Short-time bilinear estimates

6.3.1 fora= 1. For the first claim, we apply Cauchy-Schwarz in ξ2 to find Z

dτ₁ Z

(dξ₁)₁f_k^#

1,j₁(τ₁, ξ₁) Z

dτ2

Z

(dξ2)1f_k^#

2,j₂(τ2, ξ2)f_k^#

3,j₃(τ1+τ2+ Ω^a, ξ1+ξ2) .

Z dτ1

Z

(dξ1)1f_k^#

1,j₁(τ1, ξ1) Z

dτ2(1 + 2^{j3−(a−1)k1})^1/4 Z

(dξ₂)₁|f_k^#

2,j₂(τ₂, ξ₂)|²|f_k^#

3,j₃(τ₁+τ₂+ Ω^a, ξ₁+ξ₂)|² 1/2

. This estimate follows due to

∂²Ω^a

∂ξ²₂

∼2^(a−1)k1,

which is derived from Case-by-Case analysis according to the signs of the involved frequencies.

Applications of Cauchy-Schwarz inτ1,ξ1andτ2 lead to .2^j2/2(1 + 2^{j3−(a−1)k1})^1/4

3

Y

i=1

kf_ki,jik₂,

which proves the first claim form₁= 2, m₂= 3. There is no loss of generality due to the symmetry amongk_i, i= 1,2,3.

For the second claim, we argue like in Lemma 6.3.2: Let i1 = 3,i2 = 2. From the proof we shall see that this is no loss of generality.

We apply the Cauchy-Schwarz inequality inξ2to find Z

dτ1

Z

(dξ1)1f_k^#

1,j₁(τ1, ξ1) Z

dτ2

Z

(dξ2)1f_k^#

2,j₂(τ2, ξ2)f_k^#

3,j₃(τ1+τ2+ Ω^a(ξ1, ξ2), ξ1+ξ2) .

Z dτ1

Z

(dξ1)1f_k^#

1,j1(τ1, ξ1) Z

dτ2(1 + 2^j3−ak1)^1/2 Z

dξ2|f_k^#

2,j₂(τ2, ξ2)|²|f_k^#

3,j₃(τ1+τ2+ Ω^a, ξ1+ξ2)|² 1/2

.

Now the claim follows from application of Cauchy-Schwarz inequality inτ1,ξ1 and τ2.

To estimate lower order terms, we use the following estimate not exploiting the dispersion relation but following from Cauchy-Schwarz inequality:

Lemma 6.3.5. Estimate (6.3)holds with α= 2^kmin/22^jmin/2.

Proposition 6.4.1. Let T ∈(0,1]andu, v∈F_a^s,δ(T),i= 1,2.

If 1< a ≤3/2, then there areδ=δ(a, s)>0 and θ=θ(a, s)>0 so that we find the following estimates to hold:

k∂x(uv)k_N0,δ

a (T).T^θkuk_F0,δ

a (T)kvk_F0,δ

a (T), (6.7)

k∂x(uv)k_N−1/2,δ

a (T).T^θkuk_F−1/2,δ

a (T)kvk_Fs,δ

a (T) (6.8)

provided thats >3/2−a.

If 3/2< a <2, then there are δ(a)>0 and θ(a)>0 so that we find the following estimate to hold:

k∂x(uv)k_N−1/2,δ

a (T).T^θkuk_F0,δ

a (T)kvk_F−1/2,δ

a (T). (6.9)

We work withδ= 0 in the following, which will be omitted from notation. Later we shall see how the analysis yields the estimates claimed above.

The above estimates are proved after decompositions in the frequency (cf. Subsec-tion 4.8.2). This essentially reduces the estimates to

kPk3∂x(uk1uk2)kN_k3,a.α(k)kuk1kF_k1,akuk2kF_k2,a. (6.10) These estimates are proved via theL²-bilinear estimates from the previous section.

We enumerate the possible frequency interactions:

(i) High×Low→High-interaction: This case is treated in Lemma 6.4.2.

(ii) High×High→High-interaction: This case is treated in Lemma 6.4.3.

(iii) High×High→Low-interaction: This case is treated in Lemma 6.4.4.

(iv) Low×Low→Low-interaction: This case is treated in Lemma 6.4.5.

We start withHigh×Low→High-interaction:

Lemma 6.4.2. Let 1≤a≤2. Suppose thatk3≥20, k2 ≤k3−5. Then, we find (6.10)to hold with α= 1.

Proof. By the same reductions like in Chapter 4, we find that it is enough to prove 2^k3 X

j₃≥(2−a)k₃

2^−j3/2k1_D^a

k3,j3

(f_k1,j1∗f_k2,j2)k_L2 .

2

Y

i=1

2^ji/2kf_ki,jik_L2, (6.11) where supp(fk_i,j_i)⊆D^a_k

i,≤j_i fori= 2,3, and we can suppose thatji≥(2−a)k3. For the resonance function, we have the estimate from below

|Ω^a|&2^ak3+k2. Consequently, there isj_i≥ak₃+k₂−10.

Suppose thatj₃ ≥ak₃+k₂−10. Then, we apply duality and the first bound from Lemma 6.3.2 to find

X

j₃≥ak3+k₂−10

2^−j3/2k1_D^a

k3,j3

(f_k1,j1∗f_k2,j2)k_L2

.2^{−(ak3+k2)/2}2^j2/2(1 + 2^j1−ak3)^1/2

2

Y

i=1

kfk_i,j_ik2.

(6.12)

By the lower bound forj1 anda≥1, it follows (6.12).2^{−(ak3+k2)/2}2^j2/22^j1/22^{−(2−a)k3/2}

2

Y

i=1

kfk_i,j_ik2.2^−k2/2

2

Y

i=1

2^ji/2kfk_i,j_ik_L2. This yields (6.11).

Suppose thatj1 ≥ak3+k2−10. The argument forj2 ≥ak3+k2−10 is the same. An application of the second bound from Lemma 6.4.2 yields

X

j₃≥(2−a)k3

2^−j3/2k1D_k^a

3,j3(fk₁,j₁∗fk₂,j₂)kL²

. X

j3≥(2−a)k3

2^−j3/2(1 + 2^j1−ak3)^1/22^j2/2

2

Y

i=1

kf_ki,jik₂

.2^{−(2−a)k3/2}2^−ak3/2

2

Y

i=1

2^ji/2kfk_i,j_ik2= 2^−k3

2

Y

i=1

2^ji/2kfk_i,j_ik2. This completes the proof.

We turn toHigh×High→High-interaction:

Lemma 6.4.3. Let1≤a≤2. Suppose thatk₃≥50,|k1−k₂| ≤10,|k2−k₃| ≤10.

Then, we find (6.10)to hold with α= 1.

Actually, the same argument as inHigh×Low→High-interaction is applicable since there are two frequencies with group velocity difference of size 2^ak1(cf. Section 3.1). Below, we point out how to prove clearly better estimates using the resonance.

Proof. Like above it suffices to prove 2^k3 X

j₃≥(2−a)k₃

2^−j3/2k1_D^a

k3,j3

(f_k1,j1∗f_k2,j2)k_L2 .

2

Y

i=1

2^ji/2kf_ki,jik₂ (6.13)

In this case we have|Ω^a|&2^(a+1)k3. Hence, due to otherwise impossible modulation interaction, there isji ≥(a+ 1)k3−20.

Ifj3≥(a+ 1)k3−20, then we use duality and the first estimate from Lemma 6.3.4 to find

X

j₃≥(a+1)k3−10

2^−j3/2k1D^a_k

3,j3(fk₁,j₁∗fk₂,j₂)kL²

.2^−(a+1)k3/22^j1/2(1 + 2^{j2−(a−1)k3})^1/4

2

Y

i=1

kf_ki,jik₂. The claim follows even with extra smoothing.

Ifj1 ≥(a+ 1)k3−10 (or j2≥(a+ 1)k3−10, where the same estimate can be applied), then we use again the first estimate from Lemma 6.3.4 to derive

X

j3≥(2−a)k3

2^−j3/2k1D_k^a

3,j3(fk₁,j₁∗fk₂,j₂)k_L2

. X

j₃≥(2−a)k3

2^−j3/2(1 + 2^{j3−(a−1)k3})^1/42^j2/2

2

Y

i=1

kfk_i,j_ik2

.2^{−(1+ε(a))k3}

2

Y

i=1

2^ji/2kf_ki,jik₂ even for someε=ε(a)>0.

We turn toHigh×High→Low-interaction, which is dual to High×Low → High-interaction. We have to add localization in time in order to estimate the input frequencies in short-time spaces.

Lemma 6.4.4. Let k1 ≥ 30 andk3 ≤k1−5. Then, we find (6.10) to hold with α= (3k1)2^(1−a)k12^(a−3/2)k3.

Proof. Following the definition of theN_a,k-spaces, we have to estimate 2^k3 X

j₃≥(2−a)k3

2^−j3/2k1D^a_k

3,j3Ft,x(u_k1v_k2η(2^(2−a)k3(t−t_k))k_L²

τ,ξ. (6.14) The resonance is given by|Ω^a|&2^ak1+k3.

Suppose thatj3≥ak1+k3−10. Then, we find

(6.14).2^k32^{−ak1 +2k3}kuk₁vk₂η(2^(2−a)k3(t−tk))k_L2 t,x.

After adding localization in time (since we are estimating anL²_t-norm at this point), it is enough to estimate

2^k32^{(2−a)(k21−k3 )}2^{−ak1 +2k3}kuk₁vk₂η(2^(2−a)k1(t−tλ))k_L2

t,x. (6.15) Write

fk1,j1 = 1D^a_k

1,(≤)j1Ft,x[γ(2^(2−a)k1(t−tλ))uk1], fk₂,j₂ = 1D^a

k2,(≤)j2

Ft,x[γ(2^(2−a)k1+10(t−tµ))vk₂].

In the above display, the low modulations are annexed matching time localization as usual.

Then, an application of twoL⁴_t,x-Strichartz estimates gives (6.15).2^(1−a)k12^{a−12 k3}2^−k41

2

Y

i=1

X

j_i≥(2−a)k1

2^ji/2kfk_i,j_ik2

.2^(3/4−a)k12^{a−12 k3}

2

Y

i=1

X

ji≥(2−a)k1

2^ji/2kf_ki,jik₂.

This yields a first bound. Some of the above estimates are crude because the next case gives the worse bound anyway.

We turn to the sum overj3 in (6.14), wherej3≤ak1+k3−10.

By the reduction and notations from Chapter 5, we have to estimate 2^{(2−a)(k1−k3)}2^k3 X

(2−a)k3≤j3≤ak1+k₃

2^−j3/2k1D^a_k

3,≤j3(fk1,j1∗fk2,j2)k_L²

τ,ξ, wherej1, j2≥(2−a)k1.

Suppose that j1 ≥ ak1 +k3−10 by symmetry among fk₁,j₁ and fk₂,j₂. An application of Lemma 6.3.2 in conjunction with duality gives

.2^{(2−a)(k1−k3)}2^k3 X

j3≤ak1+k3

2^−j3/22^j3/2 1 + 2^j2−ak11/2

kfk₁,j₁k2kfk₂,j₂k2

.(3k1)2^(1−a)k12^(a−3/2)k3

2

Y

i=1

2^ji/2kfk_i,j_ik2,

which is inferior to the first bound. The proof is complete.

We record the estimate for Low×Low→Low-interaction which is immediate from Lemma 6.3.5:

Lemma 6.4.5. Letki≤100,i= 1,2,3. Then, we find(6.10)to hold withα(k) = 1.

Proof of Proposition 6.4.1. With the above estimates for frequency localized inter-actions at disposal, we can infer the claimed estimates: ForHigh×Low→ High-interaction Lemma 6.4.2 gives the estimates after square-summing

k∂_x(uv)k_N0

a(T).kuk_F0

a(T)kvk_F0+

a (T), k∂x(uv)k_N−1/2

a (T).kuk_F−1/2

a (T)kvk_Fs a(T), where 1< a≤3/2 ands >3/2−a.

Increasing time localization leads to extra smoothing (because the minimal size of the modulation regions will become larger). Together with Lemma 2.5.3, we deduce from the proof of Lemma 6.4.2

k∂x(uv)k_N0,δ

a (T).T^θkuk_F0,δ

a (T)kvk_F0,δ a (T), k∂x(uv)k_N−1/2,δ

a (T).T^θkuk_F−1/2,δ

a (T)kvk_Fs,δ a (T)

for someθ >0 for anyδ >0 withaandslike in the previous display.

For 3/2< a <2 the argument is analogous forHigh×Low→High-interaction.

ForHigh×High→High-interaction the estimates due to Lemma 6.4.3 are suffi-cient because of improved resonance compared toHigh×Low→High-interaction.

ForHigh×High→Low-interaction the short-time estimates become worse when increasing time localization. But there is room in the estimate from Lemma 6.4.4 to prove the estimates forδ(a)>0 chosen sufficiently small.

Im Dokument Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations (Seite 145-150)