Pseudodifferential Operators

Chapter VI Kodaira’s Projective Embedding Theorem 217 1. Hodge Manifolds 217

3. Pseudodifferential Operators

and hence theC¯αinvolve various derivatives of both metrics onEandF and of the density function ρ(x)on X. This formula suffices to define a linear differential operator L^∗: D(X, F )→D(X, E), which has automatically the property of being the adjoint ofL. Moreover, we see that the symbolσ_k(L^∗) is given by the terms in (2.6) which only differentiateτ, since all other terms give lower-order terms in the expression

|α|≤kC˜_α(x). One checks that the symbol of L^∗ as defined above is the adjoint of the symbol of the operator L by representing σ_k(L) with respect to these local frames and computing its adjoint as a linear mapping.

Q.E.D.

We have given a brief discussion of the basic elements of partial differential operators in a setting appropriate for our purposes. For more details on the subject, see Hörmander [1] for the basic theory of modern partial differential equations (principally in Rⁿ). Palais [1] has a formal presentation of partial differential operators in the context of manifolds and vector bundles, with a viewpoint similar to ours. In the next sections we shall generalize the concept of differential operators in order to find a class of operators which will serve as “inverses” for elliptic partial differential operators, to be studied in Sec. 4.

(j = 1, . . . , n) and D^α = (−i)^|α|D^α₁¹· · ·D_n^αn replaces ξ₁^α1· · · ξ_n^αn in the polynomial p(x, ξ ). By using the Fourier transform we may write, for u∈D(U ),

(3.1) P u(x)=p(x, D)u(x)=

$

p(x, ξ )u(ξ )eˆ ^ix,ξdξ, where x, ξ =n

j=1x_jξ_j is the usual Euclidean inner product, andu(ξ )ˆ = (2π )⁻ⁿ+

u(x)e^−ix,ξdx is the Fourier transform ofu.

Thus (3.1) is an equivalent way (via Fourier transforms) to define the action of a differential operator p(x, D) defined by a polynomial p(x, ξ ) on functions in the domainU. We use compact supports here so that there is no trouble with the integral existing near∂U, and sinceD(U ) is dense in most interesting spaces, it certainly suffices to know how the operator acts on such functions. Of course, P (x, D): D(U ) → D(U ), since differential operators preserve supports.

To define the generalization of differential operators we are interested in, we can consider (3.1) as thedefinitionof differential operator and generalize the nature of the function p(x, ξ )which appears in the integrand.

To do this, we shall define classes of functions which possess, axiomatically, several important properties of the polynomials considered above.

Definition 3.1: Let U be an open set inRⁿ and let m be any integer.

(a) Let S˜^m(U ) be the class ofC^∞ functions p(x, ξ )defined onU×Rⁿ, satisfying the following properties. For any compact set K in U, and for any multiindices α, β, there exists a constantC_α,β,K, depending onα, β, K, and p so that

(3.2) |D_x^βD^α_ξp(x, ξ )| ≤C_α,β,K(1+ |ξ|)^m−|α|, x ∈K, ξ ∈Rⁿ. (b) Let S^m(U ) denote the set of p∈ ˜S^m(U ) such that (3.3) The limit σ_m(p)(x, ξ )= lim

λ→∞

p(x, λξ )

λ^m exists forξ =0, and, moreover,

p(x, ξ )−ψ (ξ )σ_m(p)(x, ξ )∈ ˜S^m−1(U ),

where ψ ∈ C^∞(Rⁿ) is a cut-off function with ψ (ξ ) ≡0 near ξ =0 and ψ (ξ )≡1 outside the unit ball.

(c) LetS˜₀^m(U )denote the class ofp∈ ˜S^m(U )such that there is a compact set K ⊂U, so that for any ξ ∈ Rⁿ, the function p(x, ξ ), considered as a function of x ∈ U, has compact support in K [i.e., p(x, ξ ) has uniform compact support in the x-variable]. Let S₀^m(U )=S^m(U )∩ ˜S₀^m(U ).

We notice that ifp(x, ξ )is a polynomial of degreem(as before), then both properties (a) and (b) in Definition 3.1 above are satisfied. If the coefficients of p have compact support in U, then p∈S₀^m(U ). Property (a) expresses the growth in the ξ variable near ∞, whereas σm(p)(x, ξ )is the mth order homogeneous part of the polynomial p, the lower-order terms having gone

to zero in the limit. We shall also be interested in negative homogeneity, and the cut-off function in (b) is introduced to get rid of the singularity near ξ =0, which occurs in this case.

A second example is given by an integral transform with a smooth kernel.

LetK(x, y)be aC^∞function inU×U with compact support in the second variable. Then the operator

Lu(x)=

$

Rⁿ

K(x, y)u(y) dy

can be written in the form (3.1) for an appropriate function p(x, ξ ); i.e., for u∈D(U ),

Lu(x)=

$

p(x, ξ )u(ξ )eˆ ^ix,ξdξ,

where p∈ ˜S^m(U ) for all m. To see this, we write, by the Fourier inversion formula,

Lu(x)=

$

K(x, y)

$

e^iy,ξu(ξ ) dξˆ dy

=

$ e^ix,ξ

$

e^iy−x,ξK(x, y)dy u(ξ )dξˆ and we let

p(x, ξ )=

$

e^iy−x,ξK(x, y) dy, which we rewrite as

p(x, ξ )=e^−ix,ξ

$

e^iy,ξK(x, y) dy.

Thus p(x, ξ )is (except for the factor e^−ix,ξ) for each fixed x the Fourier transform of a compactly supported function, and then it is easy to see (by integrating by parts) that p(x, ξ ), as a function of ξ, is rapidly decreasing at infinity; i.e.,

(1+ |ξ|)^N|p(x, ξ )| −→0

as|ξ| → ∞for all powers ofN(this is the classSintroduced by Schwartz [1]).

It then follows immediately that p ∈ ˜S^m(U ) for all m. Such an operator is often referred to as a smoothing operator with C^∞ kernel. The term smoothing operator refers to the fact that it is an operator of order −∞, i.e., takes elements of any Sobolev space toC^∞ functions, which is a simple consequence of Theorem 3.4 below and Sobolev’s lemma (Proposition 1.1).

Lemma 3.2: Suppose that p∈S^m(U ). Then σ_m(p)(x, ξ )is a C^∞ function on U×(Rⁿ− {0})and is homogeneous of degree min ξ.

Proof: It suffices (by the Arzela-Ascoli theorem) to show that for any compact subset of the formK×L, whereKis compact inUandLis compact in Rⁿ− {0}, we have the limit in (3.3) converging uniformly and that all

derivatives in x andξ ofp(x, λξ )/λ^m are uniformly bounded onK×Lfor λ∈(1,∞). But this follows immediately from the estimates in (3.2) since

D_x^βD^α_ξ

p(x, λξ ) λ^m

=D_x^βD_ξ^αp(x, λξ )•λ^|α| λ^m, and hence, for all multiindices α, β,

D^β_xD_ξ^α

p(x, λξ ) λ^m

≤C_α,β,K(1+λ|ξ|)^m−|α|•λ^|α| λ^m

≤C_α,β,K(λ⁻¹+ |ξ|)^m−|α|

≤C_α,β,Ksup

ξ∈L

(1+ |ξ|)^m−|α|<∞.

Therefore all derivatives are uniformly bounded, and in particular the limit in (3.3) is uniform. Showing homogeneity is even simpler. We write, for ρ >0,

σ_m(x, ρξ )= lim

λ→∞

p(x, λρξ ) λ^m·

= lim

λ→∞

p(x, λρξ ) (ρλ)^m •ρ^m

= lim

λ→∞

p(x, λ^′ξ )

(λ^′)^m •ρ^m (λ^′=ρλ)

=σ_m(x, ξ )•ρ^m.

Q.E.D.

We now define the prototype (local form) of our pseudodifferential operator by using (3.1). Namely, we set, for anyp∈ ˜S^m(U ) andu∈D(U ),

(3.4) L(p)u(x)=

$

p(x, ξ )u(ξ )eˆ ^ix,ξ dξ,

and we call L(p) a canonical pseudodifferential operator of order m.

Lemma 3.3: L(p)is a linear operator mapping D(U ) intoE(U ).

Proof: Since u∈D(U ), we have, for any multiindexα, ξ^αu(ξ )ˆ =(2π )⁻ⁿ

$

D^αu(x)e^−ix,ξdx,

and hence, since u has compact support, |ξ^α|| ˆu(ξ )| is bounded for any α, which implies that for any large N,

| ˆu(ξ )| ≤C(1+ |ξ|)^−N,

i.e., u(ξ )ˆ goes to zero at∞ faster than any polynomial. Then we have the estimate for any derivatives of the integrand in (3.4),D_x^βp(x, ξ )u(ξ )ˆ ≤C(1+ |ξ|)^m(1+ |ξ|)^−N,

which implies that the integral in (3.4) converges nicely enough to differenti- ate under the integral sign as much as we please, and henceL(p)(u)∈E(U ).

It is clear from the same estimates that L(p)is indeed a continuous linear mapping from D(U )→E(U ).

Q.E.D.

Our next theorem tells us that the operators L(p)behave very much like differential operators.

Theorem 3.4: Suppose thatp∈ ˜S₀^m(U ). ThenL(p)is an operator of orderm.

Remark: We introduce functions with compact support to simplify things somewhat. Our future interest is compact manifolds and the functions p which will arise will be of this form due to the use of a partition of unity.

Proof: We must show that ifu∈D(U ), then, for some C >0,

(3.5) L(p)us≤Cus+m,

where •^s= •^s,Rn, as in (1.1), First we note that

(3.6) L(p)u(ξ ) =

$ ˆ

p(ξ−η,η)u(η) dη,ˆ

wherep(ξˆ −η,η)denotes the Fourier transform ofp(x,η)in the first variable evaluated at the point ξ−η. Sincep has compact support in thex-variable and because of the estimate (3.2), we have (as before) the estimate, for any large N,

(3.7) | ˆp(ξ−η,η)| ≤C_N(1+ |ξ−η|²)^−N(1+ |η|²)^m/2 for (ξ,η)∈Rⁿ×Rⁿ. We have to estimate

Lu²s =

$

|L(p)u(ξ ) |²(1+ |ξ|²)^sdξ in terms of

u²_s+m=

$

| ˆu(ξ )|²(1+ |ξ|²)^m+sdξ.

We shall need Young’s inequality, which asserts that iff∗gis the convolution of an f ∈L¹(Rⁿ)and g∈L^p(Rⁿ), then

f ∗gL^p≤ fL¹gL^p

(see Zygmund [1]).

Proceeding with the proof of (3.5) we obtain immediately from (3.6) and the estimate (3.7), letting C denote a sufficiently large constant in each estimate,

|L(p)u(ξ ) | ≤C

$

(1+ |ξ−η|²)^−N(1+ |η|²)^m/2| ˆu(η)|dη

≤C

$ (1+ |ξ−η|²)^−N

(1+ |η|²)^s/2 (1+ |η|²)^(m+s)/2| ˆu(η)|dη.

Now, using (1.2), we easily obtain

|L(p)u(ξ ) | ≤C(1+ |ξ|²)^−s/2

$

(1+ |ξ−η|²)^−N+s/2

×(1+ |η|²)^(m+s)/2| ˆu(η)|dη.

Assume now that N is chosen large enough so thatf (ξ )=(1+ |ξ|²)^−N+s/2 is integrable, and we see that

|L(p)u(ξ ) |(1+ |ξ|²)^s/2≤C

$

(1+ |ξ−η|²)^−N+s/2

×(1+ |η|²)^(m+s)/2| ˆu(η)|dη.

By Young’s inequality we obtain immediately L(p)us≤Cf0 •us+m

≤Cus+m.

Q.E.D.

We now want to define pseudodifferential operators in general. First we consider the case of operators on a differentiable manifold X mapping functions to functions.

Definition 3.6: Let Lbe a linear mappingL: D(X)→E(X). Then we say thatLis apseudodifferential operator onX if and only if for any coordinate chart U ⊂ X and any open set U^′ ⊂ ⊂ U there exists a p ∈ S₀^m(U ) (consideringUas an open subset ofRⁿ) so that ifu∈D(U^′), then [extending u by zero to be inD(X)]

Lu=L(p)u;

i.e., by restricting to the coordinate patch U, there is a functionp∈S₀^m(U ) so that the operator is a canonical pseudodifferential operator of the type introduced above.

More generally, if E and F are vector bundles over the differentiable manifold X, we make the natural definition.

Definition 3.7: Let Lbe a linear mappingL:D(X, E)→E(X, F ). ThenL is a pseudodifferential operatoronX if and only if for any coordinate chart U with trivializations of E and F over U and for any open setU^′⊂ ⊂U there exists a r×p matrix (p^ij), p^ij ∈S₀^m(U ), so that the induced map

LU: D(U^′)^p−→E(U )^r with u∈D(U^′)^p

L_U

// Lu, extendinguby zero to be an element ofD(X, E) [where p=rank E, r=rank F, and we identify E(U )^p with E(U, E) and E(U )^r with E(U, F )], is a matrix of canonical pseudodifferential operators L(p^ij), i=1, . . . , r, j=1, . . . , p, defined by (3.4).

We see that this definition coincides (except for the restriction to the relatively compact subset U^′) with the definition of a differential operator given in Sec. 2, where all the corresponding functions p^ij are polynomials in S^m(U ).

Remark: The additional restriction in the definition of restricting the action ofL_U to functions supported inU^′⊂ ⊂U is due to the fact that in general a pseudodifferential operator is not alocal operator; i.e., it does not preserve supports in the sense that suppLu⊂suppu(which is easy to see for the case of a smoothing operator, for instance). In fact, differential operators can be characterized by the property of localness (a result of Peetre [1]). Thus the symbol of a pseudodifferential operator will depend on the choice ofU^′ which can be considered as a choice of a cutoff function. The difference of two such local representations for pseudodifferential operators onU^′⊂ ⊂U andU^′′⊂ ⊂U will be an operator of order−∞acting on smooth functions supported in U^′∩U^′′.

Definition 3.8: The local m-symbol of a pseudodifferential operator L: D(X, E) → E(X, F ) is, with respect to a coordinates chart U and trivializations of E and F over U, the matrix†

σ_m(L)_U(x, ξ )= [σ_m(p^ij)(x, ξ )], i=1, . . . , r, j =1, . . . , p.

Note that in all these definitions the integer m may depend on the coordinate chart U. If X is not compact, then the integer m may be unbounded on X. We shall see that the smallest possible integer m in some sense will be the order of the pseudodifferential operator on X. But first we need to investigate the behavior of the local m-symbol under local diffeomorphisms in order to obtain aglobalm-symbol of a global operatorL.

The basic principle is the same as for differential operators. If a differential operator is locally expressed asL=

|α|≤mc_α(x)D_x^α and we make a change of coordinatesy =F (x), then we can express the same operator in terms of these new coordinates using the chain rule and obtain

˜

L=

|α|≤m

˜ cα(y)D^α_y

and L(u(F (x))˜ =

|α|≤m

˜

c_α(F (x))D_y^αu(F (x)).

Under this process, the order is the same, and, in particular, we still have a differential operator. Moreover, the mth order homogeneous part of the polynomial,

|α|=mc_αξ^α, transforms by the Jacobian of the transformation y =F (x)in a precise manner. We want to carry out this process for pseudo- differential operators, and this will allow us to generalize the symbol map given by Proposition 2.3 for differential operators. For simplicity we shall carry out this program here only for trivial line bundles over X, i.e., for

†σm(L)U will also depend on U^′⊂ ⊂U, which we have suppressed here.

pseudodifferential operators mapping functions to functions, leaving the more general case of vector bundles to the reader.

The basic result we need can be stated as follows.

Theorem 3.9: Let U be open and relatively compact in Rⁿ and let p ∈ S₀^m(U ). Suppose thatF is a diffeomorphism ofU onto itself [in coordinates x =F (y), x, y∈Rⁿ]. Suppose thatU^′⊂ ⊂U and define the linear mapping

˜

L: D(U^′)−→E(U ) by setting

˜

Lv(y)=L(p)(F⁻¹)^∗v(F (y)).

Then there is a function q ∈S^m₀(U ), so that L˜ =L(q), and, moreover, σ_m(q)(y,η)=σ_m(p)

% F (y),

_t

∂F

∂y

−1

η

&

.

Here (F⁻¹)^∗: E(U )→E(U ) is given by(F⁻¹)^∗v=v◦F⁻¹, and the basic content of the theorem is that pseudodifferential operators are invariant under local changes of coordinates and that the local symbols transform in a precise manner, depending on the Jacobian (∂F /∂y) of the change of variables. Before we prove this theorem, we shall introduce a seemingly larger class of Fourier transform operators, which will arise naturally when we make a change of coordinates. Then we shall see that this class is no larger than the one we started with.

Let p∈S₀^m(U ) for U open in Rⁿ. Then we see easily that from (3.4) we obtain the representation

(3.8) L(p)u(x)=(2π )⁻ⁿ

$ $

e^iξ,x−zp(x, ξ )u(z) dz dξ,

using the Fourier expression foru. We want to generalize this representationˆ somewhat by allowing the function p above to also depend on z. Suppose that we consider functions q(x, ξ, z) defined andC^∞ on U×Rⁿ×U, with compact support in thex- andz-variables and satisfying the following two conditions (similar to those in Definition 3.1):

(a) D_ξ^αD_x^βD_z^γq(x, ξ, z)≤C_α,β,γ(1+ |ξ|)^m−|α|. (b) The limit lim

λ→∞

q(x, λξ, x)

λ^m =σ_m(q)(x, ξ, x), ξ =0, (3.9)

exists and ψ (ξ )σ_m(q)(x, ξ, x)−q(x, ξ, x)∈ ˜S^m−1(U ).

Proposition 3.10: Let q(x, ξ, z) satisfy the conditions in (3.9) and let the operator Qbe defined by

(3.10) Qu(x)=(2π )⁻ⁿ

$

e^iξ,x−zq(x, ξ, z)u(z) dz dξ

foru∈D(U ). Then there exists ap∈S₀^m(U )such thatQ=L(p). Moreover, σ_m(p)(x, ξ )= lim

λ→∞

q(x, λξ, x)

λ^m , ξ =0.

This proposition tells us that this “more general” type of operator is in fact one of our original class of operators, and we can compute its symbol.

Proof: Let q(x, ξ, ζ )ˆ denote the Fourier transform of q(x, ξ, z) with respect to the z-variable. Then we obtain, from (3.10),

Qu(x)=

$$

e^iξ,xq(x, ξ, ξˆ −η)u(η) dηˆ dξ

=

$ e^iη,x

0$

e^iξ−η,xq(x, ξ, ξˆ −η) dξ 1

ˆ u(η) dη.

Thus, if we set

p(x,η)=

$

e^iξ−η,xq(x, ξ, ξˆ −η) dξ

=

$

e^iζ,xq(x, ζˆ +η, ζ ) dζ, (3.11)

we have the operator Q represented in the form (3.4). First we have to check that p(x,η)∈ ˜S^m(U ), but this follows easily by differentiating under the integral sign in (3.11), noting that q(x, ζˆ +η, y) decreases very fast at

∞ due to the compact support of q(x, ξ, z) in thez-variable. We now use the mean-value theorem for the integrand:

(3.12) q(x, ζˆ +η, ζ )= ˆq(x,η, ζ )+

|α|=1

D^α_ηq(x,ˆ η+ζ0, ζ )ζ^α

for a suitable ζ0 lying on the segment in Rⁿ joining 0 to ζ. We have the estimate

|D_η^αq(x,ˆ η+ζ0, ζ )| ≤C_N(1+ |η+ζ0|)^m−1(1+ |ζ|)^−N

for sufficiently large N, and since|ζ₀(ζ )| ≤ |ζ|, we see that we obtain, with a different constant,

|D_η^αq(x,ˆ η+ζ₀, ζ )| ≤ ˜C_N(1+ |η|)^m−1(1+ |ζ|)^−N+m−1.

By inserting (3.12) in (3.11), choosing N sufficiently large, and integrating the resulting two terms we obtain

p(x,η)=q(x,η, x)+E(x,η), where

(3.13) |E(x,η)| ≤C(1+ |η|)^m−1. Therefore

λ→∞lim

p(x, λη) λ^m = lim

λ→∞

q(x, λη, x) λ^m + lim

λ→∞

E(x, λη)

λ^m , η=0.

It follows that the limit on the left exists and that σ_m(p)(x,η)= lim

λ→∞

q(x, λη, x)

λ^m , η=0,

since the last term above has limit zero because of the estimate (3.13).

The fact that p(x, ξ )−ψ (ξ )σm(p)(x, ξ )∈ ˜S^m−1(U ) follows easily from the hypothesis on q(x,η, x) and the growth ofE(x,η).

Q.E.D.

We shall need one additional fact before we can proceed to our proof of Theorem 3.9. Namely, suppose that we rewrite (3.8) formally as

L(p)u(x)=(2π )⁻ⁿ

$ 0$

e^iξ,x−zp(x, ξ ) dξ 1

u(z) dz and let

K(x, x−z)=

$

e^iξ,x−zp(x, ξ ) dξ.

Then we have the following proposition.

Proposition 3.11: K(x, w) is a C^∞ function of x and w provided that w=0.

Proof: Suppose first that m <−n, then we have the estimate

|p(x, ξ )| ≤C(1+ |ξ|)⁻ⁿ⁻¹ from (3.2), and thus the integral

K(x, w)=

$

e^iξ,wp(x, ξ ) dξ

converges. Integrating repeatedly by parts, and assuming that, for instance, w₁=0, we obtain

K(x, w)=(−1)^N

$ e^iξ,w w^N₁ D^N_ξ

1p(x, ξ ) dξ for any positive integer N. Hence

D_x^αD_w^βK(x, w)=(−1)^N

$ ξ^βe^iξ,w

w^N₁ D_x^αD_ξ^N₁p(x, ξ ) dξ.

Using the estimates (3.2) we see that the integral on the right converges for N sufficiently large. ThusK(x, w)isC^∞ for w=0, provided thatm <−n.

Suppose now that mis arbitrary; then we write, choosing ρ > m+n, K(x, w)=

$

e^iξ,w(1+ |ξ|²)^−ρ(1+ |ξ|²)^ρp(x, ξ ) dξ and we see that we have (letting _w=

D_j² be the usual Laplacian in the w variable)

K(x, w)=$ )(1−_w)^ρe^iξ,w*p(x, ξ )(1+ |ξ|²)^−ρdξ which is the same as

(1−_w)^ρ

$

e^iξ,wp(x, ξ )(1+ |ξ|²)^−ρdξ

if the integral converges. But this then follows from the case considered above, since

p(x, ξ )(1+ |ξ|²)^−ρ ∈S⁻₀⁽ⁿ⁺¹⁾(U ).

So, forw=0, the above integral isC^∞, and thusK(x, w)isC^∞forw=0.

Q.E.D.

We shall now use these propositions to continue our study of the behavior of a pseudodifferential operator under a change of coordinates.

Proof of Theorem 3.9: Now p(x, ξ ) has compact support inU (in the x-variable) by hypothesis. Let ψ (x)˜ ∈D(U ) be chosen so that ψ˜ ≡1 on supp p∪U^′ and set ψ (y)= ˜ψ (F (y)). We have, as in (3.8),

L(p)u(x)=(2π )⁻ⁿ

$$

e^iξ,x−zp(x, ξ )u(z) dz dξ,

foru∈D(U^′)⊂D(U ). We writez=F (w)andυ(w)=u(F (w))and obtain L(p)u(F (y))=(2π )⁻ⁿ

$$

e^{iξ,F (y)−F (w)}p(F (y), ξ )

∂F

∂w

υ(w) dw dξ, where |∂F /∂w| is the determinant of the Jacobian matrix ∂F /∂w. By the mean-value theorem we see that

F (y)−F (w)=H (y, w)•(y−w),

where H (y, w) is a nonsingular matrix for w close to y and H (w, w) = (∂F /∂w)(w). Letχ₁(y, w) be a smooth nonnegative function ≡1 near the diagonal in U ×U and with support on a neighborhood of where H (y, w) is invertible. Let χ2=1−χ1. Thus we have

L(p)u(F (y))=(2π )⁻ⁿ

$$

e^{iξ,H (y,w)•(y−w)}p(F (y), ξ )

∂F

∂w

v(w) dw dξ, which we may rewrite, setting ζ =^tH (y, w)ξ,

L(p)u(F (y))=(2π )⁻ⁿ$$

e^iζ,y−wp(F (y),[^tH (y, w)]⁻¹ζ )

×

∂F

∂w

ψ (w) χ1(y, w)

|H (y, w)|v(w) dw dζ +Eu(F (y))

=(2π )⁻ⁿ

$$

e^iζ,y−wq₁(y, ζ, w)v(w) dw dζ+Eu(F (y)) . Here E is the term corresponding to χ2 and

q1(y, ζ, w)=p(F (y),[^tH (y, w)]⁻¹ζ )

∂F

∂w

χ₁(y, w)

|H (y, w)|ψ (w),

whileψ∈D(U )as chosen above is identically 1 on a neighborhood of supp v(y). Thus q1(y, ζ, w) has compact support in the y and w variables, and it follows readily that conditions (a) and (b) of (3.9) are satisfied. Namely,

(a) will follow from the estimates for p by the chain rule, whereas for (b) we have

σ (q1)(y, ξ, y)= lim

λ→∞

q₁(y, λξ, y) λ^m = lim

λ→∞

p(F (y),[^t(∂F /∂y)]⁻¹λξ ) λ^m

=σ (p)

% F (y),

_t

∂F

∂y

−1

ζ

&

, ξ =0; moreover the desired growth of

ψ (ξ )σ (q₁)(y, ξ, y)−q₁(y, ξ, y) follows easily from the hypothesized growth of

ψ (ξ )σ (p)(x, ξ )−p(x, ξ ).

We still have to worry about the term E, which we claim is a smoothing operator of infinite order (see the example following Definition 3.1) and will give no contribution to the symbol. In fact, we have

Eu(x)=

$

e^iξ,x−zp(x, ξ )χ₂(x, z)u(z) dz dξ

=

$ 0$

e^iξ,x−zp(x, ξ )χ₂(x, z) dξ 1

u(z) dz

=

$

χ₂(x, z)K(x, x−z)u(z) dz

=

$

W (x, z)u(z) dz, where

K(x, w)=

$

e^iξ,wp(x, ξ ) dξ.

But we have seen earlier that K(x, w) is a C^∞ function of x and w for w=0.† Also, χ2(x, z) vanishes identically near x−z=0, so the product χ2(x, z)K(x, x−z)=W (x, z) is a smooth function onU×Rⁿ, andEu(x) is then a smoothing operator with C^∞ kernel, which we can write in terms of the new coordinates y=F⁻¹(x), w=F⁻¹(z),

Eu(F (y))=

$

W (F (y), F (w))u(F (w))

∂F

∂w dw

=

$

W₁(y, w)F^∗u(w) dw,

where W1 is a C^∞ function onU×U, which we rewrite as

=

$

W₁(y, w)ψ (w)F^∗u(w) dw,

†Note thatK(x, w)has compact support in the first variable, sincep(x, ξ )=ϕ(x)p(x, ξ ) for an appropriate ϕ∈D(U ).

where we note that ψ≡1 on supp F^∗u. Then we have Eu(F (y))=

$

W2(y, w)υ(w) dw, where v(w)=F^∗u(w), as before, and

W₂(y, w)=W (F (y), F (w))

∂F

∂w ψ (w),

which is a smoothing operator of order −∞withC^∞ kernel with compact support in both variables, as discussed following Definition 3.1, Thus, by Proposition 3.11,

Eu(F (y))=

$

e^iw,ξq₂(y, ξ )υ(ξ ) dξ,ˆ where

q₂(y, ξ )=

$

e^iy−w,ξW₂(y, w) dw,

and q2∈ ˜S₀^r(U )for all r. This implies easily thatσ_m(q2)(y, ξ )≡0. Thus we can let q =q1+q2, and we have

˜

Lv(y)=L(q),

and the symbols behave correctly [here we let q₁(y, ξ, w) be replaced by q₁(y, ξ ), as given by Proposition 3.10].

Q.E.D.

We are now in a position to define the global symbol of a pseudodiffer- ential operator on a differentiable manifold X. Again we treat the case of functions first, and we begin with the following definition.

Definition 3.12: Let X be a differentiable manifold and let L: D(X)−→

E(X)be a pseudodifferential operator. ThenLis said to be a pseudodiffer- ential operator of order m on X if, for any choice of local coordinates chart U ⊂ X, the corresponding canonical pseudodifferential operator L_U is of order m; i.e., L_U = L(p), where p ∈ S^m(U ). The class of all pseudodifferential operators on X of order m is denoted by PDiffm(X).

Proposition 3.13: Suppose that X is a compact differentiable manifold. If L∈PDiffm(X), then L∈OPm(X).

Proof: This is immediate from Theorem 3.4 and the definition of Sobolev norms on a compact manifold, using a finite covering of X by coordinate charts.

Q.E.D.

This proposition tells us that the two definitions of “order” of a pseudo- differential operator are compatible. We remark that ifp∈S^m(U ), for some U ⊂Rⁿ, thenp∈S^m+k(U )for any positivek; moreover, in this case,σm+k(p)

=0, k >0. Thus we have the natural inclusion PDiffm(X)⊂PDiffm+k(X), k≥0. Denote Smblm(X×C, X×C) by Smblm(X) for simplicity.

Proposition 3.14: There exists a canonical linear map σ_m: PDiff_m(X)−→Smbl_m(X), which is defined locally in a coordinate chart U⊂X by

σ_m(L_U)(x, ξ )=σ_m(p)(x, ξ ),

where LU =L(p) and where (x, ξ )∈ U×(Rⁿ− {0}) is a point in T^′(U ) expressed in the local coordinates of U.

Proof: We merely need to verify that the local representation ofσ_m(L) defined above transforms correctly so that it is indeed globally a homomor- phism of T^′(X)×CintoT^′(X)×C, which is homogeneous in the cotangent vector variable of orderm(see the definition in Sec. 2). But this follows eas- ily from the transformation formula forσ_m(p)given in Theorem 3.9, under a local change of variables. The linearity of σ_m is not difficult to verify.

Q.E.D.

This procedure generalizes to pseudodifferential operators mapping sections of vector bundles to sections of vector bundles, and we shall leave the formal details to the reader. We shall denote by PDiffm(E, F )the space of pseudodifferential operators of ordermmappingD(X, E)intoE(X, F ).

Moreover, there is an analogue to Proposition 3.14, whose proof we omit.

Proposition 3.15: Let E and F be vector bundles over a differentiable manifold X. There exists a canonical linear map

σ_m: PDiff_m(E, F )−→Smbl_m(E, F ), which is defined locally in a coordinate chart U⊂X by

σ_m(L_U)(x, ξ )= [σ_m(p^ij)(x, ξ )],

where LU = [L(p^ij)] is a matrix of canonical pseudodifferential operators, and where (x, ξ )∈U×(Rⁿ− {0})is a point inT^′(U ) expressed in the local coordinates of U.

One of the fundamental results in the theory of pseudodifferential operators on manifolds is contained in the following theorem.

Theorem 3.16: LetEandFbe vector bundles over a differentiable manifold X. Then the following sequence is exact,

(3.15) 0−→K_m(E, F )−→^j PDiff_m(E, F )−→^σm Smbl_m(E, F )−→0, whereσmis the canonical symbol map given by Proposition 3.15,Km(E, F ) is the kernel of σm, and j is the natural injection. Moreover, Km(E, F )⊂ OPm−1(E, F )if X is compact.

Proof: We need to show thatσmis surjective and thatσm(L)=0 implies that L is an operator of order m−1. Doing the latter first, we note that σ_m(L) = 0 for some L ∈ PDiffm(E, F ) means that in a local trivializing coordinate chart, L has the representation L_U = [L(p^ij)], p^ij ∈ S^m(U ).

Since σ_m(L)|^U = σ_m(L_U) = [σ_m(p^ij)] = 0, by hypothesis, it follows that p^ij ∈ ˜S^m−1(U ), by hypothesis on the class S^m. Hence L_U is an operator of order m−1, and thus Lwill be an operator of order m−1. To prove that σ_m is surjective, we proceed as follows. Let{U_µ}be a locally finite cover of X by coordinate charts U_µ over which E and F are both trivializable. Let {φ_µ}be a partition of unity subordinate to the cover {U_µ} and let{ψ_µ}be a family of functions ψ_µ∈D(U_µ), where ψ_µ≡1 on supp φ_µ. We then let χ be a C^∞ function onRⁿ withχ ≡0 near 0∈Rⁿ and χ≡1 outside the unit ball. Let s ∈ Smbl_m(E, F ) be given, and write s =

µφ_µs =

µs_µ; supp s_µ ⊂supp φ_µ⊂U_µ. Then with respect to a trivialization of E and F over U, we see that sµ = [s_µ^ij], a matrix of homogeneous functions s_µ^ij: Uµ×Rⁿ− {0} −→C, and s_µ^ij(x, ρξ )=ρ^ms_µ^ij(x, ξ ), for ρ >0. We let p^ij_µ(x, ξ )=χ (ξ )s_µ^ij(x, ξ ). It follows from the homogeneity that p_µ^ij∈S₀^m(U ) and that σ_m(p_µ^ij)=s_µ^ij. We now let

L_µ: D(U_µ)^p−→E(U_µ)^r

be defined byL_µu= [L(p_µ^ij)]u, with the usual matrix action of the matrix of operators on the vectoru. Ifu∈D(X, E), then we letu=

φ_µu= u_µ, considering each u_µas a vector in D(U_µ)^pby the trivializations. We then define

Lu=

µ

ψ_µ(L_µu_µ), and it is clear that

L: D(X, E)−→E(X, F )

is an element of PDiffm(E, F ), since locally it is represented by a matrix of canonical pseudodifferential operators of orderm. Note that it is necessary to multiply byψµin order to sum, sinceLµuµisC^∞, where we considerLµuµ

as an element of E(U_µ, F )in U_µ, but that it does not necessarily extend in aC^∞manner to aC^∞section ofF overX. Thus we have constructed from s a pseudodifferential operatorL in a noncanonical manner; it remains to show that σ_m(L)=s. But this is simple, since (ψ_µL_µ)∈PDiffm(E, F ) and

σ_m(ψ_µL_µ)_U(x, ξ )=σ_m(ψ_µ(x)p^ij_µ(x, ξ ))=ψ_µ(x)lim

λ→∞p^ij_µ(x, λξ )/λ^m

=ψ_µ(x)s_µ^ij(x, ξ )=s_µ^ij(x, ξ ), since ψ_µ≡1 on supp s_µ. It follows that

σm(ψµLµ)=sµ, and by linearity of the symbol map

σm µ

ψµLµ

=

µ

sµ=s.

Q.E.D.