Scaling-sharp dispersive estimates for the Korteweg-de Vries group

Estimées dispersives invariantes d’échelle pour le groupe de Korteweg-de Vries

Scaling-sharp dispersive estimates for the Korteweg-de Vries group

Raphaël Côte Luis Vega

Abstract

We prove weighted estimates on the linear KdV group, which are scaling sharp.

This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations.

Nous prouvons des inégalités à poids pour le groupe linéaire de KdV, qui sont optimales vis-à-vis du changement d’échelle. Ce type d’inégalité suit l’esprit de celles utilisées pour montrer que les solutions des équations de KdV généralisées dont les données sont petites dispersent linéairement.

The purpose of this short note is to give a simple proof of two dispersive estimates which are heavily used in the proof of small data scattering for the generalized Korteweg- de Vries equations [2].

The proof of these estimates can be easily extended to other dispersive equations.

Denote U (t) the linear Korteweg-de Vries group, i.e v = U (t)φ is the solution to v

+ v

= 0,

v(t = 0) = φ, i.e. U \ (t)φ = e

φ ˆ or (U (t)φ)(x) = 1 t

Z Ai

x − y t

φ(y)dy,

where Ai is the Airy function

Ai(z) = 1 π

Z

0

cos ξ

3 + ξz

dξ.

Theorem 1. Let φ, ψ ∈ L

, such that xφ, xψ ∈ L

. Then

kU (t)φk

≤ 2k Ai k

t

kφk

kxφk

, (1) kU (t)φU (−t)ψ

k

≤ Ct

(kφk

kxψk

+ kψk

kxφk

). (2) Furthermore, the constant 2k Ai k

in the first estimate is optimal.

Remark 1. These estimates are often used with φ replaced by U (−t)φ : denoting J(t) = U (t)xU(−t), they take the form

kφk

≤ Ct

kφk

kJ (t)φk

, (3)

kφψ

k

≤ Ct

(kφk

kJ(t)ψk

+ kψk

kJ(t)φk

). (4)

Proof. Due to a scaling argument (and representation in term of the Airy function), we are reduced to show that

kU (1)φk

≤ Ckφk

kxφk

, and similarly for the second inequality. Hence we consider

(U (1)φ)(x) = Z

Ai(x − y)φ(y)dy,

and we recall that the Airy function satisfies | Ai(x)| ≤ C(1 + |x|)

and | Ai

(x)| ≤ C(1 + |x|)

. Then

|U (1)φ|

(x) = Z Z

Ai(x − y)φ(y) Ai(x − z) ¯ φ(z) y − z y − z dydz

= Z

Ai(x − y)yφ(y)

Z Ai(x − z) y − z φ(z)dz ¯

| {z }

Hz7→y(Ai(x−z)φ(z))(y)

dy

− Z

Ai(x − z)z φ(z) ¯

Z Ai(x − y)

y − z φ(y)dy

| {z }

−Hy7→z(Ai(x−y)φ(y))(z)

dz

= 2<

Z

Ai(x − y)yφ(y)H

(Ai(x − z)φ(z))(y)dy,

where H denotes the Hilbert transform (and with the slight abuse of notation

for vp

). As H : L

→ L

is isometric and hence continuous (with norm 1), and Ai ∈ L

, we get

|U (1)φ|

(x) ≤ 2k Ai(x − y)yφ(y)k

kH(Ai(x − ·)φ)(y)k

(5)

≤ 2k Ai k

kyφk

kφk

. (6) This is the first inequality. Let us now prove that the constant is sharp.

First consider the minimizers in the following Cauchy-Schwarz inequality :

Z

yψ(y)H(ψ)(y)dy

≤ kyψ(y)k

kψk

. (7) There is equality if yψ(y) = λH(ψ)(y) for some λ ∈ C . Then a Fourier Transform shows that ∂

ψ(ξ) = ˆ λ sgn ξ ψ(ξ), hence ˆ ψ(ξ) = ˆ C exp(−λ|x|), or equivalenty, one has equality in (7) as soon as

ψ(y) = C

1 + (y/λ)

for some λ, C ∈ C . (8)

(Notice that all the functions involved lie in L

)

We now go back to (6). Let x

∈ R where | Ai | reaches its maximum. Now as

Ai(x

) 6= 0, let ε > 0 such that for all y ∈ [−ε, ε], | Ai(x

− y)| ≥ | Ai(x

)|/2, and

consider the sequence of functions

Denote ψ

(y) =

. As Ai(x

− y)φ

(ny) = √

nψ

(ny),

|U (1)φ

|

(x

) = 2 Z

y √

nψ

(ny)H

( √

nψ

(nz))(y)dy = 2 n

Z

yψ

(ny)H(ψ

)(ny)dy.

One easily sees that ψ

(y) →

in L

and yψ

(y) →

in L

, and hence, in view of (8), as H is homogeneous of degree 0 and L

isometric, we have

|U (1)φ

|

(x

) ∼ 2 n

Z y

1 + |y|

H( 1

1 + | · |

)(y)dy ∼ 2 n k y

1 + |y|

k

k 1 1 + |y|

k

∼ 2ky √

nψ

(ny)k

k √

nψ

(ny)k

∼ 2ky Ai(x

− y)φ

(y)k

k Ai(x

− y)φ

(y)k

. As φ

concentrates at point 0, we deduce

|U (1)φ

|

(x

) ∼ 2| Ai(x

)|

kyφ

(y)k

kφ

(y)k

as n → ∞, (9) which proves that the sharp constant in the first inequality is 2k Ai k

.

For the second inequality (estimate of the derivative), we have as for the first in- equality :

(U (1)φU (1) ¯ ψ

)(x)

= Z Z

Ai(x − y)φ(y)Ai

(x − z) ¯ ψ(z) y − z y − z dydz

= Z

Ai

(x − z) ¯ ψ(z)

Z Ai(x − y)yφ(y) y − z dy

dz −

Z

Ai

(x − z)z ψ(z) ¯

Z Ai(x − y)φ(y) z − y dy

dz

= Z

Ai

(x − z) ¯ ψ(z)H

(Ai(x − y)yφ(y))(z)dz − Z

Ai

(x − z)z ψ(z)H ¯

(Ai(x − y)φ(y))(z)dz.

Denote ω

(z) = √

1+|x−z|

: ω

∈ A

(with the notation of [4]), so that there exists C not depending on x such that

∀v, Z

|Hv|

ω

≤ C Z

|v|

ω

.

Recall the well-known asymptotic |Ai

(x)| ≤ C(1 + |x|

). Then

Z

Ai

(x − z) ¯ ψ(z)H

(Ai(x − y)yφ(y))(z)dz

≤ Z

|Ai

(x − z) ¯ ψ(z)|

ω

(z)dy

Z

|H

(Ai(x − y)yφ(y))(z)|

ω

(z)dz

≤ Ckψk

Z

|Ai(x − y)yφ(y)|

ω

(y)dy

≤ Ckψk

kyφ(y)k

.

In the same way,

Z

Ai

(x − z)z ψ(z)H(Ai(x ¯ − ·)φ)(z)dz

≤ Z

|Ai

(x − z)z ψ(z)| ¯

ω

(z)dz

Z

|H(Ai(x − ·)φ)(z)|

ω

(z)dz

≤ Ckzψ(z)k

Z

|Ai(x − y)φ(y)|

ω

(y)dy

≤ Ckyψ(y)k

kφk

. Thus:

kU (1)φU(1) ¯ ψ

k

≤ C(kφk

kxψk

+ kψk

kxφk

).

Up to scaling and replacing ψ by ψ, this is the second inequality. ¯

Remark 2. This proof (especially (5)) is reminiscent of that in [3] (see also [1])

kφk

≤ kφk

kφ

k

,

where the constant is sharp and minimizers are Ce

. This has application to the Schrödinger group U (t) (i.e U \ (t)φ = e

φ). We have the following Schrödinger version ˆ of estimate (3) (notice that U (t)xU (−t) = e

2 4t it

2

∂

e

2 4t

) :

kψk

= ke

2

4t

ψk

≤ ke

2

4t

ψk

ke

2 4t

∂

e

2

4t

ψk

≤ 2

t kψk

kU (t)xU (−t)ψk

. From Theorem 1, we can easily obtain the optimal decay in a scaling sharp Besov like space. Let ϕ ∈ D( R ) be non-negative with support in ] −2, 2[ and such that ϕ equals 1 in a neighbourhood of [−1.5, 1.5]. Denote ψ(x) = ϕ(2x) − ϕ(x) and ψ

(x) = ψ(x/2

).

Finally introduce

kφk

=

∞

X

j=−∞

2

kψ

U (−t)φk

.

Corollary 1. We have :

kφk

≤ Ct

kφk

. Proof. Notice that |xψ

(x)| ≤ 2

ψ

(x). As φ = P

j

U (t)ψ

U (−t)φ, we have : kφk

≤ X

j

kU (t)ψ

U (−t)φk

≤ Ct

X

j

kU(t)ψ

U (−t)φk

kU (t)xU(−t)U (t)ψ

U (−t)φk

≤ Ct

X

j

kU(t)ψ

U (−t)φk

kxψ

U (−t)φk

≤ Ct

X

j

kU(t)ψ

U (−t)φk

L²

2

kU (t)ψ

U (−t)φk

L²

≤ Ct

kφk

.

References

[1] J. Duoandikoetxea, L. Vega, Some weighted Gagliardo-Nirenberg inequalities and applications, Proc. Amer. Math. Soc. 135 (2007) 2795–2802 (electronic).

[2] N. Hayashi, P. I. Naumkin, Large time asymptotics of solutions to the generalized Korteweg-De Vries equation, J. Funct. Anal. 159 (1998) 110–136.

[3] E. H. Lieb, M. Loss, Analysis, American Mathematical Society, Providence, RI, 2001.

[4] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, N.J., 1970.

Raphaël Côte

Centre de Mathématiques Laurent Schwartz, École polytechnique 91128 Palaiseau Cedex, France

cote@math.polytechnique.fr

Luis Vega

Departamento de Matemáticas, Universidad del País Vasco Aptdo. 644, 48080 Bilbao, España

luis.vega@ehu.es