Global boundedness theorems for Fourier integral operators associated with canonical transformations

GLOBAL BOUNDEDNESS THEOREMS FOR FOURIERINTEGRAL OPERATORS ASSOCIATED WITH

CANONICAL TRANSFORMATIONS

MICHAEL RUZHANSKY AND MITSURU SUGIMOTO

Abstract. We present global L2-boundedness theorems for a class

of Fourier integral operators, which appears naturally in many quan-

titative studies of partial differential equations. As a consequence,

we will improve known global results for several classes of pseudo-

differential and Fourier integral operators, and also extend previ-

ous results of Asada and Fujiwara [1] or Kumano-go [12]. As an

application, we show a global smoothing estimate to generalized

Schrodinger equations which extends the results of Kato and Ya-

jima [10], Walther [19]. We also present a result on some non-linear

problem.

This survey paper is based on the talk by the second author. Theproofs of the theorems below have appeared or will appear in [13], [14],[15], [16] and so on.

1. Fourier integral operators

We consider the following Fourier integral operator:

(1.1) Tu(x) =∫

Rn

∫Rn

eiφ(x,y,ξ)a(x, y, ξ)u(y) dydξ

(x ∈ Rn), where a(x, y, ξ) is an amplitude function and φ(x, y, ξ) is a

real phase function of the form

φ(x, y, ξ) = x · ξ + ϕ(y, ξ).

2000 Mathematics Subject Classification. Primary 35S30; Secondary 35B65.

Key words and phrases. Fourier integral operator, Smoothing effect.

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”.

1

2 M. RUZHANSKY AND M. SUGIMOTO

Note that, by the equivalence of phase function theorem, Fourier in-tegral operators with the local graph condition can always be written inthis form locally.

Local L2 mapping property of (1.1) has already been established byEskin [7] and Hormander [9]. We present here the global L2-boundednessproperties of operators (1.1), which appears naturally in many problems.

2. When is T globally L2-bounded?

Here is a known answer to this question:

• (Asada and Fujiwara [1]) Assume that all the derivatives ofa(x, y, ξ) and all the derivatives of each entry of the matrix

D(φ) =

(∂x∂yφ ∂x∂ξφ

∂ξ∂yφ ∂ξ∂ξφ

)

are bounded. Also assume that |detD(φ)| ≥ C > 0. Then T is L2(Rn)-bounded.

The result above was used to construct the fundamental solution ofSchrodinger equation in the way of Feynman’s path integral.

(The result of Kumano-go [12] was used to construct the fundamentalsolution of hyperbolic equations, and it requires that

J(y, ξ) = φ(x, y, ξ) − (x− y) · ξsatisfies ∣∣∣∂α

y ∂βξ J(y, ξ)

∣∣∣ ≤ Cαβ(1 + |ξ|)1−|β|

for all α and β.)However, there one had to make a quite restrictive and not always

natural assumption on the boundedness of ∂ξ∂ξφ, which fails in manyimportant cases.

The case we have in mind is

(2.1) φ(x, y, ξ) = x · ξ − y · ψ(ξ),

where ψ(ξ) is a smooth function of growth order 1. If we take ψ(ξ) = ξ,then we have φ(x, y, ξ) = x · ξ − y · ξ, and the operator T defined by itis a pseudo-differential operator.

BOUNDEDNESS THEOREMS FOR FOURIER INTEGRAL OPERATORS 3

We cannot use Asada-Fujiwara’s result with our example (2.1), be-cause the boundedness of the entries of ∂ξ∂ξφ fails generally. (We cannotuse Kumano-go’s either by the same reason.)

3. Why is the phase function (2.1) important?

Because it is used to represent a canonical transformations. In fact,if we take a(x, y, ξ) = 1, we have

(3.1) Tu(x) = F−1[(Fu)(ψ(ξ))](x)

henceT · σ(D) = (σ ψ)(D) · T.

Especially, for a positive and homogeneous function p(ξ) ∈ C∞(Rn \ 0)of degree 1, we have the relation

(3.2) T · (−) · T−1 = p(D)2

if we take

(3.3) ψ(ξ) = p(ξ)∇p(ξ)|∇p(ξ)|

and assume that the hypersurface

Σ = ξ; p(ξ) = 1has non-vanishing Gaussian curvature.

The curvature condition on Σ means that the Gauss map

∇p|∇p| : Σ → Sn−1

is a global diffeomorphism and its Jacobian never vanishes. (See, forexample, Kobayashi and Nomizu [11].) Hence, we can construct theinverse C∞-map ψ−1(ξ) of ψ(ξ) defined by (3.3). On account of (3.1),the inverse T−1 can be given by replacing ψ by ψ−1.

The L2-property of the Laplacian − is well known in various situ-ations. Our objective is to know the L2-property of the operator T , sothat we can extract the L2-property of the operator p(D)2 from that ofthe Laplacian.


4. Main results

The following is our main result, which is expected to have manyapplications. For m ∈ R, we set

〈x〉m =(1 + |x|2)m/2

.

Let L2m(Rn) be the set of functions f such that the norm

‖f‖L2m(Rn) =

(∫Rn

|〈x〉mf(x)|2 dx)1/2

is finite.

Theorem 4.1. Let φ(x, y, ξ) = x · ξ + ϕ(y, ξ). Assume that

|det ∂y∂ξϕ(y, ξ)| ≥ C > 0,

and all the derivatives of entries of ∂y∂ξϕ are bounded. Also assumethat ∣∣∂α

ξ ϕ(y, ξ)∣∣ ≤ Cα〈y〉 for all |α| ≥ 1,∣∣∣∂α

x ∂βy ∂

γξ a(x, y, ξ)

∣∣∣ ≤ Cαβγ〈x〉−|α|

for all α, β, and γ. Then T is bounded on L2m(Rn) for any m ∈ R.

Theorem 4.1 says that, if amplitude functions a(x, y, ξ) have somedecaying properties with respect to x, we do not need the boundednessof ∂ξ∂ξφ for the L2-boundedness, as required in Asada-Fujiwara [1], andwe can have weighted estimates, as well.

The same is true when both phase and amplitude functions have somedecaying properties with respect to y.

Theorem 4.2. Let φ(x, y, ξ) = x · ξ + ϕ(y, ξ). Assume that

|det ∂y∂ξϕ(y, ξ)| ≥ C > 0.

Also assume that∣∣∣∂αy ∂

βξ ϕ(y, ξ)

∣∣∣ ≤ Cα〈y〉1−|α| for all α, |β| ≥ 1,∣∣∣∂αx ∂

βy ∂

γξ a(x, y, ξ)

∣∣∣ ≤ Cαβγ〈y〉−|β|

for all α, β, and γ. Then T is bounded on L2m(Rn) for any m ∈ R.


If the amplitude a(x, y, ξ) is independent of the variable x or y, thedecaying property can be automatically satisfied. Furthermore, we canreduce the regularity assumptions for amplitude and phase functions inthis case. In order to state it precisely, we introduce the Besov spacesB

(s,s′)p,q of product type on R

nx × R

nξ :

Fix the partition of n:

n = n1 + . . .+ nN = n′1 + . . .+ n′N ′ .

We always split variables in Rn following this partition as

x = (x1, . . . , xN ), ξ = (ξ1, . . . , ξN ′),

and let s = (s1, . . . , sN ), s′ = (s′1, . . . , s′N ′) be the corresponding regu-larity indices. We set

Φj,k(y, η) =Θj1(y1) · · ·ΘjN (yN )

· Θk1(η1) · · ·ΘkN′ (ηN ′),

where j = (j1, . . . , jN ), k = (k1, . . . , kN ′) are multi-indices, and eachΘi(z) forms a dyadic system of the corresponding dimension. Then wedefine the space B(s,s′)

p,q by the norm

||f ||B

(s,s′)p,q

=

⎧⎨⎩∑j,k

(∥∥∥2j·s+k·s′F−1Φj,kFf(x, ξ)∥∥∥

Lpx,ξ

)q⎫⎬⎭

1/q

.

Theorem 4.3. Let φ(x, y, ξ) = x · ξ + ϕ(y, ξ) and a(x, y, ξ) = a(x, ξ).Assume that

|det ∂y∂ξϕ(y, ξ)| ≥ C > 0

and each entry h(y, ξ) of ∂y∂ξϕ(y, ξ) satisfies

∣∣∂αy h(y, ξ)

∣∣ ≤ Cα,∣∣∣∂β

ξ h(y, ξ)∣∣∣ ≤ Cβ

for |α|, |β| ≤ 2n + 1. Also assume that a(x, ξ) belongs to the classB

(1/2)(n,n′)∞,1 . Then T is L2(Rn)-bounded.


We note that the symbol class S00,0 is contained in B

(1/2)(n,n′)∞,1 . For

more relations between symbol classes and Besov spaces we refer to [17]and [18]. Especially, we have the following corollary:

Corollary 4.4. Let φ(x, y, ξ) = x · ξ + ϕ(y, ξ) and a(x, y, ξ) = a(x, ξ).Assume that

|det ∂y∂ξϕ(y, ξ)| ≥ C > 0

and each entry h(y, ξ) of ∂y∂ξϕ(y, ξ) satisfies∣∣∂αy h(y, ξ)

∣∣ ≤ Cα,∣∣∣∂β

ξ h(y, ξ)∣∣∣ ≤ Cβ

for |α|, |β| ≤ 2n+ 1. Also assume

∂αx ∂

βξ a(x, ξ) ∈ L∞(Rn

x × Rnξ )

for one of the following:

(i) α, β ∈ 0, 1n, (ii) |α|, |β| ≤ [n/2] + 1,

(iii) |α| ≤ [n/2] + 1, β ∈ 0, 1n, (iv) α ∈ 0, 1n, |β| ≤ [n/2] + 1.

Then T is L2(Rn)-bounded.

Note that Corollary 4.4 is contained in Theorem 4.3 with p = ∞(N = N ′ = n for condition (i); N = N ′ = 1 for (ii); N = 1, N ′ = n for(iii); N = n,N ′ = 1 for (iv)). Although Theorem 4.3 is more general,conditions of Corollary 4.4 are easier to be checked.

Corollary 4.4 with ϕ(y, ξ) = −y ·ξ is a refined version of known resultson the L2-boundedness of pseudo-differential operators with non-regularsymbols: (i) with α, β ∈ 0, 1, 2, 3n is due to Calderon and Vaillancourt[3], (ii) is due to Cordes [6], and conditions (iii) with |α| ≤ [n/2] + 1,β ∈ 0, 1, 2n, is due to Coifman and Meyer [5].

Theorem 4.5. Let φ(x, y, ξ) = x · ξ + ϕ(y, ξ) and a(x, y, ξ) = a(y, ξ).Assume that ∣∣∣∂α

y ∂βξ a(y, ξ)

∣∣∣ ≤ Cαβ ,

for |α|, |β| ≤ 2n+ 1. Also assume that

|det ∂y∂ξϕ(y, ξ)| ≥ C > 0


and each entry h(y, ξ) of ∂y∂ξϕ(y, ξ) satisfies∣∣∂αy h(y, ξ)

∣∣ ≤ Cα,∣∣∣∂β

ξ h(y, ξ)∣∣∣ ≤ Cβ

for |α|, |β| ≤ 2n+ 1. Then the operator T is L2(Rn)-bounded.

5. An example of how to use our results

Kato and Yajima [10] showed that the classical Schrodinger equation(i∂t + x)u(t, x) = 0,u(0, x) = g(x)

has the global smoothing estimate

(5.1)∥∥∥〈x〉−1〈D〉1/2u

∥∥∥L2(Rt×Rn

x)≤ C‖g‖L2(Rn

x),

where n ≥ 3.From this fact, we can extract a similar estimate for generalized

Schrodinger equations

(5.2)

(i∂t − p(D)2

)u(t, x) = 0,

u(0, x) = g(x).

Assume that the function p(ξ) ∈ C∞(Rn \ 0) is positively homogeneousof order 1, p(ξ) > 0, and the hypersurface Σ = ξ; p(ξ) = 1 has non-vanishing Gaussian curvature.

Remember that we have the relation

T−1 · p(D)2 = (−) · T−1

by (3.2). Operating T−1 from the left hand side of equation (5.2), wehave, by this relation,

(i∂t −)T−1u(t, x) = 0,T−1u(0, x) = T−1g(x).

Hence, from (5.1) and Theorem 4.1, we obtain the following conclusion:

Theorem 5.1. Suppose n ≥ 3. Under the assumption above, the solu-tion u(t, x) to generalized Schrodinger equation (5.2) has the same globalsmoothing estimate (5.1) as the classical one has.


Remark 5.1. Walther [19] consider the case of radially symmetric p(ξ)2.Theorem 5.1 says that we can treat more general case.

6. Smoothing effect with a structure

By using the idea in the previous sections, we can have more refinedglobal smoothing estimates. In order to state them, we introduce somenotations:

• Classical orbit determined by p(D)2:

(6.1)

x(t) = ∇ξp

2(ξ(t)), ξ(t) = 0

x(0) = 0, ξ(0) = k.

• The set of the path of all classical orbits:

Γp =(x(t), ξ(t)); sol. of (6.1), t ∈ R, k ∈ Rn \ 0

=(t∇p2(ξ), ξ

); ξ ∈ R

n \ 0, t ∈ R.

• Homogeneity:

σ(x, ξ) ∼ |x|a|ξ|b ⇐⇒σ(x, ξ) ∈ C∞((Rn

x \ 0) × (Rnξ \ 0)

),

σ(λx, τξ) = λaτ bσ(x, ξ) ; (λ, τ > 0).

Theorem 6.1. Let n ≥ 2. Suppose σ(x, ξ) ∼ |x|−1/2|ξ|1/2. Suppose alsothe structure condition

(6.2) σ(x, ξ) = 0 if (x, ξ) ∈ Γp and x = 0.

Then the solution u to equation (5.2) satisfies

(6.3) ‖σ(X,D)u‖L2(Rt×Rnx) ≤ C‖g‖L2(Rn

x).

Remark 6.1. Without the structure condition (6.2), we have estimate(6.3) in Theorem 6.1 with the following:

· σ(x, ξ) = 〈x〉−s|ξ|1/2 (s > 1/2) (Ben-Artzi and Klainerman [2])

· σ(x, ξ) = |x|α−1|ξ|α (0 < α < 1/2) (Kato and Yajima [10])


We have a similar result for inhomogeneous equations, as well. Forexample, we have an estimate for the equation

(6.4)

(i∂t + x)u(t, x) = f(t, x)

u(0, x) = g(x).

Note that

(6.5) Ωij(x, ξ) = (xi/〈x〉)ξj − (xj/〈x〉)ξi,where x = (x1, . . . , xn), ξ = (ξ1, . . . , ξn), is a typical example satisfyingstructure condition (6.2) in the case p(D)2 = −x.

Theorem 6.2. Suppose n ≥ 2, s, s ≥ 0, and −1 < k < 1. Then thesolution u to (6.4) satisfies the estimate∥∥∥〈x〉−1/2+kΩij(X,Dx)u

∥∥∥Hs

t (H sx)

≤ C∥∥∥〈x〉〈Dx〉2s+s+1/2g

∥∥∥L2(Rn

x)+ C

∥∥∥〈x〉1/2+kf∥∥∥

Hst (H s

x),

where Ωij(x, ξ) is given by (6.5).

7. Derivative Nonlinear Schrodinger Equation

Finally, we refer to further applications. We consider the followingnonlinear Schrodinger equation:

(7.1)

(i∂t + x)u(t, x) =|∇u(t, x)|Nu(0, x) = g(x), t ∈ R, x ∈ R

n.

What is the condition of the initial data g(x) for equation (7.1) to havetime global solution? There are some answers:

• N ≥ 3 (Chihara [4]). Smooth, rapidly decay, and sufficiently small.• N ≥ 2 (Hayashi, Miao and Naumkin [8]). g ∈ H [n/2]+5, rapidly

decay, and sufficiently small.

Question: Can we weaken the smoothness assumption for g(x)?

Answer: Yes if the non-linear term has a “structure”!


Instead of (7.1), we consider

(7.2)

(i∂t + x)u(t, x) =|Ωij(X,Dx)u|Nu(0, x) = g(x), t ∈ R, x ∈ R

n,

where Ωij(x, ξ) is given by (6.5). With the aid of Theorem 6.2, we havethe following result.

Theorem 7.1. Suppose n ≥ 2, s > (n+3)/2, and N ≥ 4. Assume that〈x〉〈Dx〉sg ∈ L2 and its L2-norm is sufficiently small. Then equation(7.2) has a time global solution.

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Department of Mathematics, Imperial College, 180 Queen’s Gate, Lon-

don SW7 2BZ, UK

E-mail address: [email protected]

Department of Mathematics, Graduate School of Science, Osaka Uni-

versity, Toyonaka, Osaka 560-0043, Japan

E-mail address: [email protected]

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Global boundedness theorems for Fourier integral operators associated with canonical transformations