Advances in pseudo-differential operators, 65–75
Oper. Theory Adv. Appl., 155, Birkhäuser, Basel, 2004.
A new proof of global smoothing estimates for
dispersive equations
Michael Ruzhansky and Mitsuru Sugimoto
Abstract. The aim of this article is to provide a new method to prove global
smoothing estimates for dispersive equations such as Schrödinger equations.
For the purpose, the Egorov-type theorem via canonical transformation in the
form of a class of Fourier integral operators is established, and their weighted
L2 -boundedness is also proved. The boundedness result is not covered by
previous one such as Asada and Fujiwara [1]. By using them, a different proof
for the result obtained by Ben-Artzi & Klainerman [2] is provided. This new
idea gives a clear understanding of smoothing effects of dispersive equations,
and further developments are also expected. In fact, some extended results
based on the same idea are also announced.
1. Introduction
We consider Fourier integral operators, which can be globally written in the form
Z Z
−n
eiφ(x,y,ξ) p(x, y, ξ)u(y) dydξ (x ∈ Rn ),
(1.1)
T u(x) = (2π)
Rn
Rn
where p(x, y, ξ) is an amplitude function and φ(x, y, ξ) is a real phase function.
Especially, if p(x, y, ξ) = 1 and φ(x, y, ξ) satisfies the graph condition
Λ = {(x, φx , y, −φy ); φξ = 0}
= {(x, ξ), χ(x, ξ)} ⊂ T ∗ Rn × T ∗ Rn ,
Received by the editors December 31, 2003.
1991 Mathematics Subject Classification. Primary 35Q40; Secondary 35B65.
Key words and phrases. Dispersive equation, Smoothing effect, Canonical transformation.
This work was completed with the aid of “UK-Japan Joint Project Grant” by “The Royal Society”
and “Japan Society for the Promotion of Science”.
2
M. Ruzhansky and M. Sugimoto
we have the relation
T ∙ A(X, D) = B(X, D) ∙ T + (lower),
B(x, ξ) = (A ◦ χ)(x, ξ).
In this way, Fourier integral operators are recognized as a tool of the realization
of the canonical transformation. This fact is known as Egorov’s theory, and by
taking phase function appropriately, properties of the operator B(X, D) can be
extracted from those of the well known operator A(X, D).
If we take
φ(x, y, ξ) = x ∙ ξ − y ∙ ψ(ξ)
(1.2)
as a special case, we have
T u(x) = F −1 [(F u)(ψ(ξ))](x),
where F (F −1 resp.) denotes the (inverse resp.) Fourier transformation. Hence we
have the relation
(1.3)
T ∙ σ(D) = (σ ◦ ψ)(D) ∙ T,
a(D) = (σ ◦ ψ)(D),
for constant variable operators σ(D) and a(D). For example, for a positive and
positively homogeneous function a(ξ) of order 2, we have
T ∙ (−4) = a(D) ∙ T,
if we take
p
∇a(ξ)
.
|∇a(ξ)|
Since the property of the Laplacian is well known in various situations, we can
expect the same for a(D).
By using Egorov’s theorem, many qualitative properties of solutions of partial differential equations (propagation of singularity for example) has been investigated. Our main interest is to have quantitative (L2 -property for example) by the
same idea. As an example, we present a new proof of global smoothing estimates
for Schrödinger equations in this article.
σ(η) = |η|2 ,
ψ(ξ) =
a(ξ)
2. Smoothing effect of Schrödinger equations
We consider the following Schrödinger equation:
(
(i∂t + 4x ) u(t, x) = 0,
(2.1)
u(0, x) = ϕ(x) ∈ L2 (Rn ).
By Plancherel’s theorem, we know that the solution operator eit4 preserves the
L2 -norm, that is, we have
ku(t, ∙)kL2 (Rnx ) = kϕkL2 (Rn ) ,
Smoothing estimates for dispersive equations
3
for fixed t ∈ R. On the other hand, the extra gain of regularity of order 1/2 in x
can be observed if we integrate the solution in t. For example, in the case n = 1,
we have
≤ CkϕkL2 (R) ,
|Dx |1/2 u(∙, x) 2
L (Rt )
for any fixed x ∈ R. This result was given by Kenig, Ponce and Vega [5], and will
play an important role later in Section 4 (Proposition 4.1).
In the higher dimensional case n ≥ 2, similar smoothing properties were
given:
kAukL2 (Rt ×Rn ) ≤ CkϕkL2 (Rn ) ,
(2.2)
x
x
where A is one of the followings:
(1)
(2)
(3)
A = hxi−s |D|1/2 ; s > 1/2,
A = hxi−s hDi1/2 ; s ≥ 1
α−1
A = |x|
(s > 1 if n = 2),
α
|D| ; 1 − n/2 < α < 1/2.
The type (1) was given by Ben-Artzi and Klainerman [2] (n ≥ 3), and Chihara
[3] (n ≥ 2). The type (2) was given by Kato and Yajima [4] (n ≥ 3), and Walther
[13] (n ≥ 2) which also showed that it is not true for s < 1 (s ≤ 1 if n = 2).
The type (3) was given by Kato and Yajima [4] (n ≥ 3, 0 ≤ α < 1/2 or n = 2,
0 < α < 1/2 ), and Sugimoto [12] (n ≥ 2). Watanabe [15] showed that it is not
true for α = 1/2.
Each proof was carried out by proving one of the following estimates (or their
variants):
√
d
(2.3)
(Restriction theorem),
A∗ f |ρS n−1 2 n−1 ≤ C ρkf kL2 (Rn )
L (ρS
where, ρS
(2.4)
n−1
)
= {ξ; |ξ| = ρ}, (ρ > 0).
2
sup |(R(ζ)A∗ f, A∗ f )| ≤ Ckf kL2 (Rn )
(Resolvent estimate),
Im ζ>0
−1
where R(ζ) = (−4 − ζ) .
Estimate (2.3) implies the dual one of estimate (2.2). Estimate (2.4) implies
(2.2) since the resolvent R(ζ) is the Laplace transform of the solution operator
eit4 of equation (2.1):
Z
1 ∞ it4 iζt
e e dt (Im ζ > 0).
R(ζ) =
i 0
The fact that (2.4) implies (2.3) due to the formula
2
1
ˆ
n−1
f
Im R(ρ2 + i0)f, f =
2 n−1
4(2π)n−1 ρ |ρS
L (ρS
)
was also used.
Following the idea in Introduction, we present a completely different idea to
prove them. In this note, we give another proof of the result given by Ben-Artzi and
4
M. Ruzhansky and M. Sugimoto
Klainerman [2] which treated the type (1) above. In order to treat more general
equations, we consider the following dispersive equation:
(
(2.5)
(i∂t − a(D)) u(t, x) = 0,
u(0, x) = ϕ(x) ∈ L2 (Rn ).
where a(ξ) ∈ C ∞ (Rn \ 0) is real-valued and positively homogeneous of order 2.
Equation (2.5) with a(ξ) = |ξ|2 is the Schrödinger equation (2.1).
Theorem 2.1. Suppose n ≥ 3 and s > 1/2. Assume that the real-valued function
a(ξ) ∈ C ∞ (Rn \ 0) satisfies a(λξ) = λ2 a(ξ) and ∇a(ξ) 6= 0 for all λ > 0 and
ξ ∈ Rn \ 0. Then the solution u to (2.5) satisfies
hxi
−s
|D|1/2 u(t, x) ∈ L2 (Rt × Rnx ).
Remark 2.2. Recently Chihara [3] proved Theorem 2.1 in a general setting by
proving the restriction theorem (2.3) or the resolvent estimates (2.4). But we can
prove more general result by using our idea. See Theorem 5.1 and Remark 5.2 in
Section 5.
3. Main tool
Based on the argument in Introduction, we introduce the main tool for the proof
of Theorem 2.1.
Let ψ, ψ −1 : Rn \ 0 → Rn \ 0 be C ∞ -maps satisfying ψ ◦ ψ −1 (ξ) =
ψ −1 ◦ ψ (ξ) = ξ, ψ(λξ) = λψ(ξ), and ψ −1 (λξ) = λψ −1 (ξ) for all λ > 0 and
ξ ∈ Rn \ 0. Then we have
C −1 ≤ |det ∂ψ(ξ)| ≤ C, C −1 ≤ det ∂ψ −1 (ξ) ≤ C
for some C > 0. We set
Iu(x) = (2π)
−n
(3.1)
I −1 u(x) = (2π)
Z
Rn
−n
Z
Z
ei(x∙ξ−y∙ψ(ξ)) u(y) dydξ,
Rn
Rn
Z
ei(x∙ξ−y∙ψ
−1
(ξ))
u(y) dydξ.
Rn
We remark that we have
Iu(x) = F −1 [(F u)(ψ(ξ))](x),
I −1 u(x) = F −1 (F u)(ψ −1 (ξ)) (x).
Since I −1 ∙ I = I ∙ I −1 = id, we have the formula
(3.2)
a(D) = I ∙ σ(D) ∙ I −1 ,
a(ξ) = (σ ◦ ψ)(ξ)
Smoothing estimates for dispersive equations
5
by (1.3). By Plancherel’s theorem, the operators I and I −1 are L2 -bounded. Furthermore, we have the boundedness on weighted spaces. For k ∈ R, let L2k (Rn ) be
the set of functions f such that the norm
Z
1/2
k
k/2
hxi f (x)2 dx
kf kL2 (Rn ) =
; hxik = 1 + |x|2
k
Rn
is finite.
Proposition 3.1. Suppose n ≥ 3 and 0 ≤ k ≤ 1. Then the operators I and I −1
defined by (3.1) are L2−k (Rn )-bounded.
Proof. We prove the boundedness of I. The boundedness of I −1 can be given by
replacing ψ by ψ −1 . If we use the equality
1 + iy ∙ ∂ξ −iy∙ξ
e−iy∙ξ =
e
,
2
hyi
change of variables, and integration by parts, we can justify the following:
Iu(x)
= (2π)−n
= (2π)−n
= (2π)
−n
Z Z
Z Z
Z Z
ei(x∙ψ
−1
ei(x∙ψ
−1
e
(ξ)−y∙ξ)
d(ξ)u(y)dydξ
!
d(ξ) 1 + x∂ψ −1 (ξ)t y − i(∂d)(ξ) ∙ y
(ξ)−y∙ξ)
hyi
2
1 + xA(ξ)t y + a(ξ) ∙ y
i(x∙ξ−y∙ψ(ξ))
hyi
where
2
!
u(y)dydξ
u(y)dydξ,
−1
Hence we have
(3.3)
˜
A(ξ) = (∂ψ) (ξ), a(ξ) = −i(∂d)(ψ(ξ))d(ξ),
˜ = |det ∂ψ(ξ)|.
d(ξ) = det ∂ψ −1 (ξ), d(ξ)
I=I
t
1
hxi
2
+ xA(D)
x
hxi
2
+ a(D) ∙ I
x
hxi
2.
We remember here that I is L2 -bounded by Plancherel’s theorem. We remark that
all entries of |D|a(D) and A(D) are L2 -bounded, and |D|−1 is bounded from L2
to L2−1 . Using them, we obtain the L2−1 -boundedness of I from (3.3). Then, by the
interpolation, we have the desired result. To justify these boundedness, use the
following results of Kurtz and Wheeden [6], Stein and Weiss [11]:
Lemma 3.2 ([6]). Suppose −n/2 < δ < n/2.
X
δ
|x| m(D)u 2 n ≤ C
sup
L (R )
|γ|≤n
ξ∈Rn
Then we have
|γ| γ
|ξ| ∂ m(ξ)|x|δ uL2 (Rn ) .
6
M. Ruzhansky and M. Sugimoto
Lemma 3.3 ([11]). Suppose γ < n/2, δ < n/2, m < n, and γ + δ + m = n. Then
we have
−γ
|x| |D|m−n u 2 n ≤ C |x|δ u 2 n .
L (R )
L (R )
4. Proof of Theorem 2.1
By the formula (3.2), we may show the result by replacing a(D) by σ(D). In fact,
applying I −1 to equation (2.5) we have
(
(i∂t − σ(D)) v(t, x) = 0
(4.1)
v(0, x) = g(x),
where
v = I −1 u,
Suppose that we can show
−s 1/2
(4.2)
hxi |D| v(t, x)
g = I −1 ϕ.
L2 (Rt ×Rn
x)
≤ CkgkL2 (Rn ) .
x
1/2
Substituting v = I −1 u and g = I −1 ϕ and noticing |ψ(D)|
= I ∙ |D|1/2 ∙ I −1 by
(3.2)Cwe have
−s −1
≤ C I −1 ϕL2 (Rn ) .
hxi I |ψ(D)|1/2 u(t, x) 2
n
L (Rt ×Rx )
x
Furthermore, for 0 ≤ s ≤ 1, |ψ(D)|1/2 |D|−1/2 is L2−s -bounded by Lemma 3.2, and
I, I −1 by Proposition 3.1. Hence we have
−s 1/2
≤ CkϕkL2 (Rn ) ,
hxi |D| u(t, x) 2
x
n
L (Rt ×Rx )
which means that we have the same estimates for equation (2.5).
By the microlocalization and the rotation, we may assume that the initial
data ϕ in equation (2.5) satisfies supp ϕ̂ ⊂ Γ, where Γ ⊂ Rn \ 0 is a sufficiently
small conic neighborhood of en = (0, . . . 0, 1). We remark that the argument so
far can be justified for microlocal C ∞ -maps ψ : Γ → Γ̃ and ψ −1 : Γ̃ → Γ, where
Γ, Γ̃ ⊂ Rn \ 0 are cones. All our task is to find ψ : Γ → Γ̃ and σ(η) such that
det ∂ψ(en ) 6= 0, a(ξ) = (σ ◦ ψ)(ξ) (ξ ∈ Γ), and prove the estimate (4.2) for equation
(4.1). We also remark that we have supp ĝ ⊂ Γ̃ if we notice g = I −1 ϕ and (3.2).
The assumption ∇a(en ) 6= 0 implies the following two cases:
(i): ∂n a(en ) 6= 0. Then, by Euler’s identity a(ξ) = (ξ/2) ∙ ∇a(ξ), we have
a(en ) 6= 0. Hence, in this case, we may assume
a(ξ) > 0 (ξ ∈ Γ),
∂n a(en ) 6= 0.
Smoothing estimates for dispersive equations
7
(ii): ∂n a(en ) = 0. Then, by assumption ∇a(en ) 6= 0, there exits j 6= n such
that ∂j a(en ) 6= 0. Hence, in this case, we may assume
∂1 a(en ) 6= 0.
Case (i)
We take
p
σ(η) = ηn2 , ψ(ξ) = (ξ1 , . . . , ξn−1 , a(ξ)).
Then we have a(ξ) = (σ ◦ ψ)(ξ) and
En−1
0 p
det ∂ψ(en ) = 6= 0,
∂n a(en )/ 2 a(en ) ∗
where En−1 is the identity matrix of order n − 1. The estimate for σ(D) = Dn2 was
given by the following (Kenig, Ponce and Vega [5, p.56]):
Proposition 4.1. In the case n = 1, we have
2
sup |Dx |1/2 eitDx f (x)
2
L (Rt )
x∈R
Hence we have
−s
hxi |Dn |1/2 eitσ(D) g(x)
≤ Ckf kL2 (Rx ) .
L2 (Rt ×Rn
x)
≤ CkgkL2 (Rn )
x
−s
−s
for s > 1/2. Here we have used the trivial inequality hxi ≤ hxn i , Schwarz’s
inequality, and Plancherel’s theorem. We remark that ψ maps Γ to another small
conic neighborhood Γ̃ of en . Since supp ĝ ⊂ Γ̃ and |ξn | is equivalent to |ξ| on Γ̃,
we have the estimate
−s 1/2 itσ(D)
g(x)
≤ CkgkL2 (Rn ) ,
hxi |D| e
2
n
x
L (Rt ×Rx )
that is, estimate (4.2).
Case (ii)
We take
σ(η) = η1 ηn ,
ψ(ξ) =
Then we have a(ξ) = (σ ◦ ψ)(ξ) and
a(ξ)
, ξ2 , . . . , ξ n
ξn
∂ a(e )
det ∂ψ(en ) = 1 n
0
∗ 6= 0.
En−1 The estimate for σ(D) = D1 Dn was given by the following (Linares and
Ponce [7, p.528]):
Proposition 4.2. In the case n = 2, we have
sup |Dx |1/2 eitDx Dy f (x, y)
2
y∈R
L (Rt ×Rx )
≤ Ckf kL2 (R2 ) .
x,y
8
M. Ruzhansky and M. Sugimoto
Hence, similarly in the case (i), we have
−s 1/2 itσ(D)
g(x)
hxi |D| e
2
L (Rt ×Rn
x)
≤ CkgkL2 (Rn )
x
for s > 1/2, that is, estimate (4.2).
5. Announcement of further results
Finally, we mention further results. By developing the argument in Section 4, we
can have more generalized results on smoothing effects for dispersive equations.
We exhibit here some of them without the proofs which will appear in [9] and [10].
Let consider equation (2.5), but we assume that real-valued function a(ξ) is
of order m. Let am (ξ) ∈ C ∞ (Rn \0) be a positively homogeneous function of order
m satisfying ∇am (ξ) 6= 0 (ξ 6= 0). We assume one of the followings:
(H): a(ξ) = am (ξ).
m−1−|α|
(L): a(ξ) ∈ C ∞ (Rn ), ∇a(ξ) 6= 0, and |∂ α (a(ξ) − am (ξ))| ≤ Chξi
for
large |ξ|.
For example a(ξ1 , ξ2 ) = ξ13 + ξ23 + ξ1 satisfies (L).
Under the assumption above, we have the following results corresponding to
the types (1), (2), and (3) respectively in Section 2:
Theorem 5.1 (Type (1)). Suppose m > 0 and n ≥ 2. Assume (H) or (L). Then the
solution u to (2.5) satisfies
for s > 1/2.
hxi
−s
|D|(m−1)/2 u(t, x) ∈ L2 (Rt × Rnx )
Remark 5.2. Chihara [3] proved Theorem 5.1 with m > 1 under the assumption
(H).
Theorem 5.3 (Type (2)). Suppose m > 1 and n > m + 1, or n > m > 1 in the
case a(ξ) > 0. Assume (H) or (L). Then the solution u to (2.5) satisfies
hxi
−m/2
hDi
(m−1)/2
u(t, x) ∈ L2 (Rt × Rnx ).
Remark 5.4. Walther [14] proved Theorem 5.3 with a(ξ) = |ξ|m and n > m > 1.
The optimality of the orders −m/2 and (m − 1)/2 are also mentioned there.
For the proofs of Theorems 5.1 and 5.3, we use the following result which is
a generalized version of Proposition 3.1:
Proposition 5.5. Suppose n ≥ 2 and 0 ≤ |k| ≤ n/2. Then the operators I and I −1
defined by (3.1) are L2k (Rn )-bounded.
Smoothing estimates for dispersive equations
9
In order to state one more result, we introduce some notations. Let
{(x(t), ξ(t)); t ∈ R}
be the classical orbit, that is, the solutions of the ordinary differential equation
(
˙ =0
ẋ(t) = (∇a)(ξ(t)), ξ(t)
x(0) = 0,
ξ(0) = ω,
and consider the set of the path of all classical orbits
Γa = (x(t), ξ(t)); t ∈ R, ω ∈ S n−1
= {(λ∇a(ξ), ξ); ξ ∈ Rn \ 0, λ ∈ R}.
In the case a(ξ) = |ξ|2 for example, we have
Γa = {(x, ξ) ∈ T ∗ Rn \ 0; |x ∧ ξ| = 0}.
For the symbol σ(x, ξ) of the pseudo-differential operator σ(X, D), we use the
6 0, ξ 6= 0 and
notation σ(x, ξ) ∼ |x|a |ξ|b if the symbol σ(x, ξ) is smooth in x =
satisfies
σ(λx, ξ) = λa σ(x, ξ), σ(x, λξ) = λb σ(x, ξ)
(λ > 0).
Theorem 5.6 (Type (3)). Suppose m ∈ N and n ≥ 2. Assume that (H), a(ξ) > 0,
and the Gaussian curvature of Σa = {ξ; a(ξ) = 1} never vanishes. Also assume
that
τ (x, ξ) ∼ |x|−1/2 |ξ|(m−1)/2
and
(5.1)
τ (x, ξ) = 0
if (x, ξ) ∈ Γa and x 6= 0.
Then the solution u to (2.5) satisfies
τ (X, D)u(t, x) ∈ L2 (Rt × Rnx ).
Remark 5.7. In the case m = 2, Theorem 5.6 corresponds to the critical case
α = 1/2 of Kato and Yajima [4] which cannot be allowed without the additional
structure condition (5.1).
For the proof of Theorem 5.6, we need more refined boundedness theorem of
Fourier integral operators. We should mention here that the global L2 -boundedness
with the phase function (1.2) is not covered by previous results, for example, Asada
and Fujiwara [1], The result of [1] is motivated by the construction of fundamental
solution of Schrödinger equation in the way of Feynman’s path integral, and it
requires the boundedness of all the derivatives of entries of the matrix
∂ x ∂y φ ∂ x ∂ ξ φ
D(φ) =
.
∂ ξ ∂ y φ ∂ ξ ∂ξ φ
10
M. Ruzhansky and M. Sugimoto
• (Asada and Fujiwara [1]) Assume that |det D(φ)| ≥ C > 0 and and all
the derivatives of p(x, y, ξ) and those of each entry of D(φ) are bounded. Then T
defined by (1.1) is bounded on L2 (Rn ).
With our phase function (1.2), the boundedness of the entries of ∂ξ ∂ξ φ fails.
But the following result will appear in [8]:
Theorem 5.8. Let the operator T be defined by (1.1) with φ(x, y, ξ) = x∙ ξ + ϕ(y, ξ).
Assume that
|det ∂y ∂ξ ϕ(y, ξ)| ≥ C > 0,
and all the derivatives of entries of ∂y ∂ξ ϕ are bounded. Also assume that
α
∂ξ ϕ(y, ξ) ≤ Cα hyi (∀|α| ≥ 1),
α β γ
∂x ∂y ∂ξ p(x, y, ξ) ≤ Cαβγ hxi−|α| (∀α, β, γ)
or
α β
∂y ∂ξ ϕ(y, ξ) ≤ Cα hyi1−|α| (∀α, |β| ≥ 1),
α β γ
∂x ∂y ∂ξ p(x, y, ξ) ≤ Cαβγ hyi−|β| (∀α, β, γ)
Then T is bounded on L2k (Rn ) for any k ∈ R.
This theorem says that, if amplitude functions p(x, y, ξ) have some decaying
properties with respect to x or y, we do not need the boundedness of ∂ξ ∂ξ φ for the
L2 -boundedness, as required in [1], and we can have weighted estimates, as well.
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Smoothing estimates for dispersive equations
11
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Math. 14 (1991), 239–250.
Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7
2BZ, UK
E-mail address: [email protected]
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
E-mail address: [email protected]