_iui = ui for i ∈ {1,2,3}. Then, we find (5.71) to hold with D(α, k) = 2^{(α/2−1+ε)k1}2^(1−α)k4 for anyε >0.

Proof. Contrary to the previous cases, we have to add localization in time in order to estimateu_k1 andu_k2 inF_k^α

1 orF_k^α

2, respectively.

For this purpose letγ :R→[0,1] be a smooth function supported in [−1,1] with the property

X

m∈Z

γ³(x−m)≡1.

We find the left hand-side to be dominated by C2^k4 X

|m|.2^[α(k1−k4 )]

sup

t_k4∈R

kF_t,x[u₁η₀(2^αk4(t−t_k4))γ(2^αk1(t−t_k4)−m)]∗

Ft,x[u2γ(2^αk1(t−tk)−m)]∗ Ft,x[u3γ(2^αk1(t−tk)−m)]^˜kX_k4.

With the additional localization in time available, we can annex the modulation variable for ji ≤[αk^∗₁], i = 1,2,3 and denote fk_i = F[uiγ(2^k1(t−tk)−m)] and with additional localization in modulation we write

fk_i,j_i =

(η_≤ji(τ−n³)fk_i, ji= [αk₁^∗], η_ji(τ−n³)f_ki, j_i>[αk₁^∗].

With the reduction steps from above, we have to prove 2^α(k1−k4)2^k4 X

j₄≥[αk₄]

2^−j4/2 X

j₁,j₂,j₃≥[αk^∗₁]

k1D˜_k4,j4(f_k1,j1∗f_k2,j2∗f_k3,j3)^˜k_L2 τ`²_n

.2^{(α/2−1+ε)k1}2^(1−α)k4

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfki,jikL²_τ`²_n.

(5.77) Like in the proof of Lemma 5.2.8, the resonance is localized to

2^2k∗1 .|Ω|.2^2k∗1+k∗3,

and we introduce additional localization P_Ω^l for |Ω| ∼ 2^2k∗1+l, where 0 ≤ l ≤k₃^∗. Correspondingly, we decomposef_k1,j1 into intervalsIof length 2^l, which allows an almost orthogonal decomposition of the output.

Lastly, we split the sum overj₄ into j₄≤2k^∗₁+l andj₄ ≥2k^∗₁+l. For fixedl we find from (5.68)

2^α(k1−k4)2^k4 X

[αk4]≤j4≤[2k^∗₁+l], j_i≥[αk^∗₁]

2^−j4/2

× X

I

kP_Ω^l1D˜k4,j4(fk₁,j₁∗fk₂,j₂∗f_k^I_3,j3)^˜k²_L2 τ`²_n

!1/2

.2^k42^α(k1−k4) X

[αk₄]≤j₄≤[2k^∗₁+l], j_i≥[αk^∗₁]

2^−j4/22^−j1∗/22^l/22^−[αk1]/22^j4/2

3

Y

i=1

2^ji/2kfk_i,j_ik2

.k₁^∗2^αk1/22^(1−α)k42^−k1

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ik2

.2(α/2−1+ε/2)k12^(1−α)k4

3

Y

i=1

X

ji

2^ji/2kfk_i,j_ik_L2.

Likewise, we find for the contribution ofj4≥2k^∗₁+l the bound 2(α/2−1+ε/2)k12^(1−α)k4

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ik2. Summation overl yields the claim.

At last, we turn toHigh×High×High→Low-interaction:

Lemma 5.2.12. Letk1≥20,k1≤k2≤k3,k1≥k3−10,k4≤k1−10and suppose that Pk_iui = ui for i ∈ {1,2,3}. Then, we find (5.71) to hold with D(α, k) = 2^{(α−3/2+ε)k1}2^(1−α)k4 for anyε >0.

Proof. As in the proof of Lemma 5.2.11, we have to add localization in time accord-ing tok^∗₁. With the notation and conventions from Lemma 5.2.11, we have to show the estimate

2^k42^α(k1−k4) X

j₄≥[αk4]

2^−j4/2 X

j₁,j₂,j₃≥[αk^∗₁]

k1D˜_k4,j4(fk1,j1∗fk2,j2∗fk3,j3)^˜kL²_τ`²_n

.^ε2^(1−α)k42^{(α−3/2+ε)k1}

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ik_L2 τ`²_n.

(5.78) The resonance function impliesj^∗₁ ≥3k₁^∗−15. We split the sum in (5.78) over j₄up into [αk₄]≤j₄≤3k^∗₁ andj₄≥3k₁^∗.

The first part is estimated by an application of (5.63) 2^k42^α(k1−k4) X

[αk₄]≤j4≤3k^∗₁

2^−j4/2 X

j_i≥[αk^∗₁]

k1D˜_k4,j4(f_k1,j1∗f_k2,j2∗f_k3,j3)^˜kL²_τ`²_n

.^ε2^αk12^(1−α)k4 X

[αk₄]≤j4≤3k^∗₁

2^−j4/2 X

j₁,j₂,j₃≥[αk₁^∗]

2^−j1∗/22^εk∗12^j4/2

3

Y

i=1

2^ji/2kfk_i,j_ikL²_τ`²_n

.2^{(α+ε−3/2)k1}2^k4(1−α)(3k₁^∗)2^(α−2)k1/2

3

Y

i=1

X

j_i≥[αk^∗₁]

2^ji/2kfk_i,j_ik_L2 τ`²_n.

The estimate forj₄≥3k₁^∗follows similarly. This proves the claim together with the standard modification of lowering the modulation regularity slightly.

We record the estimate for the interaction of low frequencies, which follows in a straight-forward manner from Cauchy-Schwarz inequality.

Lemma 5.2.13. Letk₁, . . . , k₄≤200. Then, we find (5.71)to hold withD(α, k) = 1.

We summarize the regularity thresholds for which we can show the trilinear estimate (5.72) by splitting up the frequencies and using the estimate (5.71)

1. High×Low×Low→High-interaction: Lemma 5.2.8 provides us with the regularity thresholds=−(α/4)+.

2. High×High×Low→High-interaction: Lemma 5.2.9 provides us with the regularity thresholds=−(1/2)+.

3. High×High×High →High-interaction: Lemma 5.2.10 provides us with the regularity thresholds= (1−α)/4.

4. High×High×Low→Low-interaction: Lemma 5.2.11 provides us with the regularity thresholds=−(1/6)+ forα= 1.

5. High×High×High→Low-interaction: Lemma 5.2.12 provides us with the regularity thresholds=−(1/6)+ forα= 1.

6. Low×Low×Low→Low-interaction: There is no threshold.

This proves the following proposition:

Proposition 5.2.14. Let T ∈ (0,1]. For 0 < s < 1/2, there is α(s) < 1 and θ=θ(s)>0 or fors= 0,α= 1 andθ= 0 we find the following estimate holds:

kN(u1, u₂, u₃)kN^s,α(T).T^θ

3

Y

i=1

kuikF^s,α(T).

Furthermore, there is δ⁰ >0 so that for any 0 < δ < δ⁰ there is s=s(δ)<0 and θ >0 so that

kN(u₁, u₂, u₃)k_Ns,1+δ(T).T^θ

3

Y

i=1

ku_ik_Fs,1+δ(T).

We do not quantify the estimates in detail for negative Sobolev regularity be-cause in this case, we can only prove a conditional result.

Im Dokument Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations (Seite 108-115)