L p -smoothing estimates for elliptic phase functions with variable co-

where the estimate was concluded by the following estimate X

θ

kPθfk^p_Lp

!^1/p

.^p kfkL^p,

which holds for 2 ≤p < ∞. This can be inferred from Plancherel’s theorem for p= 2 and from Young’s inequality for p =∞. The remaining cases follow from interpolation.

7.4 L

^p

-smoothing estimates for elliptic phase

Proposition 7.4.3. Let p >2 +_n+1⁴ andφbe a normalized elliptic phase function giving rise to the H¨ormander operatorT^λ. Then, for anyε >0, we find the following estimate to hold:

kT^λfˆk_Lp(Rⁿ⁺¹).^ελn(1/2−1/p)+εkfk_Lp(Rⁿ). (7.52) For a phase functionφsatisfyingH1) letq(t, x;ξ) =∂tφ(t, x; [∂xφ(t, x;·)]⁻¹(ξ)) denote the local parametrization of the surface associated to the adjoint Fourier re-striction operator. To prove theL^p-smoothing estimate for variable-coefficient phase functions, crucial use is made of the following bilinear estimate using transversality, which is a generalization of the theorem from [Tao03]:

Theorem 7.4.4 ([Lee06b, Theorem 1.1, p. 58]). For i = 1,2, let φi be smooth functions satisfyingH1). Suppose that the Hessian∂_ξξ²qi satisfies

det∂_ξξ² qi(t, x;∂xφi(t, x;ξi))6= 0 on the support ofai and for(t, x;ξi)∈supp(ai)

|h∂_xξ² φi(t, x;ξi)δ(t, x;ξ1, ξ2),

[∂_xξ² φi(t, x;ξi)]⁻¹[∂²_ξξqi(t, x;ui)]⁻¹δ(t, x;ξ1, ξ2)i| ≥c >0 fori= 1,2, whereui=∂xφi(t, x;ξi)and

δ(t, x;ξ1, ξ2) =∂ξq1(t, x;u1)−∂ξq2(t, x;u2).

Then, for anyε >0, there is a constantC=C(ε)such that forp≥(n+ 3)/(n+

1),

kT₁^λf T₂^λgk_Lp(R×Rⁿ)≤Cελ^εkfk2kgk2,

whereT_i^λ denotes the H¨ormander-operator associated to the data(φ_i, a_i).

For normalized elliptic phase functions this yields the following corollary:

Corollary 7.4.5. Let φ be a normalized elliptic phase function and suppose for f1, f2 ∈L²(Rⁿ) with supp(fi)⊆Q(2). Then, for any ε >0 there is Cε such that forp≥(n+ 3)/(n+ 1)we find the following estimate to hold:

kT^λf1T^λf2kL^p(Rⁿ×R)≤Cελ^εkf1k2kf2k2.

Proof. We check that the assumptions of the more general Theorem 7.4.4 are sat-isfied: Indeed, from normalization we infer ∂_xξ² φ ∼ I_n, δ(x, t;ξ₁, ξ₂) ∼ ξ₁ −ξ₂,

∂²_ξξq_i∼I_n and consequently,

|h∂²_xξφ(x, t;ξi)δ(x, t;ξ1, ξ2),[∂_xξ² φ(x, t;ξi)]⁻¹[∂_ξξ²qi(x, t;ui)]⁻¹δ(x, t;ξ1, ξ2)i|

∼ |ξ1−ξ2|²≥c². This yields the claim.

Due to a localization property of the kernel, this yields the followingL^p -estima-tes.

Lemma 7.4.6. Let p > ²⁽ⁿ⁺³⁾_n+1 , B1, B2 ⊆ Q(1) balls with dist(B1, B2) ≥ c > 0 andφbe a normalized elliptic phase function. Then, forf1, f2 with supp( ˆfi)⊆Bi, i= 1,2, we find the following estimate to hold:

kT^λfˆ1T^λfˆ2k_Lp/2(R×Rⁿ).^p,ε λ^{n(1−2/p)+ε}kf1kL^p(Rⁿ)kf2kL^p(Rⁿ). Proof. First, we note the localization property for

K(t, x;y) =a^λ₀(t, x) Z

Rⁿ

e^i(φλ(t,x;ξ)−hy,ξi)β(ξ)dξ.

In fact,

∂ξ[φ^λ(t, x;ξ)− hy, ξi] =∂ξφ^λ(0, x;ξ) + Z t/λ

0

∂t∂ξφ(t⁰, x/λ;ξ)dt⁰−y

=x−y+λ Z t/λ

0

dt⁰ Z 1

0

ds∂t∂_ξξ²φ(t⁰, x/λ;sξ)·ξ

=x−y+O(λ).

Consequently, integration by parts yields that there is a constantCwhich depends only on the constants in the definition of normalized elliptic phase functions so that for |y| ≥ Cλ we find |K(t, x;y)| ≤ CNλ^−N|x−y|^−N from integration by parts.

LetχQ be a smooth function which is essentially supported onQ(Cλ) with Fourier support inB(0, c/10) and decomposefi=χQfi+fi2=fi1+fi2.

From trivial kernel estimates we find

kT^λfˆ_i1kL^p(R×Rⁿ)≤λ^C0kfikL^p(Rⁿ),

whereC⁰=C(n, p) and from the kernel estimate and Young’s inequality kT^λfˆ_i2kL^p(R×Rⁿ)=

Z

K(t, x;y)f_i2(y)dy _Lp(

R×Rⁿ)

≤C_Nλ^−Nkf_i2k_Lp(Rⁿ)≤C_Nλ^−Nkf_ik_Lp(Rⁿ). We split

kT^λfˆ₁T^λfˆ₂k_Lp/2(R×Rⁿ)≤ kT^λfˆ₁₁T^λfˆ₂₁k_Lp/2(R×Rⁿ)+kT^λfˆ₁₂T^λfˆ₂₁k_Lp/2(R×Rⁿ)

+kT^λfˆ11T^λfˆ22k_Lp/2(R×Rⁿ)+kT^λfˆ12T^λfˆ22k_Lp/2(R×Rⁿ)

≤I+II+III+IV.

II,III,IV are estimated by the above elementary estimates. For instance, II≤ kT^λfˆ12kL^pkT^λfˆ21kL^p≤CNλ^−Nkf1kL^pλ^C0kf2kL^p.

Only for I we have to invoke the bilinear estimate from above. Due to the support properties ofχ_Q we can still use Corollary 7.4.5

kT^λfˆ11T^λfˆ21k_Lp/2(Rⁿ×R).^ελ^εkχ[Qf1kL²kχ[Qf2kL²

≤Cελ^εkχQf1k_L2kχQf2k_L2

≤C_ελ^ελ^n(1−2/p)kf1kL^p(Rⁿ)kf2kL^p(Rⁿ), and the proof is complete.

Lemma 7.4.6 is the most important building block for further arguments. To utilize it efficiently, we carry out a Whitney decomposition of T^λfˆ1T^λfˆ2 in terms of frequency support and separation.

Precisely, let

B(1)×B(1)\D= [

0≤j≤j⁰

[

dist(Q^j_k

1,Q^j_k

2)∼2^−j

Q^j_k

1×Q^j_k

2, where for 0≤j < j⁰= ¹₂log₂λ^1/2the cubesQ^j_k

1, Q^j_k

2have side length comparable to 2^−j and are separated with a distance comparable to 2^−j. Forj=j⁰the separation is less or equal to 2^−j.

Adapted to the Whitney-decomposition, we write (T^λfˆ)²= X

0≤j≤j⁰

X

k₁,k₂

T^λ(β_k^j

1

fˆ)T^λ(β_k^j

2

fˆ)

=: X

0≤j≤j⁰

B˜j[f, f],

where theβ^j_k denote a smooth partition of unity adapted to the Whitney-decom-position.

Forj < j⁰ we prove the following estimate by scaling:

Lemma 7.4.7. Let φ be a normalized elliptic phase function andf_i have Fourier support in cubes of side length2^−j which are also separated with a distance of2^−j. Then, we find the following estimate to hold provided that1≤2^2j ≤λ:

kT^λfˆ1T^λfˆ2k_Lp/2.^ελ^ε2^4jp(λ/2^2j)^n(1−2/p)kf1k_Lp(Rⁿ)kf2k_Lp(Rⁿ). (7.53) Proof. We use parabolic rescaling to write

kT^λfˆ1T^λfˆ2k_Lp/2(Rⁿ⁺¹)

= 2^2jnp 2^−2jn2^4jpk Z

e^{iφλ(22jt0,2jy,ξ0+2−jξ0)}a^λ(2^2jt⁰,2^jy;ξ₀+ 2^−jξ⁰) ˆf₁(ξ₀+ 2^−jξ⁰)dξ⁰ Z

e^{iφλ(22jt,2jy;ξ0+2−jξ0)}a^λ(2^2jt⁰,2^jy;ξ0+ 2^−jξ⁰) ˆf2(ξ0+ 2^−jξ⁰)dξ⁰k_Lp/2(Rⁿ⁺¹)

= 2^−2jn2^2jnp 2^4jpk Z

e^iφ˜λ/22

j(t⁰,y;ξ⁰)˜a^λ/22j(t⁰, y;ξ⁰)ˆg1(ξ⁰)dξ⁰ Z

e^iφ˜λ/22

j(t⁰,y;ξ⁰)˜a^λ/22j(t⁰, y;ξ⁰)ˆg2(ξ⁰)dξ⁰k_Lp/2(Rⁿ×R).

(7.54) Due to ˆgi(ξ) = ˆfi(ξ0+ 2^−jξ), we havekgikL^p= 2^nj2^−njpkfikL^p. We argue that the latter expression is amenable to Lemma 7.4.6: Note that

supp(˜a^λ/22j(t,·;ξ))⊆Q(λ/2^j).

However, we want to apply Lemma 7.4.6 with parameterλ/2^2j. For this purpose we use again the localization property of the kernel. Note that this holds for ˜φ^λ/22j independently of x and in particular, there is no renormalization with an affine change of coordinates necessary for this property to hold.

CoverQ(λ/2^j) byλ/2^2j cubes Ql and letg=P

lχQ_lg, whereχQ_l are smooth functions essentially supported onQl withP

χQ_l= 1 and

supp(χdQ_l)⊆B(0, c) (7.55)

for somec1. This is possible due to the Poisson summation formula andλ/2^2j≥ 1.

Effectively, we work with the decomposition g=

N

XχQ_lg+χQ^∗g, χQ^∗g= X

Ql∩Q(λ^1+ε/2^j)=∅

χQ_lg.

Like in the proof of Lemma 7.4.6, we find that the terms containingχQ^∗gcan safely be neglected.

In fact, we find the main contribution of (7.54) raised top/2 to be given by X

Q_k,l,m:

Q_l∩λ^εQ_k6=∅, Qm∩λ^εQk6=∅

T˜^λ/22j(χ\Q_lg1) ˜T^λ/22j(χ\Q_mg2)

p/2

L^p/2([−λ/2^2j,λ/2^2j]×Qk) (7.56)

The conditions onQk, l, min the above display are denoted byQk, l, m: (∗) in the following.

From (7.55) the supports of χ\Q_lg1and χ\Q_mg2 are still separated of unit order, and after the change of variablesx→x0+x, we find

kT˜^λ/22j(χ\Q_lg1) ˜T^λ/22j(χ\Q_mg2)k_Lp/2([−λ/2^2j,λ/2^2j]×Qk)

=kT˜_k^λ/22j(e^iφ˜χ\Q_lg1) ˜T_k^λ/22j(e^iφ˜χ\Q_mg2)k_Lp/2([−λ/2^2j,λ/2^2j]×Q0),

(7.57)

whereφ_k= ˜φ(t, x+x₀;ξ)−φ(0, x˜ ₀;ξ).

Note that theφ_kare normalized elliptic phase functions after an additional affine transformation, which is close to the identity mapping. Hence, (7.57) is amenable to Lemma 7.4.6 with parameter λ/2^2j, and we conclude the bound by means of Cauchy-Schwarz:

X

Qk,l,m:(∗)

kT˜^λ/22j(χ\_Qlg₁) ˜T^λ/22j(χ\_Qmg₂)k^p/2_Lp/2

([−λ/2^2j,λ/2^2j]×Qk)

.^ε X

Q_k,l,m:(∗)

(λ/2^2j)[n(1−2/p)+ε](p/2)kχQlg₁k^p/2_LpkχQmg₂k^p/2_Lp

.ε(λ/2^2j)[n(1−2/p)+2ε](p/2) X

l

kχQlg₁k^p_Lp

!^1/2 X

m

kχQmg₂k^p_Lp

!^1/2

.^ε(λ/2^2j)[n(1−2/p)+2ε](p/2)kg1k^p/2_Lpkg2k^p/2_Lp, and the proof is complete.

For possibly vanishing separation for cubes with side lengthλ^−1/2 we have the following lemma:

Lemma 7.4.8. Let supp( ˆf)⊆B, whereB is a ball of radiusλ^−1/2. Then, we find the following estimate to hold:

kT^λfˆk_Lp(R×Rⁿ).^ελ^ελ^1/pkfk_Lp(Rⁿ).

Proof. We use parabolic rescaling to write like in the proof of Lemma 7.4.7 kT^λfkˆ _Lp(R×Rⁿ).λ^−n/2λ^n/2pλ^1/pkT˜¹gkˆ _Lp(R×Rⁿ),

wherekgkL^p=λ^n/2λ^−n/2pkfkL^p. It is enough to prove kT˜¹ˆgk_Lp(R×Rⁿ).ελ^εkgk_Lp(Rⁿ).

Again, we use the localization property of the kernel to consider cubes of unit size and adapted functions with localized Fourier transformχ_Q. Decompose like in the proof of Lemma 7.4.7

g= X

l:Ql∩Q(λ^1/2+ε)6=∅

χ_Qlg+χ_Q^∗g.

Like in the proof of Lemma 7.4.7 one finds kT˜¹ˆgk^p_Lp([−1,1]×Q).X

Q,l

kT˜¹χ[Q_lgk^p_Lp([−1,1]×Q)+CNλ^−NkgkL^p. The proof is concluded following along the lines of the proof of Lemma 7.4.7.

To separate the contribution of the differentki, we use a natural orthogonality of the oscillatory integrals, which was used in a similar context in [Lee06b].

Lemma 7.4.9([Lee06b, Lemma 3.3, p. 82]). Letφ^λ be a normalized elliptic phase function, 1 ≤ p/2 ≤ 2 and j ≥ ¹₂log₂λ. Then, we find the following estimate to hold:

X

dist(Q^j_k

1,Q^j_k

2)∼2^−j

T^λ(β_k^j

1

fˆ)T^λ(β^j_k

2

f)ˆ _p/2

≤C_ελ^ε



 X

k₁,k₂

kT^λ(β^j_k

1

fˆ)T^λ(β_k^j

2

fˆ)k^p/2_p/2





2/p

+CNλ^−Nkfk²_p.

For the error term recall that we can always suppose thatf in position space is localized to a cube of lengthλfrom the kernel estimate. Hence, using the estimate from [Lee06b, Lemma 3.3] withr= 2 yields the claim after applications of H¨older’s inequality and Plancherel’s theorem.

We prove the following bound for ˜B_j.

Proposition 7.4.10. Let 0 ≤j ≤ ¹₂log₂(λ) and B˜ like above. Then, we find the following estimate to hold:

kB˜j[f, f]k_Lp/2 .^ε

( λ^ε2^4j/p(λ/2^2j)^n(1−2/p)kfk²_Lp, 2 +ⁿ⁺⁴_n+1 ≤p≤4,

λ^ε2^4j/p(λ/2^2j)^n(1−2/p)kfk²_Lp, 4< p <∞. (7.58)

Proof. We have to distinguish between 1 ≤ p/2 ≤ 2 and p > 4. In fact, for 1 ≤p/2 ≤2 we can apply Lemma 7.4.9, but for p >4 we use instead the trivial observation

X

dist(Q^j_k

1,Q^j_k

2)∼2^−j

T^λ(β^j_k

1

f)Tˆ ^λ(β_k^j

2

fˆ) _L∞

.2^jn sup

k₁,k₂

kT^λ(β_k^j

1

fˆ)T^λ(β_k^j

2

fˆ)k_L^∞

and interpolation yields kX

T^λ(β^j_k

1

fˆ)T^λ(β_k^j

2

fˆ)k_Lp/2 ≤C_ελ^ε2^jn(1−4/p)



 X

k₁,k₂

kT^λ(β_k^j

1

fˆ)T^λ(β_k^j

2

fˆ)k^p/2_Lp/2





2/p

.

(7.59)

Consequently, for 1≤p/2≤2:

k X

dist(Q^j_k

1,Q^j_k

2)∼2^−j

T^λ(β_k^j

1

fˆ)T^λ(β_k^j

2

fˆ)k_Lp/2

.^ελ^ε X

dist∼2^−j

kT^λ(β_k^j

1

fˆ)T^λ(β^j_k

2

f)kˆ ^p/2_Lp/2

!^2/p

.ελ^2ε2^4j/p(λ/2^2j)^n(1−2/p)



 X

k1,k2

kP_k^j

1fk^p/2_LpkP_k^j

2fk^p/2_Lp





2/p

.^ελ^2ε2^4j/p(λ/2^2j)^n(1−2/p) X

k₁

kP_k^j

1fk^p_Lp

!1/p

X

k₂

kP_k^j

2fk^p_Lp

!1/p

≤Cελ^2ε2^4j/p(λ/2^2j)^n(1−2/p)kfkL^pkfkL^p

and forp >4 by the above means, but appealing to (7.59) than Lemma 7.4.9,

X

dist(Q^j_k

1,Q^j_k

2)∼2^−j

T^λ(β_k^j

1

fˆ)T^λ(β^j_k

2

fˆ) _Lp/2

.ελ^2ε2^jn(1−4/p)

2^4j/p(λ/2^2j)^(1−2/p)kfk²_Lp.

Proof of Proposition 7.4.3. We conclude that for small p the main contribution comes fromj= 0 and summing a geometric series yields the bound

kT^λfkˆ L^p=k(T^λfˆ)²k^1/2_Lp/2 ≤



 X

0≤j≤j⁰

kB˜_j[f, f]k_Lp/2





1/2

.ε



 X

0≤j≤j⁰

λ^ελ^n(1−2/p)2^4j/p2−2j(n(1−2/p))kfk²_Lp





1/2

.λ^ελ^n(1/2−1/p)kfkL^p.

The same argument works forp→ ∞and interpolation between the bounds for smallpand largepproves the claim.

Conclusively, we argue how Proposition 7.4.3 implies Theorem 7.1.3.

Proposition 7.4.3⇒Theorem 7.1.3. Let ˆf be supported in a ρ⁻¹-ball centered at ρ0. It follows from parabolic rescaling (cf. Section 7.2)

e^{iλφ(Φξ0(ρ2}t/λ,ρx/λ);ξ₀)T^λfˆ◦Φ^λ_ξ

0◦D_ρ= ˜T^λ/ρ2ˆg, where

T˜^λ/ρ2g(t, x) =ˆ Z

Rⁿ

e^iφ˜λ/ρ

2(t,x;ξ)a˜^λ/ρ2(t, x;ξ)ˆg(ξ)dξ, and the phase ˜φis given by

hx, ξi+ Z 1

0

(1−r)h∂_ξξ² φ(Φ_ξ0(t, D_ρ⁰−1x), ξ₀+rξ/ρ)ξ, ξidξ.

A change of spatial variables gives

kT^λfˆkL^p(R×Rⁿ).^φρ^n+2p ρ⁻ⁿkT˜^λ/ρ2ˆgk_Lp((Φ^λ_ξ

0◦Dρ)⁻¹(R×Rⁿ), where like abovekgkL^p =ρⁿρ^−n/pkfkL^p.

The support of ˜T^λ/ρ2ˆgis essentially aλ/ρ×. . .×λ/ρ×λ/ρ²-ellipse. As argued in Section 7.2, ˜φis up to an affine transformation a normalized elliptic phase function.

Note that in the current context, we chooseρ depending on φ so that the phase function, we arrive at after rescaling and an additional affine change of coordinates, is actually normalized. Moreover, the magnitude of the Jacobian of Φ_ξ0 andL(see Section 7.2 for notation) depend on φ. Thus, the implicit constant depends on φcontrary to the applications of parabolic rescaling for normalized elliptic phase functions. To utilize Proposition 7.4.3, we perform an additional decomposition of the ellipse intoλ/ρ²-cubes and use again the orthogonality property which follows from the localization property of the kernel. The argument is concluded like in the proofs of Lemma 7.4.7 and 7.4.8. In order to avoid repetition, the details are omitted.

In the case of constant coefficients it was shown in [RS10] how a localization property of the kernel and an interpolation argument yield globalization.

We point out that the bilinear approach depicted above for variable coefficients also applies to the hyperbolic Schr¨odinger equation (cf. [Lee06a, Var05]) in two dimensions:

i∂_tu(t, x) + (∂_xx−∂_yy)u(t, x) = 0, (t, x)∈R×R²,

u(0, x) =u₀(x)∈L^p_β(R²), (7.60) and using the localization and interpolation argument from [RS10, Section 4] which applies mutatis mutandis yields the following result:

Theorem 7.4.11. Forp >10/3we find the following estimate to hold:

ke^{it(∂xx−∂yy)}u0k_Lp([0,1]×R²).ku0k_L^p

β(R²), β≥4(1 2 −1

p)−2

p (7.61)

This covers the endpoint Sobolev regularity, which is not covered in [Rog08].

Im Dokument Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations (Seite 192-200)