Kazuaki Taira. Boundary Value Problems and Markov Processes

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3.2 Fourier Integral Operators 65

The next lemma will play a fundamental role in deﬁning oscillatory inte-

grals.

Lemma 3.3. If ϕ(x, θ) is a phase function on Ω ×



\{0}



, then there

exists a ﬁrst-order diﬀerential operator

L =



j=1

(x, θ)

∂

∂θ



k=1

(x, θ)

∂

∂x

+ c(x, θ)

such that

L(e

iϕ

)=e

iϕ

and that its coeﬃcients a

(x, θ), b

(x, θ), c(x, θ) enjoy the following properties:

(x, θ) ∈ S

1,0

; b

(x, θ),c(x, θ) ∈ S

−1

1,0

Furthermore, the transpose L



of L has coeﬃcients a



(x, θ), b



(x, θ), c



(x, θ)

inthesamesymbolclassesasa

(x, θ), b

(x, θ), c(x, θ), respectively.

For example, if ϕ(x, y, ξ) is a phase function as in Example 3.5

ϕ(x, y, ξ)=(x − y) · ξ, (x, y) ∈ U × U, ξ ∈ (R

\{0}) ,

then the operator L is given by the formula

L =

√

−1

1 − ρ(ξ)

2+|x − y|



j=1

− y

)

∂

∂ξ



k=1

|ξ|

∂

∂x



k=1

−ξ

|ξ|

∂

∂y

(

+ ρ(ξ),

where ρ(ξ) is a function in C

∞

) such that ρ(ξ)=1for|ξ|≤1.

3.2.3 Oscillatory Integrals

If Ω is an open subset of R

,welet

∞

ρ,δ



Ω × R





m∈R

ρ,δ



Ω × R



If ϕ(x, θ) is a phase function on Ω ×



\{0}



, we wish to give a meaning

to the integral

(aw)=



Ω×R

iϕ(x,θ)

a(x, θ)u(x) dx dθ, u ∈ C

∞

(Ω), (3.2)

for each symbol a(x, θ) ∈ S

∞

ρ,δ



Ω × R



66 3 L

Theory of Pseudo-Diﬀerential Operators

By Lemma 3.3, we can replace e

iϕ

in formula (3.2) by L(e

iϕ

). Then a

formal integration by parts gives us that

(au)=



Ω×R

iϕ(x,θ)



(a(x, θ)w(x, y)) dx dθ.

However, the properties of the coeﬃcients of the transpose L



imply that L



maps S

ρ,δ

continuously into S

r−η

ρ,δ

for all r ∈ R,whereη =min(ρ, 1 − δ). By

continuing this process, we can reduce the growth of the integrand at inﬁnity

until it becomes integrable, and give a meaning to the integral (3.2) for each

symbol a(x, θ) ∈ S

∞

ρ,δ

(Ω × R

More precisely, we have the following:

Theorem 3.4. (i) The linear functional

−∞



Ω × R



 a −→ I

(au) ∈ C

extends uniquely to a linear functional  on S

∞

ρ,δ



Ω × R



whose restriction

to each S

ρ,δ



Ω × R



is continuous. Furthermore, the restriction of the linear

functional  to S

ρ,δ



Ω × R



is expressed as the formula

(a)=



Ω×R

iϕ(x,θ)



)

(a(x, θ)w(x, y)) dx dθ,

where k>(m + N )/η and η =min(ρ, 1 − δ).

(ii) For any ﬁxed a(x, θ) ∈ S

ρ,δ



Ω × R



, the mapping

∞

(Ω)  u −→ I

(au)=(a) ∈ C (3.3)

is a distribution of order ≤ k for k>(m + N )/η with η =min(ρ, 1 − δ).

We call the linear functional  on S

∞

ρ,δ

an oscillatory integral, but use the

standard notation as in formula (3.2). The distribution (3.3) is called the

Fourier integral distribution associated with the phase function ϕ(x, θ)and

the amplitude a(x, θ), and will be denoted as follows:

K(x)=



iϕ(x,θ)

a(x, θ) dθ.

If u is a distribution on Ω, then the singular support of u is the smallest

closed subset of Ω outside of which u is smooth. The singular support of u is

denoted by sing supp u.

The next theorem estimates the singular support of a Fourier integral

distribution.

Theorem 3.5. If ϕ(x, θ) is a phase function on the space Ω ×



\{0}



and if a(x, θ) is in S

∞

ρ,δ



Ω × R



, then the distribution

K(x)=



iϕ(x,θ)

a(x, θ) dθ ∈D



(Ω)

satisﬁes the condition

sing supp K ⊂



x ∈ Ω : d

ϕ(xθ)=0for some θ ∈ R

\{0}



3.3 Pseudo-Diﬀerential Operators 67

3.2.4 Fourier Integral Operators

Let U and V be open subsets of R

and R

, respectively. If ϕ(x, y, θ)isa

phase function on U × V × (R

\{0})andifa(x, y, θ) ∈ S

∞

ρ,δ

(U ×V × R

then there is associated a distribution K ∈D



(U ×V ) deﬁned by the formula

K(x, y)=



iϕ(x,y,θ)

a(x, y, θ) dθ.

By applying Theorem 3.5 to our situation, we obtain that

sing supp K ⊂



(x, y) ∈ U × V : d

ϕ(x, y, θ)=0forsomeθ ∈ R

\{0}



The distribution K ∈D



(U ×V ) deﬁnes a continuous linear operator

A : C

∞

(V ) −→ D



(U)

by the formula

Av, u = K, u ⊗ v,u∈ C

∞

(U),v∈ C

∞

(V ).

The operator A is called the Fourier integral operator associated with the

phase function ϕ(x, y, θ) and the amplitude a(x, y, θ), and will be denoted as

follows:

Av(x)=



V ×R

iϕ(x,y,θ)

a(x, y, θ)v(y) dy dθ, v ∈ C

∞

(V ).

The next theorem summarizes some basic properties of the operator A:

Theorem 3.6. (i) If d

y,θ

ϕ(x, y, θ) =0on U × V ×



\{0}



, then the

operator A maps C

∞

(V ) continuously into C

∞

(U).

(ii) If d

x,θ

ϕ(x, y, θ) =0on U × V ×



\{0}



, then the operator A

extends to a continuous linear operator on E



(V ) into D



(U).

(iii) If d

y,θ

ϕ(x, y, θ) =0and d

x,θ

ϕ(x, y, θ) =0on U × V ×



\{0}



then we have, for all v ∈E



(V ),

sing supp Av

⊂



x ∈ U : d

ϕ(x, y, θ)=0for some y ∈ sing supp v and θ ∈ R

\{0}



3.3 Pseudo-Diﬀerential Operators

Let Ω

and Ω

be open subsets of R

and R

, respectively. If K(x

)is

a distribution in D



(Ω

× Ω

), we can deﬁne a continuous linear operator

A ∈ L(C

∞

(Ω

), D



(Ω

))

by the formula

68 3 L

Theory of Pseudo-Diﬀerential Operators

Aψ, ϕ = K, ϕ ⊗ ψ,ϕ∈ C

∞

(Ω

),ψ∈ C

∞

(Ω

We then write A =Op(K). Since the tensor space C

∞

(Ω

) ⊗ C

∞

(Ω

)is

sequentially dense in C

∞

(Ω

× Ω

), it follows that the mapping



(Ω

× Ω

)  K −→ Op (K) ∈ L(C

∞

(Ω

), D



(Ω

))

is injective. The next theorem asserts that it is also surjective:

Theorem 3.7 (the Schwartz kernel theorem). If A is a continuous lin-

ear operator on C

∞

(Ω

) into D



(Ω

), then there exists a unique distribution

) in D



(Ω

× Ω

) such that A =Op(K).

The distribution K

is called the kernel of A. Formally we have the formula

Aψ(x



) ψ(x

) dx

,ψ∈ C

∞

(Ω

Now we give some important examples of distributions kernels (see Exam-

ple 3.2):

Example 3.6. (a) Riesz potentials: Ω

= Ω

= R

,0<α<n.

(−Δ)

−α/2

u(x)=R

∗ u(x)

Γ ((n − α)/2)

n/2

Γ (α/2)



|x − y|

n−α

u(y) dy, u ∈ C

∞

(b) Newtonian potentials: Ω

= Ω

= R

, n ≥ 3.

(−Δ)

−1

u(x)=N ∗u(x)

Γ ((n − 2)/2)

4π

n/2



|x − y|

n−2

u(y) dy, u ∈ C

∞

= Ω

= R

, α>0.

(I − Δ)

−α/2

u(x)=G

∗ u(x)=



(x − y) u(y) dy, u ∈ C

∞

(d) Riesz operators: Ω

= Ω

= R

,1≤ j ≤ n.

u(x)=R

∗ u(x)

√

−1

Γ ((n +1)/2)

(n+1)/2

v. p.



− y

|x − y|

n+1

u(y) dy, u ∈ C

∞

(e) The Calder´on–Zygmund integro-diﬀerential operator: Ω

= Ω

= R

(−Δ)

1/2

u(x)=

√

−1



j=1



∂u

∂x



(x)

Γ ((n +1)/2)

(n+1)/2



j=1

v. p.



− y

|x − y|

n+1

∂u

∂y

(y) dy,

u ∈ C

∞

3.3 Pseudo-Diﬀerential Operators 69

Let Ω be an open subset of R

and m ∈ R.Apseudo-diﬀerential operator

of order m on Ω is a Fourier integral operator of the form

Au(x)=



Ω×R

i(x−y)·ξ

a(x, y, ξ)u(y) dydξ, u ∈ C

∞

(Ω), (3.4)

with some a(x, y, ξ) ∈ S

ρ,δ

(Ω ×Ω ×R

). In other words, a pseudo-diﬀerential

operator of order m is a Fourier integral operator associated with the phase

function ϕ(x, y, ξ)=(x −y) ·ξ and some amplitude a(x, y, ξ) ∈ S

ρ,δ

(Ω ×Ω ×

We let

ρ,δ

(Ω) = the set of all pseudo-diﬀerential operators of order m on Ω.

By applying Theorems 3.5 and 3.6 to our situation, we obtain the following

three assertions:

(1) A pseudo-diﬀerential operator A maps C

∞

(Ω) continuously into C

∞

(Ω)

and extends to a continuous linear operator A : E



(Ω) →D



(Ω).

(2) The distribution kernel K

(x, y) of a pseudo-diﬀerential operator A sat-

isﬁes the condition

sing supp K

⊂{(x, x):x ∈ Ω},

that is, the kernel K

is smooth oﬀ the diagonal {(x, x):x ∈ Ω} in Ω×Ω.

(3) sing supp Au ⊂ sing supp u, u ∈E



(Ω). In other words, Au is smooth

whenever u is. This property is referred to as the pseudo-local property.

We set

−∞

(Ω)=



m∈R

ρ,δ

(Ω).

The next theorem characterizes the class L

−∞

(Ω).

Theorem 3.8. The following three conditions are equivalent:

(i) A ∈ L

−∞

(Ω).

(ii) A is written in the form (3.4) with some a ∈ S

−∞

(Ω × Ω ×R

(iii) A is a regularizer, or equivalently, its distribution kernel K

(x, y) is in

the space C

∞

(Ω × Ω).

We recall that a continuous linear operator A : C

∞

(Ω) →D



(Ω)issaid

to be properly supported if the following two conditions are satisﬁed:

(a) For any compact subset K of Ω, there exists a compact subset K



of Ω

such that

supp v ⊂ K =⇒ supp Av ⊂ K



(b) For any compact subset K



of Ω, there exists a compact subset K ⊃ K



of Ω such that

supp v ∩ K = ∅ =⇒ supp Av ∩ K



= ∅.

70 3 L

Theory of Pseudo-Diﬀerential Operators

If A is properly supported, then it maps C

∞

(Ω) continuously into E



(Ω)

and extends to a continuous linear operator on C

∞

(Ω)intoD



(Ω).

The next theorem states that every pseudo-diﬀerential operator can be

written as the sum of a properly supported operator and a regularizer.

Theorem 3.9. If A ∈ L

ρ,δ

(Ω), then we have the decomposition

A = A

+ R,

where A

∈ L

ρ,δ

(Ω) is properly supported and R ∈ L

−∞

(Ω).

If p(x, ξ) ∈ S

ρ,δ

(Ω×R

), then the operator p(x, D), deﬁned by the formula

p(x, D)u(x)=

(2π)



ix·ξ

p(x, ξ) "u(ξ) dξ, u ∈ C

∞

(Ω), (3.5)

is a pseudo-diﬀerential operator of order m on Ω,thatis,p(x, D) ∈ L

ρ,δ

(Ω).

The next theorem asserts that every properly supported pseudo-diﬀerential

operator can be reduced to the form (3.5).

Theorem 3.10. If A ∈ L

ρ,δ

(Ω) is properly supported, then we have the as-

sertion

p(x, ξ)=e

−ix·ξ

A(e

ix·ξ

) ∈ S

ρ,δ

(Ω × R

and

A = p(x, D).

Furthermore, if a(x, y, ξ) ∈ S

ρ,δ

(Ω ×Ω ×R

) is an amplitude for A,thenwe

have the following asymptotic expansion:

p(x, ξ) ∼



α≥0

α!

∂

(a(x, y, ξ))



y=x

The function p(x, ξ) is called the complete symbol of A.

We extend the notion of a complete symbol to the whole space L

ρ,δ

(Ω).

If A ∈ L

ρ,δ

(Ω), then we choose a properly supported operator A

∈ L

ρ,δ

(Ω)

such that A − A

∈ L

−∞

(Ω), and deﬁne

σ(A) = the equivalence class of the complete symbol of A

ρ,δ

(Ω × R

)/S

−∞

(Ω × R

In view of Theorems 3.10 and 3.11, it follows that σ(A) does not depend on

the operator A

chosen. The equivalence class σ(A) is called the complete

symbol of A. It is easy to see that the mapping

ρ,δ

(Ω)  A −→ σ(A) ∈ S

ρ,δ

(Ω × R

)/S

−∞

(Ω × R

)

3.3 Pseudo-Diﬀerential Operators 71

induces an isomorphism

ρ,δ

(Ω)/L

−∞

(Ω) −→ S

ρ,δ

(Ω × R

)/S

−∞

(Ω × R

We shall often identify the complete symbol σ(A) with a representative

in the class S

ρ,δ

(Ω ×R

) for notational convenience, and call any member of

σ(A) a complete symbol of A.

A pseudo-diﬀerential operator A ∈ L

1,0

(Ω)issaidtobeclassical if its

complete symbol σ(A) has a representative in the class S

(Ω × R

We let

(Ω) = the set of all classical pseudo-diﬀerential operators of order m

on Ω.

Then the mapping

(Ω)  A −→ σ(A) ∈ S

(Ω × R

)/S

−∞

(Ω × R

)

induces an isomorphism

(Ω)/L

−∞

(Ω) −→ S

(Ω × R

)/S

−∞

(Ω × R

Also we have the assertion

−∞

(Ω)=



m∈R

(Ω).

If A ∈ L

(Ω), then the principal part of σ(A) has a canonical representa-

tive σ

(x, ξ) ∈ C

∞

(Ω ×(R

\{0})) which is positively homogeneous of degree

m in the variable ξ. The function σ

(x, ξ) is called the homogeneous principal

symbol of A.

The next two theorems assert that the class of pseudo-diﬀerential operators

forms an algebra closed under the operations of composition of operators and

taking the transpose or adjoint of an operator.

Theorem 3.11. If A ∈ L

ρ,δ

(Ω), then its transpose A



and its adjoint A

∗

are

both in L

ρ,δ

(Ω), and the complete symbols σ(A



) and σ(A

∗

) have respectively

the following asymptotic expansions:

σ(A



)(x, ξ) ∼



α≥0

α!

∂

(σ(A)(x, −ξ)) ,

σ(A

∗

)(x, ξ) ∼



α≥0

α!

∂



σ(A)(x, ξ)



Theorem 3.12. If A ∈ L



,δ



(Ω) and B ∈ L



,δ



(Ω) where 0 ≤ δ



<ρ



≤ 1

and if one of them is properly supported, then the composition AB is in

72 3 L

Theory of Pseudo-Diﬀerential Operators





ρ,δ

(Ω) with ρ =min(ρ



,ρ



), δ =max(δ



,δ



), and we have the following

asymptotic expansion:

σ(AB)(x, ξ) ∼



α≥0

α!

∂

(σ(A)(x, ξ)) · D

(σ(B)(x, ξ)) .

A pseudo-diﬀerential operator A ∈ L

ρ,δ

(Ω)issaidtobeelliptic of order

m if its complete symbol σ(A) is elliptic of order m. In view of Theorem 3.2,

it follows that a classical pseudo-diﬀerential operator A ∈ L

(Ω) is elliptic if

and only if its homogeneous principal symbol σ

(x, ξ) does not vanish on the

space Ω × (R

\{0}).

The next theorem states that elliptic operators are the “invertible” ele-

ments in the algebra of pseudo-diﬀerential operators.

Theorem 3.13. An operator A ∈ L

ρ,δ

(Ω) is elliptic if and only if there exists

a properly supported operator B ∈ L

−m

ρ,δ

(Ω) such that:



AB ≡ I mod L

−∞

(Ω),

BA ≡ I mod L

−∞

(Ω).

Such an operator B is called a parametrix for A. In other words, a para-

metrix for A is a two-sided inverse of A modulo L

−∞

(Ω). We observe that a

parametrix is unique modulo L

−∞

(Ω).

The next theorem proves the invariance of pseudo-diﬀerential operators

under change of coordinates.

Theorem 3.14. Let Ω

and Ω

be two open subsets of R

and let χ : Ω

→

be a C

∞

diﬀeomorphism. If A ∈ L

ρ,δ

(Ω

),where1 − ρ ≤ δ<ρ≤ 1,then

the mapping

: C

∞

(Ω

) −→ C

∞

(Ω

)

v −→ A(v ◦ χ) ◦ χ

−1

is in L

ρ,δ

(Ω

), and we have the asymptotic expansion

σ(A

)(y, η) ∼



α≥0

α!



∂

σ(A)



(x,



(x) · η) · D



ir(x,z,η)





z=x

(3.6)

with

r(x, z, η)=χ(z) − χ(x) − χ



(x) · (z −x),η.

Here x = χ

−1

(y),χ



(x) is the derivative of χ at x and



(x) is its transpose.

Remark 3.1. Formula (3.6) shows that

σ(A

)(y, η) ≡ σ(A)





(x) · η



mod S

m−(ρ−δ)

ρ,δ

Note that the mapping

3.3 Pseudo-Diﬀerential Operators 73

× R

 (y, η) −→





(x) · η



∈ Ω

× R

is just a transition map of the cotangent bundle T

∗

). This implies that

the principal symbol σ

(A)ofA ∈ L

ρ,δ

) can be invariantly deﬁned on

∗

)when1−ρ ≤ δ<ρ≤ 1.

The situation may be represented by the following diagram:

∞

(Ω

)

−−−−→ C

∞

(Ω

)

∗

⏐

∗

∞

(Ω

) −−−−→

∞

(Ω

)

Here χ

∗

v = v ◦ χ is the pull-back of v by χ and χ

∗

u = u ◦ χ

−1

is the push-

forward of u by χ, respectively.

A diﬀerential operator of order m with smooth coeﬃcients on Ω is con-

tinuous on H

s,p

loc

(Ω)(resp.B

s,p

loc

(Ω)) into H

s−m,p

loc

(Ω)(resp.B

s−m,p

loc

(Ω)) for all

s ∈ R. This result extends to pseudo-diﬀerential operators:

Theorem 3.15 (the Besov space boundedness theorem). Every prop-

erly supported operator A ∈ L

1,δ

(Ω), 0 ≤ δ<1, extends to a continuous

linear operator

A : H

s,p

loc

(Ω) −→ H

s−m,p

loc

(Ω)

for all s ∈ R and 1 <p<∞, and also it extends to a continuous linear

operator

A : B

s,p

loc

(Ω) −→ B

s−m,p

loc

(Ω)

for all s ∈ R and 1 ≤ p ≤∞.

For a proof of Theorem 3.15, the reader might refer to Bourdaud [Bo,

Theorem 1] (see also [Ta5, Appendix A]).

Now we deﬁne the concept of a pseudo-diﬀerential operator on a manifold,

and transfer all the machinery of pseudo-diﬀerential operators to manifolds.

Let M be an n-dimensional compact smooth manifold without boundary. The-

orem 3.14 leads us to the following:

Deﬁnition 3.1. Let 1 − ρ ≤ δ<ρ≤ 1. A continuous linear operator A :

∞

(M) → C

∞

(M) is called a pseudo-diﬀerential operator of order m ∈ R if

it satisﬁes the following two conditions:

(i) The distribution kernel K

(x, y)ofA is smooth oﬀ the diagonal {(x, x):

x ∈ M} in M × M.

(ii) For any chart (U, χ)onM , the mapping

: C

∞

(χ(U)) −→ C

∞

(χ(U))

u −→ A(u ◦ χ) ◦ χ

−1

belongs to the class L

ρ,δ

(χ(U)).

74 3 L

Theory of Pseudo-Diﬀerential Operators

We let

ρ,δ

(M) = the set of all pseudo-diﬀerential operators of order m on M,

and set

−∞

(M)=



m∈R

ρ,δ

(M).

Some results about pseudo-diﬀerential operators on R

stated above are

also true for pseudo-diﬀerential operators on M. In fact, pseudo-diﬀerential

operators on M are deﬁned to be locally pseudo-diﬀerential operators on R

For example, we have the following ﬁve assertions:

(1) A pseudo-diﬀerential operator A extends to a continuous linear operator

A : D



(M) →D



(M).

(2) sing supp Au ⊂ sing supp u, u ∈D



(M).

(3) A continuous linear operator A : C

∞

(M) →D



(M) is a —em regularizer

ifandonlyifitisintheclassL

−∞

(M).

(4) The class L

ρ,δ

(M) is stable under the operations of composition of oper-

ators and taking the transpose or adjoint of an operator.

(5) (The Besov space boundedness theorem) A pseudo-diﬀerential op-

erator A ∈ L

1,δ

(M), 0 ≤ δ<1, extends to a continuous linear operator

A : H

s,p

(M) −→ H

s−m,p

(M)

for all s ∈ R and 1 <p<∞, and also it extends to a continuous linear

operator

A : B

s,p

(M) −→ B

s−m,p

(M)

for all s ∈ R and 1 ≤ p ≤∞.

A pseudo-diﬀerential operator A ∈ L

1,0

(M)issaidtobeclassical if, for

any chart (U, χ)onM, the mapping A

: C

∞

(χ(U)) → C

∞

(χ(U)) belongs

to the class L

(χ(U)).

We let

(M) = the set of all classical pseudo-diﬀerential operators of order

m on M.

We observe that

−∞

(M)=



m∈R

(M).

Let A ∈ L

(M). If (U, χ)isachartonM , there is associated a ho-

mogeneous principal symbol σ

∈ C

∞

(χ(U) × (R

\{0})). In view of

Remark 3.1, by smoothly patching together the functions σ

we can ob-

tain a smooth function σ

(x, ξ)onT

∗

(M) \{0} = {(x, ξ) ∈ T

∗

(M):ξ =0},

which is positively homogeneous of degree m in the variable ξ. The function

(x, ξ) is called the homogeneous principal symbol of A.