5.3 Coordinate Transformations and Pseudodifferential Operators

5.3 Coordinate Transformations and Pseudodifferential Op- erators on Manifolds

In this section we will show that for every suitable smooth diffeomorphism κ : Rⁿ→ Rⁿ the operator Q : S(Rⁿ) → S(Rⁿ) defined by

Qu(x) = (p(x, Dx)w)(κ(x)), w(x) = u(κ⁻¹(x)), u ∈ S(Rⁿ) (5.17) is again a pseudodifferential operator. This is important to obtain a definition of pseudodifferential operators on a manifold in a way that is essentially independent of the choice of local charts. We note that

Qu= κ^∗p(x, Dx)κ^∗,−1u, where (κ^∗v)(x) = v(κ(x)) and (κ^∗,−1u)(x) = u(κ⁻¹(x)).

More precisely, we assume that κ : Rⁿ → Rⁿ is a smooth function such that

∂_xjκ∈ C_b^∞(Rⁿ) for all j = 1, . . . , n and

0 < c ≤ | det Dκ(x)| ≤ C < ∞ (5.18) for all x ∈ Rⁿand some constants c, C > 0. In particular, this implies that κ⁻¹: Rⁿ→ Rⁿ is a again smooth and ∂xjκ⁻¹ ∈ C_b^∞(Rⁿ) for all j = 1, . . . , n. In the following

∇xκ the total derivative of κ : Rⁿ → Rⁿ. The main result of this section is:

THEOREM 5.11 Let p∈ S_1,0^m(Rⁿ× Rⁿ), m ∈ R, let κ : Rⁿ → Rⁿ be as above and Q: S(Rⁿ) → S(Rⁿ) be defined by (5.17). Then there is some q ∈ S_1,0^m(Rⁿ × Rⁿ) such that Qu= q(x, Dx)u for all u ∈ S(Rⁿ). Moreover, q(x, Dx) has the asymptotic expansion

q(x, ξ) ∼ X

α∈Nⁿ0

1

α!∂^α_ξD_y^αq(x, ξ, y)|˜ _y=x, where

˜

q(x, ξ, y) = p(κ(x), A(x, y)^−Tξ)| det A(x, y)|⁻¹| det ∇yκ(y)| and A(x, y) =

Z 1 0

∇yκ(x + t(y − x)) dt for all x, y close enough

in the sense that for any N ∈ N₀ q(x, ξ) − X

|α|≤N

1

α!∂_ξ^αD_y^αq(x, ξ, y)|˜ _y=x ∈ S_1,0^{m−N −1}(Rⁿ× Rⁿ).

In particular, we have

q(x, ξ) = p(κ(x), (∇yκ(x))^−Tξ) + r(x, ξ) with r∈ S_1,0^m−1.

Proof: We only prove the theorem under the additional assumption that sup

x∈Rⁿ

|∇κ(x) − I| ≤ 1 2.

For the proof in the general case we refer to [KG74, Theorem 6.3]. Here |.| is any matrix norm, which is induced by some vector norm. This ensures that

|A(x, y) − I| ≤ 1

2 for all x, y ∈ Rⁿ

and therefore A(x, y)^−T = (A(x, y)⁻¹)^T exists for all x, y ∈ Rⁿ. Moreover, let χ ∈ S(Rⁿ) with χ(0) = 1 and χε(ξ) = χ(εξ) for ε > 0, ξ ∈ Rⁿ.

Under these assumptions we have Qu = lim

ε→0

Z

Rn×Rⁿ

ei(κ(x)−y)·ξχε(ξ)p(κ(x), ξ)u(κ⁻¹(y))d(y, ξ) (2π)ⁿ

= lim

ε→0

Z

Rn×Rⁿ

e^{i(κ(x)−κ(x′))·ξ}χε(ξ)p(κ(x), ξ)u(x^′)| det ∇x^′κ(x^′)|d(y, ξ) (2π)ⁿ for all u ∈ S(Rⁿ). On the other hand, since

κ(x) − κ(x^′) =

Z 1 0

∇x^′κ(x^′+ t(x − x^′)) dt



(x − x^′) = A(x, x^′)(x − x^′), we obtain by the change of variable ξ = A(x, x^′)^−Tη

Qu = lim

ε→0

Z

Rn×Rⁿ

e^{i(x−x′)·η}χε(A(x, x^′)^−Tη)˜q(x, x^′, η)u(x^′) d(x^′, η)

= lim

ε→0Os–

Z Z

e^−iy·ηχε(A(x, x + y)^−Tη)˜q(x, x + y, η)u(x + y) d(x^′, η) due to (κ(x) − κ(x^′)) · ξ = (x − x^′) · A(x, x)⁻¹ξ, where

˜

q(x, x^′, η) = p(κ(x), A(x, x^′)^−Tη)| det A(x, x^′)|⁻¹| det ∇x^′κ(x^′)|.

Since {χε(A(x, x^′)^−Tη) : ε ∈ (0, 1)} is bounded in A⁰₀(Rⁿ× Rⁿ) with respect to (x^′, η) and

χε(A(x, x^′)^−Tη) →ε→0 1 for all x, x^′, η∈ Rⁿ

∂_x^α′∂_η^βχ_ε(A(x, x^′)^−Tη) →ε→0 0 for all x, x^′, η∈ Rⁿ,|α| + |β| > 0, we can apply Corollary 4.10 to conclude that

Qu(x) = lim

ε→0

Z

Rn×Rⁿ

e^{i(x−x′)·η}q(x, x˜ ^′, η)u(x^′) d(x^′, η)

= lim

ε→0Os–

Z Z

e^−iy·ηχε(A(x, x + y)^−Tη)˜q(x, x + y, η)u(x + y) d(x^′, η).

Finally, an application of Theorem 4.27 finishes the proof.

Next we define pseudodifferential operators on smooth compact manifolds. To this end recall:

Definition 5.12 M is a smooth compact manifold if M is a topological compact space and the following conditions hold:

1. There are finitely many open sets Ω1, . . . ,ΩN ⊂ M such that M =SN j=1Ωj. 2. For every j ∈ {1, . . . , N} there is an open set Uj ⊂ Rⁿ and a continuous

bijective functions κj: Ωj → U_j with continuous inverse, called charts.

3. For every j, k ∈ {1, . . . , N} such that Ωj,k := Ωj ∩ Ωk 6= ∅ the mapping κj,k: κk(Ωj,k) → κj(Ωj,k)

can be extended to some is C^∞-diffeomorphism with ˜κj,k: Vx → Vy for some open Vx ⊃ κk(Ωj,k), Vy ⊃ κj(Ωj,k).

A function f : M → C is called smooth if for every j = 1, . . . , N as above uj :=

u◦ κ⁻¹_j ∈ C^∞(κj(Ωj)). Moreover, if Ω ⊂ M is open

C₀^∞(Ω) = {f ∈ C^∞(M) : supp f ⊂ Ω}.

Before we define pseudodifferential operators on a compact manifold we the defi- nition of certain remainder operators:

Definition 5.13 Let M be a smooth compact manifold as above and let P : C^∞(M) → C^∞(M) be a linear operator. Then P has a C^∞-kernel representation if for every j, k ∈ {1, . . . , N} there is some Kj,k ∈ C^∞(Uk× Uj) such that for every u ∈ C₀^∞(Ωj)

(P u)(κ⁻¹_k (x)) = Z

U_j

Kj,k(x, x^′)uj(x^′) dx^′ for all x ∈ Uk, where uj(x) = u(κ⁻¹_j (x)) in Uj.

For the following let Φj, j = 1, . . . , N be a smooth partition of unity subordinate to Ωj, j = 1, . . . , N, i.e.,

0 ≤ Φj ∈ C₀^∞(Ωj),

N

X

j=1

Φj(x) = 1 for all x ∈ M (5.19) Moreover, let Ψj ∈ C₀^∞(Ωj), j = 1, . . . , N, be such that

Ψj(x) = 1 for all x ∈ supp Φj, j = 1, . . . , N. (5.20) Finally, we define ϕj, ψj ∈ C₀^∞(Uj) by

ϕj = Φj ◦ κ⁻¹_j , ψj = Ψj ◦ κ⁻¹_j (5.21)

Definition 5.14 Let M be a smooth compact manifold as above and let P : C^∞(M) → C^∞(M) be a linear operator. Then P is a pseudodifferential operator of class S_1,0^m(M) with symbols (pj)^N_j=1with respect to the charts κj, j = 1, . . . , N, if the following holds true: Let Φj,Ψj ∈ C^∞(M), j = 1, . . . , N satisfy (5.19) and (5.20) and let

(P u)(x) =

N

X

j=1

(ΦjPΨju)(x) +

N

X

j=1

(ΦjP(1 − Ψj)u)(x) for all x ∈ M. (5.22)

Then, using ϕj, ψj defined in (5.21), ΦjPΨju can be written in the form (ΦjPΨju)(κ⁻¹_j (x)) = ϕjpj(x, Dx)(ψjuj)(x) for all x ∈ Uj

where uj(x) = u(κ⁻¹_j (x)), x ∈ Uj, for all u ∈ C^∞(M), j = 1, . . . , N and (ΦjP(1 − Ψj)u) has a C^∞-kernel representation.

Remark 5.15 If P is a differential operator, supp P (1 − Ψj)u ⊆ supp(1 − Ψj).

Therefore (ΦjP(1 − Ψj)u)(x) = 0 for all x ∈ M. Hence

(P u)(x) =

N

X

j=1

(ΦjPΨju)(x) for all x ∈ M

in this case.

The following theorem shows that the definition of pseudodifferential operators on M are independent of the choice of charts κ₁, . . . , κN.

THEOREM 5.16 LetM be a smooth compact manifolds as above and letΩ^′_j, U_j^′, κ^′_j, κ^′_j,k, j, k = 1, . . . , N^′ be another sequence satisfying conditions 1.-3 in Definition 5.12 such that

˜

κj,j^′ := κj◦ κ⁻¹_j′ : κ^′_j′(Ωj ∩ Ω^′_j′) → κj(Ωj ∩ Ω^′_j′), j ∈ {1, . . . , N}, j^′ ∈ {1, . . . , N^′}, extends to a C^∞-diffeomorphim on some open neighborhoods of κ^′_j′(Ωj ∩ Ω^′_j′) and κj(Ωj ∩ Ω^′_j′). If P : C^∞(M) → C^∞(M) is a pseudodifferential operator in S_1,0^m(M) with respect to {Ωj, Uj, κj, κj,k : j, k = 1, . . . , N}, then P is in S_1,0^m(M) with respect to {Ω^′_j, U_j^′, κ^′_j, κ^′_j,k : j, k = 1, . . . , N^′}.

Proof: For simplicity we assume that ˜κj,j^′ can be extended to a C^∞-diffeormphism

˜

κj,j^′: Rⁿ → Rⁿ such that ∇˜κj,j^′,∇˜κ⁻¹_j,j′ ∈ C_b^∞(Rⁿ). For the proof in the general case we refer to [KG74, Theorem 7.3].

First of all, we prove that for every Ψ, Φ ∈ C^∞(M) we have:

supp Φ ∩ supp Ψ = ∅ ⇒ ΦP Ψ has a C^∞-kernel representation. (5.23)

We write

ΦP Ψ =

N

X

j=1

Φ(ΦjPΨj)Ψ +

N

X

j=1

Φ(ΦjP(1 − Ψj))Ψ.

Then Φ(ΦjP(1 − Ψj))Ψ has a C^∞-kernel representation since ΦjP(1 − Ψj) has one.

Now we have

(Φ(ΦjPΨj)Ψu)(κ⁻¹_j (x)) = (ϕϕj)(x)pj(x, Dx)(ψjψuj)(x) for all x ∈ Uj, for all u ∈ C^∞(M), j = 1, . . . , N, where uj = u ◦ κ⁻¹_j and ϕ = Φ ◦ κ⁻¹_j , ψ = Ψ ◦ κ⁻¹_j . Since supp(ψjψ) ∩ supp(ϕjϕ) = ∅, Theorem 5.8 implies that there are smooth kj: Rⁿ× (Rⁿ\ {0}) → C such that

(Φ(ΦjP(1 − Ψj))Ψu)(κ⁻¹_j (x)) = Z

Rn

(ϕϕj)(x)kj(x, x − y)ψj(y)ψ(y)uj(y) dy.

Hence (ϕϕj)(x)kj(x, x − y)ψj(y)ψ(y) ∈ C^∞(Rⁿ× Rⁿ) and Φ(ΦjP(1 − Ψj))Ψ has a C^∞-kernel representation. This proves (5.23).

Let Φ^′_j′,Ψ^′_j′ ∈ C^∞(M), j^′ = 1, . . . , N^′, satisfy (5.19)-(5.20) with Ωj replaced by Ω^′_j′. Now let us write P as

P =

N^′

X

j^′=1

Φ^′_j′PΨ^′_j′+

N^′

X

j^′=1

Φ^′_j′P(1 − Ψ^′_j′).

Because of (5.23), the second term has a C^∞-kernel representation. For the first term we use that

Φ^′_j′PΨ^′_j′ =

N

X

j=1

Φ^′_j′(ΦjPΨj)Ψ^′_j′ +

N

X

j=1

Φ^′_j′(ΦjP(1 − Ψj))Ψ^′_j′,

where again the second term has a C^∞-representation. Moreover, if Ωj ∩ Ω^′_j′ 6= ∅ by assumption

(Φ^′_j′ΦjP(ΨjΨ^′_j′u))(κ⁻¹_j (x)) = ϕ^′_j′(x)ϕj(x)(pj(x, Dx)(ψ^′_j′ψjuj))(x) for all x ∈ Uj. Hence

(Φ^′_j′ΦjP(ΨjΨ^′_j′u))(κ⁻¹_j′ (x))

= ˜ϕ^′_j′(x)

(κ⁻¹_j ◦ κ^′_j′)⁻¹
∗

ϕjpj(x, Dx)(ψ_j^′′(κ⁻¹_j ◦ κj^′)^∗( ˜ψju^′_j′))

(x) for all x ∈ Uj. where u^′_j′(x) = u((κ^′_j′)⁻¹(x)), (κ^∗v)(x) = v(κ(x)) for κ : Rⁿ→ Rⁿ, and ˜ϕ^′_j′ = Φ^′_j′◦κ⁻¹_j′ , ψ˜_j^′′ = Ψ^′_j′ ◦ κ⁻¹_j′ . Now, since ˜κ_j,j′ := κj ◦ (κ^′_j′)⁻¹ can be extended to some C^∞- diffeomorphism on Rⁿ with ∇xκ˜j,j^′,∇xκ˜⁻¹_j,j′ ∈ C_b^∞(Rⁿ), Theorem 5.11 implies that there is some qj,j^′ ∈ S_1,0^m(Rⁿ× Rⁿ) such that

qj,j^′(x, Dx)u = (κ⁻¹_j ◦ κ^′_j′)⁻¹
∗

ϕjpj(x, Dx) ψj(κ⁻¹_j ◦ κ^′_j′)^∗u

.

If we now define

qj^′(x, ξ) = X

j^′:Ω^′

j′∩Ωj6=∅

qj,j^′(x, ξ),

we have that

(Φ^′_j′PΨ^′_j′u)((κ^′_j)⁻¹(x)) = ˜ϕ^′_jqj(x, Dx)(ψ_j^′u^′_j′)(x) for all x ∈ U_j^′, which proves the theorem.