3. Pseudodifferential operators

Differential and pseudodifferential operators on manifolds

Course on analytic geometry (Jean-Pierre Demailly)

1. Differential operators on vector bundles

We first describe some basic concepts concerning differential operators (symbol, composi- tion, ellipticity, adjoint), in the general context of vector bundles. Assume given a differentiable manifold M of class C^∞, dim_RM = m, and let E, F be vector bundles on M, over the field K=R or K=C, such that rankE =r, rankF =r⁰.

1.1. Definition. A (linear) differential operator of degreeδ from E toF is a K-linear operator P :C^∞(M, E)→C^∞(M, F), u7→P u of the form

P u(x) = X

|α|6δ

a_α(x)D^αu(x),

where EΩ 'Ω×K^r, FΩ 'Ω×K^r

0 are local trivializations on an open chart Ω⊂M with local coordinates (x1, . . . , xm), and the coefficients aα(x) are r⁰×r matrices aαλµ(x)

16λ6r⁰,16µ6r

with C^∞ coefficients on Ω. One write here D^α = (∂/∂x₁)^α1· · ·(∂/∂x_m)^αm as usual, and the matrices u= (uµ)16µ6r, D^αu = (D^αuµ)16µ6r are viewed as column vectors.

If t∈K is a parameter andf ∈C^∞(M,K), u∈C^∞(M, E), an easy calculation shows that e^−tf(x)P(e^tf(x)u(x)) is a homogeneous polynomial of degree δ in t, of the form

e^−tf(x)P(e^tf(x)u(x)) =t^δσP(x, df(x))·u(x) +terms cj(x)t^j of degree j < δ, where σP is a homogeneous polynomial map T_M^? →Hom(E, F) defined by

(1.2) T_M,x^? 3ξ7→σP(x, ξ)∈Hom(Ex, Fx), σP(x, ξ) = X

|α|=δ

aα(x)ξ^α.

Then σP(x, ξ) is a C^∞ function of the variables (x, ξ) ∈T_M^? , and this function is independent of the choice of the coordinates or trivializations used for E, F. The function σP is called the principal symbol of P. The principal symbol of a composition Q◦P of differential operators P :C^∞(M, E)→C^∞(M, F), Q:C^∞(M, F)→C^∞(M, G) is simply the product

(1.3) σ_Q◦P(x, ξ) =σ_Q(x, ξ)σ_P(x, ξ),

calculated as a product of matrices (i.e. as a composition of endomorphisms). Differential operators for which the symbol is injective play a very important role :

1.4. Definition. A differential operator P is said to be elliptic if σP(x, ξ) ∈ Hom(Ex, Fx) is injective for all x∈M and ξ ∈T_M,x^? r{0}.

Let us now assume that M is oriented and assume given a C^∞ volume form dV(x) = γ(x)dx₁∧ · · · ∧dx_m onM, whereγ(x)>0is aC^∞ density ; if(M, g)is a Riemannian manifold, one can for instance take The Riemannian volume form dV =dVg =p

det(gij)dx1∧ · · · ∧dxm.

IfE is a Euclidian or Hermitian vector bundle, we can define a Hilbert spaceL²(M, E)of global sections with values in E, namely the space of forms u with measurable coefficients that are square summable for the scalar product

kuk² = Z

M

|u(x)|²dV(x), (1.5)

hhu, vii= Z

M

hu(x), v(x)idV(x), u, v∈L²(M, E).

(1.5⁰)

1.6. Definition. If P : C^∞(M, E) → C^∞(M, F) is a differential operator and if the bundles E, F are Euclidian or Hermitian, there exists a unique differential operator

P^? :C^∞(M, F)→C^∞(M, E),

called the formal adjoint of P, such that for all sections u ∈C^∞(M, E)and v ∈C^∞(M, F) one has an identity

hhP u, vii=hhu, P^?vii whenever Suppu∩Suppv bM .

Proof. The uniqueness is easy to verify, being a consequence of the density of C^∞ form with compact support inL²(M, E). By a partition of unity argument, we reduce the verification of the existence ofP^? to the proof of its existence locally onM. Now, letP u(x) =P

|α|6δaα(x)D^αu(x) be the description of P relative to the trivializations of E, F associated to an orthonormal frame and to the system of locate coordinates on an open set Ω ⊂ M. By assuming Suppu∩Suppv bΩ, an integration by parts gives

hhP u, vii= Z

Ω

X

|α|6δ,λ,µ

a_αλµD^αu_µ(x)v_λ(x)γ(x)dx₁. . . dx_m

= Z

Ω

X

|α|6δ,λ,µ

(−1)^|α|uµ(x)D^α(γ(x)aαλµvλ(x) dx1. . . dxm

= Z

Ω

hu, X

|α|6δ

(−1)^|α|γ(x)⁻¹D^α γ(x)a^∗_αv(x)

idV(x)

wherea^∗_αdenotes the adjoint (=conjugate of transpose) of matrixa_α. We thus see thatP^? exists, and is defined in a unique way by

(1.7) P^?v(x) = X

|α|6δ

(−1)^|α|γ(x)⁻¹D^α γ(x)a^∗_αv(x) .

1.8. Remark.The conditionSuppu∩SuppvbM avoids boundary terms in the integration by parts. As a consequence, if (M, ∂M) is a manifold with boundary and Suppu∩Suppv touches the boundary, the equality is no longer valid. On the other hand, under the same condition for the supports, the formula still holds whenever products D^αu_µv_λ make sense in the sense of distributions, e.g. if u∈D⁰(M, E) and v∈C^∞(M, F), or vice versa.

Formula (1.7) shows immediately that the principal symbol of P^? is (1.9) σP^?(x, ξ) = (−1)^δ X

|α|=δ

a^∗_αξ^α = (−1)^δσP(x, ξ)^?.

If rankE = rankF, the operator P is elliptic if and only if σ_P(x, ξ) is invertible for ξ 6= 0, therefore the ellipticity of P is equivalent to that of P^?.

2. Sobolev spaces 3

2. Sobolev spaces

The space of tempered distributions S⁰(R^m) is by definition the dual of the Schwartz space S(R^m) of rapidly decrasing C^∞ functions, namely the subspace of functions f ∈ C^∞(R^m) for which all semi-norms f 7→pk,`(f) = sup<sub>|α|6`</sub>(1 +|x|)^k|D^αf(x)| are finite – S(R^m) is a Fréchet space. The Fourier transform ub of a tempered distribution u ∈S⁰(R^m) is defined by the usual adjunction property u(fb ) =u(fb), where

f(ξ) :=b Z

R^m

f(x) e^−2πix·ξdλ(x), f ∈S(R^m).

Let us recall the basic Fourier transform formulas, which can be obtained by differentiating under the integral sign, resp. by an integration by parts

D_ξ^αfb(ξ) = (−2πix)^αf(x)∧

, Dd_x^αf(ξ) = (2πiξ)^αf(ξ)b for any f ∈S(R^m). Also, the Fourier inversion formula stipulates that

bb

f(x) =f(−x) ⇐⇒ f(x) = Z

R^m

fb(ξ) e^2πix·ξdλ(ξ).

By adjunction, the same formulas are still valid for all u∈ S⁰(R^m). For any real number s, we define the Sobolev space W^s(R^m)to be the Hilbert space of tempered distributions u∈S⁰(R^m) such that the Fourier transform ubis a L²_loc function satisfying the estimate

(2.1) kuk²_s =

Z

R^m

(1 +|ξ|²)^s|bu(ξ)|²dλ(ξ)<+∞.

For s∈N, we have Dd^αu(ξ) = (2πiξ)^αbu(ξ), therefore, up to equivalence of norms,

kuk²_s ∼ Z

R^m

X

|α|6s

|D^αu(x)|²dλ(x),

and W^s(R^m) is the Hilbert space of functions u such that all derivatives D^αu of order |α|6s are in L²(R^m). It is also easy to see that W^s(R^m)⊂L²(R^m) for s >0.

We now assume that M is an oriented C^∞ manifold of dimension m, equipped with a volume formdV. LetE →M be aC^∞ Hermitian vector bundle of rank r onM. We denote by

W_loc^s (M, E)

the (local) Sobolev space of sections u : M → E whose components are locally in W^s(R^m) on all open charts. More precisely, choose a locally finite covering(Ωj)of M by relatively compact open coordinate charts Ω_j ' B(0, r_j) ' R^m on which E is trivial. Consider an orthonormal frame (ej,λ)16λ6r of EΩ_j and write u in terms of its components, i.e. u= P

16λ6ruj,λej,λ in Ω_j. We then define a Fréchet topology on W_loc^s (M, E) by considering the semi-norms

(2.2) kuk²_j,s = X

16λ6r

kψjuj,λk²_s

where (ψ_j)is a “quadratic partition of unity” subordinate to(Ω_j), i.e. such thatP

ψ_j² = 1. We show below that the topology defined in this way is independent of the choices made (covering,

partition of unity, trivializations of E). If M is compact, the covering(Ω_j) can be taken finite, and one gets a Hilbert space topology associated to the global norm

(2.2⁰) kuk²_s =X

j

kuk²_j,s=X

j,λ

kψjuj,λk²_s.

When M is compact, we simply write W_loc^s (M, E) = W^s(M, E). To get the independence of the topology on the choices made, we write u = P

j,λψ²_ju_j,λe_j,λ and decompose again u with respect to another covering (Ω⁰_k), Ω⁰_k ' R^m, another quadratic partition of unity P

ψ_k⁰² = 1 and other choices of orthonormal frames (e⁰_k,λ). The new components ψ_k⁰u⁰_k,λ are finite linear combinations of the ψ_ju_j,λ, multiplied by C^∞ functions with compact support in Ω_j ∩Ω⁰_k (as all such terms involve a factor ψjψ_k⁰). It is then sufficient to apply the following lemma.

2.3. Lemma. For any f ∈D(R^m), the multiplication map u7→f u is continuous on W^s(R^m), i.e. kf uks 6Ckuks for some constant C =Cf >0.

Proof. As is well known, we have f uc =fb∗u, that is,b f u(ξ) =c

Z

R^m

fb(ξ−η)bu(η)dλ(η).

One uses the Peetre inequality asserting that

(2.4) (1 +|ξ|²)^s 62^|s|(1 +|ξ−η|²)^|s|(1 +|η|²)^s

for all ξ, η∈R^m and all s ∈R. Replacing ξ by ξ+η, the case s >0 is equivalent to 1 +|ξ+η|² 62(1 +|ξ|²)(1 +|η|²)

which is itself an easy consequence of |ξ+η|² +|ξ−η|² = 2(|ξ|² +|η|²). Now, for s < 0, the inequality is equivalent to (1 +|η|²)^|s| 62^|s|(1 +|η−ξ|²)^|s|(1 +|ξ|²)^|s|, resulting from the case

|s|=−s >0 by switching ξ and η. The Peetre inequality applied to s/2 gives (1 +|ξ|)^s/2|cf u(ξ)|62^|s|/2

Z

R^m

(1 +|ξ−η|)^|s|/2|fb(ξ−η)|(1 +|η|)^s/2|u(η)|b dλ(η).

The latter integral can be seen as a convolution of g(η) = (1 +|η|)^|s|/2|fb(η)| with h(η) = (1 +|η|)^s/2|u(η)|. The Young inequalityb kg∗hk_L² 6kgk_L¹khk_L² finally implies

kf uks6Cfkuks where Cf = 2^|s|/2 Z

R^m

(1 +|η|)^|s|/2|fb(η)|dλ(η) and Cf <+∞ since fb∈S(R^m).

We now recall two further fundamental facts, namely the Sobolev lemma and the Rellich lemma.

2.5. Sobolev lemma. For any integer k ∈ N and any real number s > k + ^m₂, we have W_loc^s (M, E)⊂C^k(M, E) and the inclusion is continuous.

Proof. It is enough to consider the case M = R^m and to show that W^s(R^m) ⊂ C^k(Rⁿ) for s > k + m/2. However, by the Fourier inversion formula, D^αu is the Fourier transform of

2. Sobolev spaces 5

(−2πiξ)^αu(−ξ). If we check thatb ξ^αu(ξ)b ∈ L¹(R^m), it will follow that D^αf is continuous and kD^αfk_∞ 6(2π)^|α|kξ^αu(ξ)kb _L¹. Now, for |α|6k, the Cauchy-Schwarz inequality implies

ξ^αu(ξ)b _L1 6

Z

R^m

|ξ|^k|u(ξ)|b dλ(ξ) = Z

R^m

(1 +|ξ|²)^−s/2|ξ|^k(1 +|ξ|²)^s/2|u(ξ)|b dλ(ξ) 6

Z

R^m

|ξ|^2k

(1 +|ξ²|)^s dλ(ξ)

1/2 Z

R^m

(1 +|ξ²|)^s|bu(ξ)|²dλ(ξ) 1/2

,

and we infer kD^αuk_∞ 6Ckuks as soon as the first integral of the right hand side is convergent, which is the case if s > k+m/2.

It follows immediately from the Sobolev lemma that

(2.6) \

s>0

W_loc^s (M, E) =C^∞(M, E).

Since W^s(R^m) and W^−s(R^m)are dual, one can also infer by duality that

(2.6⁰) [

s60

W_loc^s (M, E) =D⁰(M, E).

A continuous linear operator ϕ : F → G of Fréchet spaces is called compact if there exists a neighborhood U of 0 in E such that ϕ(U) is compact in V. If F is a Banach space, this just means that for any bounded sequence (xν) in E, one can extract a subsequence (x_ν(k)) such that ϕ(xν(k))converges in G. It is easy to see that a compositionψ◦ϕof continuous operators ϕ, ψ is compact as soon as one of them is compact.

2.7. Rellich lemma. Let Ω bM be a relatively compact open subset. Then for all t > s, the restriction morphism

W_loc^t (M, E)→W_loc^s (Ω, E)

is compact. In particular, if M is compact, the inclusion W^t(M, E),→W^s(M, E) is compact.

Proof. By definition of the topology ofW_loc^s (Ω, E), it is sufficient to show the inclusion morphism W_K^t (R^m) ,→ W^s(R^m) is compact, where W_K^t (R^m) consists of elements u ∈ W^t(R^m) with support in a given compact subset K ⊂ R^m. Now, for any bounded sequence uν ∈ W_K^t (R^m), the Fourier transforms

buν(ξ) = Z

R^m

uν(x) e^−2πix·ξdλ(x)

are C^∞ functions (they even extend as entire holomorphic functions ubν ∈ O(C^m) by taking ξ ∈C^m), since the convergence of any derivativeD^αbuν(ξ)is guaranteed by the compactness of the support of u_ν. If θ ∈ D(R^m) is a real cut-off function equal to 1 on a neighborhood of K and f ∈D(R^m)⊂S(R^m), the formula uν(f) =ubν(f^∧−1) =buν(f(−η))b implies

D^αub_ν(ξ) = (−2πi)^|α|

Z

R^m

u_ν(x) θ(x)x^αe^−2πix·ξdλ(ξ)

= (−2πi)^|α|

Z

R^m

ub_ν(η) θ(x)x^α∧

(ξ−η)dλ(η).

The Cauchy-Schwarz and Peetre inequalities finally give

|D^αubν(ξ)|6(2π)^|α|kuνkt

Z

R^m

(1 +|η|²)^−t

θ(x)x^α∧

(ξ−η)

2

dλ(η) 1/2

6(2π)^|α|2^|t|/2(1 +|ξ|²)^|t|/2kθ(x)x^αk−tkuνkt.

As ku_νk_t is bounded and θ(x)x^α ∈D(R^m)⊂ W^−t(R^m), this shows that ub_ν is equicontinuous on every ball B(0, R) ⊂ R^m. By the Ascoli theorem, we can extract a subsequence ub_ν(k) that converges uniformly on every ball B(0, R). Now

ku_ν(`)−u_ν(k)k²_s = Z

{|ξ|6R}∪{|ξ|>R}

(1 +|ξ|²)^s|bu_ν(`)(ξ)−bu_ν(k)(ξ)|²dλ(ξ) 6(1 +R²)^max(s,0)

Z

|ξ|6R

|ub_{ν(`)</sub>(ξ)−ub<sub>ν(k)</sub>(ξ)|<sup>2</sup>dλ(ξ) + (1 +R<sup>2</sup>)<sup>s−t</sup>ku<sub>ν(`)}−u_ν(k)k²_t, and by taking R > 0 large, we see that (u_ν(k)) is a Cauchy sequence in the Hilbert space W^s(R^m). Therefore (u_ν(k)) is convergent in W^s(R^m), and we infer thatW_K^t (R^m),→ W^s(R^m) is compact.

3. Pseudodifferential operators

If u∈D(R^m), the Fourier transform ofu is

(3.1) bu(ξ) =

Z

R^m

u(x) e^−2πix·ξdλ(x)

and the Fourier inversion formula gives u(x) = R

R^mu(ξ) eb ^2πix·ξdλ(ξ), thus for any differential operator P =P

|α|6δa_α(x)D^α on R^m, we have (3.2) P u(x) = X

|α|6δ

aα(x)D^αu(x) = Z

R^m

X

|α|6δ

aα(x)(2πiξ)^αu(ξ) eb ^2πix·ξdλ(ξ).

We call

σ(x, ξ) = X

|α|6δ

aα(x)(2πiξ)^α = X

|α|6δ

(2πi)^|α|aα(x)ξ^α

the (total) symbol of P. By analogy with (3.2), and in more generality, a pseudodifferential operator is defined to be an operator Op_σ of the form

(3.3) Op_σ(u)(x) = Z

R^m

σ(x, ξ)u(ξ) eb ^2πix·ξdλ(ξ), u ∈D(R^m)⊂S(R^m),

whereσ belongs to a suitable class of functions on T_R^?m. The so-calledstandard classof symbols S^δ(R^m) is defined as follows : for δ ∈R, S^δ(R^m) is the space of C^∞ functions σ(x, ξ) on T^?

R^m

such that for any α, β ∈N^m and any compact subset K ⊂R^m one has an estimate (3.4) |D^α_xD_ξ^βσ(x, ξ)|6CK,α,β(1 +|ξ|)^δ−|β|, ∀(x, ξ)∈K×R^m,

where δ ∈ R is regarded as the “degree” of σ. Since bu ∈ S(R^m), the integral (3.3) is always convergent under condition (3.4). The case whenσ(x, ξ)is a polynomial of degreeδinξ (δ∈N) precisely corresponds to the case of a differential operator of degreeδ. In general, sinceubbelongs to the class S(R^m) of rapidly decreasing functions, the integral (3.3) is convergent as well as all its derivative D_x^α, thus Op_σ(u) is a well defined C^∞ function on R^m. In the more general situation of operators acting on a bundle E and having values in a bundleF over a manifoldM, we introduce an analogous space of symbols S^δ(M;E, F). The elements of S^δ(M;E, F) are functions

T_M^? 3(x, ξ)7→σ(x, ξ)∈Hom(Ex, Fx)

3. Pseudodifferential operators 7

satisfying condition (3.4) with respect to all coordinate systems and all local trivializations of E, F. An associated global operatorOp_σ onM can be defined by taking a locally finite covering (Ω_j)of M providing coordinates and trivializations of E, F, a quadratic partition of unity (ψ_j) subordinate to (Ωj) (i.e. P

ψ_j² = 1), and by putting

(3.5) Op_σ(u) =X

j

ψjOp_σj(ψju), u∈C^∞(M, E), where σ_j ∈ C^∞(T_M|Ω^∗

j,Hom(K^r,K^r

0) is the local expression of σ on Ω_j with respect to the trivializations of E, F. This definition of Op_σ allows to reduce most properties to be checked to the case of R^m with the trivial bundles E =F =K.

3.6. Remark. The global definition ofOp_σ provided by (3.5) does depend on the choice of the partition of unity (ψj), and also on the trivializations of E, F. This can be seen already in the case of an ordinary differential operator D^α. In fact, the Leibniz formula shows that

X

j

ψjD^α(ψju) =D^αu+ X

β6α, β6=α

α β

ψjD^α−βψjD^βu

differs from the top degree term D^αu by variable lower order terms. The important fact, however, is that the principal symbol is what one expects, and it is always possible to add lower order terms to Op_σ to correct any discrepancies. One can e.g. rely on the following elementary fact about asymptotic expansions of symbols.

3.7. Asymptotic expansions. Let σν ∈S^δ−ν(M;E, F), ν ∈ N, be any sequence of symbols.

Then there exists τ ∈S^δ(M;E, F) such that τ −Pk

ν=0σν ∈S^δ−k−1(M;E, F) for all k ∈N. Proof. By a partition of unity argument, it is sufficient to prove the result in the caseM =R^m, E = F = K. Let θ(ξ) > 0 be a C^∞ cutoff function on R^m, equal to 0 for |ξ| 6 1 and to 1 for |ξ|>2. We set

τ(x, ξ) =σ₀(x, ξ) +

+∞

X

ν=1

θ(ε_νξ)σ_ν(x, ξ)

where εν ∈ ]0,1] is a sequence that decays sufficiently fast to 0. In fact, by (3.4) there is an increasing sequence of constants C_N >0 such that

|D_x^αD^β_ξσν(x, ξ)|6CN(1 +|ξ|)^{δ−ν−|β|}

for all |α|,|β|, ν 6 N and x ∈R^m such that |x| 6N. Also, since θ vanishes on B(0,1) and its derivatives vanish onR^mrB(0,2), there is a constantC_N⁰ such that|D^γθ(ξ)|6C_N⁰ |ξ|(1+|ξ|)^−|γ|

for all ξ∈R^m and |γ|6N, thus

|D^γ(ξ7→θ(ε_νξ))|=ε^|γ|_ν |D^γθ(ε_νξ)|6 C_N⁰ ε^|γ|ν ε_ν|ξ|

(1 +εν|ξ|)^|γ| 6C_N⁰ ε^1+|γ|ν (1 +|ξ|)

(εν +εν|ξ|)^|γ| 6C_N⁰ ε_ν(1 +|ξ|)^1−|γ|. By the Leibniz formula, for any |α|,|β|6N and x∈R^N with |x|6N, we get

D^α_xD_ξ^{β +∞}

X

ν=1

θ(ε_νξ)σ_ν(x, ξ)

6

+∞

X

ν=1

X

γ6β

β γ

C_N⁰ ε_ν(1 +|ξ|)^1−|γ|C_max(N,ν)(1 +|ξ|)^{δ−ν−|β−γ|}

62^NC_N⁰

+∞

X

ν=1

ε_νC_max(N,ν)(1 +|ξ|)^δ−|β|,

and the convergence of the series is achieved by taking e.g. ε_ν = 2^−νC_ν⁻¹. This already implies τ ∈S^δ(R^m). Next,

τ(x, ξ)−

k

X

ν=0

σν(x, ξ) =θk+1(εk+1ξ)σk+1(x, ξ)+

k

X

ν=1

(1−θν(ενξ))σν(x, ξ)+

+∞

X

ν=k+2

θν(ενξ))σν(x, ξ).

Here, the first term coincides with σk+1(x, ξ) for |ξ|>2/εk+1, thus it belongs toS^δ−k−1(R^m).

The same is true for the summation Pk

ν=1 which has compact support inξ. Finally, the above estimates imply

D^α_xD_ξ^β

^+∞

X

ν=k+2

θ(ε_νξ)σ_ν(x, ξ)

62^NC_N⁰

+∞

X

ν=k+2

ε_νC_max(N,ν)(1 +|ξ|)^{δ−k−1−|β|},

hence the summation P+∞

ν=k+2 is also in S^δ−k−1(R^m).

The basic results pertaining to the theory of pseudodifferential operators are summarized below.

3.8. Schwartz kernel of a pseudodifferential operator. Condition (3.4) implies that ξ 7→ σ(x, ξ) as well as all its derivatives ξ 7→ D^α_xσ(x, ξ) are of polynomial growth at infinity.

This implies that they are in the space of tempered distributions S⁰(R^m), and can be assigned a partial Fourier transformy 7→bσ_ξ(x, y)with respect to the second argument, that is in S⁰(R^m) and depends smoothly on x. The Fourier transform of σ(x, ξ)e^2πix·ξ is Kσ(x, y) =bσξ(x, y−x), and for M =R^m, the usually adjunction formula bg(u) =g(u)b implies

(3.9) Op_σ(u) =

Z

y∈M

Kσ(x, y)u(y)dV(y), ∀u∈D(M), or equivalently, viewing Op_σ(u)as a distribution,

Z

x∈M

Op_σ(u)(x)f(x)dV(x) = Z

M×M

K_σ(x, y)f(x)u(y)dV(x)dV(y), ∀f, u∈D(M).

The distributionK_σ(x, y)∈D⁰(R^m×R^m)is called the (Schwartz) kernel ofOp_σ. More generally, in the case of a manifold, we get by (3.5) a formula similar to (3.9), with a global kernel K_σ(x, y) = P

jψ_j(x)K_σj(x, y)ψ_j(y) in D⁰(M ×M). Laurent Schwartz has shown (this is the so-called Schwartz kernel theorem) that every continuous linear operator T : D(M)→ D⁰(M) is actually given by such a kernel K ∈D⁰(M ×M), i.e.

T u(x) = Z

y∈M

K(x, y)u(y)dV(y)⇔ Z

M

T u(x)f(x)dV(x) = Z

M×M

K(x, y)f(x)u(y)dV(x)dV(y)

for all f, u∈ D(M). A proof can be obtained as a direct application of Grothendieck’s theory of topological tensor products (see Chapter 9 of online book).

3.10. Action of pseudodifferential operators on Sobolev spaces. For σ ∈S^δ(M;E, F), the operator Op_σ can be extended uniquely as a continuous linear operator

Op_σ :W_loc^s (M, E)→W_loc^s−δ(M, F).

3. Pseudodifferential operators 9

Proof. Thanks to the presence of a term ψ_j on both sides in (3.5), we are reduced to the case (3.3) where M = R^m and x 7→u(x), x 7→ σ(x, ξ) have support in a compact subset K ⊂ R^m. The result is easily seen to be true when Op_σ = P

|α|6δa_α(x)D^α is a differential operator.

In fact, since multiplication by aα is continuous on W^t(R^m) by Lemma 2.3, we only have to show that D^α : W^s(R^m) → W^s−δ(R^m) is continuous for |α| 6 δ, but this is obvious from the identity Dd^αu(ξ) = (2πiξ)^αu(ξ). In general, we just compute the Fourier transform ofb Op_σ(u).

By Fubini, for u∈D(R^m)⊂S(R^m), we find Op_σ(u)^∧(η) =

Z

R^m

Z

R^m

σ(x, ξ)u(ξ) eb ^2πix·ξdλ(ξ)

e^−2πix·ηdλ(x) = Z

R^m

σb_x(η−ξ, ξ)u(ξ)b dλ(ξ) where σbx is the partial Fourier transform of σ(x, ξ) with respect to x. Inequality (3.4) implies

|(2πiη)^αD_ξ^βbσx(η, ξ)|=

Z

R^m

D^α_xD_ξ^βσ(x, ξ) e^−2πix·ηdλ(x)

6λ(K)CK,α,β(1 +|ξ|)^δ−|β|, thus we have an estimate

(3.11) |D_ξ^βσbx(η, ξ)|6Cβ,N(1 +|η|²)^−N/2(1 +|ξ|²)^{(δ−|β|)/2} for every N ∈N. At this point, we need (3.11) only for β = 0. The Peetre inequality gives

(1 +|η|²)^(s−δ)/2

Op_σ(u)^∧(η) 62^|s−δ|/2

Z

R^m

(1 +|η−ξ|²)^|s−δ|/2(1 +|ξ|²)^(s−δ)/2

σbx(η−ξ, ξ) u(ξ)b

dλ(ξ) 6C_N,s,δ⁰

Z

R^m

(1 +|η−ξ|²)^{(|s−δ|−N)/2}(1 +|ξ|²)^s/2 u(ξ)b

dλ(ξ).

We choose N >|s−δ|+m, e.g. N =b |s−δ| c+m+ 1, so thatt :=N − |s−δ|> m. Then Z

R^m

(1 +|η−ξ|²)^−t/2dλ(ξ) = Z

R^m

(1 +|ξ|²)^−t/2dλ(ξ)<+∞,

and we get a probability measure dµ_η(ξ) = c(1 + |η −ξ|²)^−t/2dλ(ξ) for a certain constant c > 0. By Cauchy-Schwarz applied to the measure dµη(ξ) and to the functions f(ξ) = 1, g(ξ) = (1 +|ξ|²)^s/2|bu(ξ)|, we find

(1 +|η|²)^s−δ

Op_σ(u)^∧(η)

2 6C_s,δ⁰⁰ Z

R^m

c(1 +|η−ξ|²)^−t/2(1 +|ξ|²)^s bu(ξ)

2 dλ(ξ).

By the Fubini theorem, a final integration with respect to η implies kOp_σ(u)k²_s−δ 6C_s,δ⁰⁰ kuk²_s. Since D(R^m) is dense in W^s(R^m) (exercise !), the existence of a unique continuous extension Op_σ :W^s(R^m)→W^s−δ(R^m) follows.

3.12. Regularizing operators. In particular, if σ ∈ S^−∞(M;E, F) := T

δ<0S^δ(M;E, F), then Op_σ is a continuous operator sending an arbitrary distribution D⁰(M, E) into C^∞(M, F).

More precisely, since ξ7→σ(x, ξ)is rapidly decreasing locally uniformly inxwith all derivatives D_x^α the Schwartz kernelK_σ(x, y) =bσ_ξ(x, y−x)is then easily seen to be smooth, and conversely, if we are given a kernel K(x, y) in C^∞(M ×M) with “proper support” in M ×M in the sense

that for every point x₀ ∈ X, there is a neighborhood W of x₀ and a compact set L ⊂M such that K(x, y) = 0 on W ×(M rL), the formula

(3.13) Ru(x) =

Z

M

K(x, y)u(y)dV(y), u∈D⁰(M)

defines a continuous operator R:D⁰(M)→C^∞(M). In the case M =R^m, we have Ru(x) =

Z

M

Kby(x,−ξ)u(ξ)b dV(ξ), ∀u ∈D⁰(M),

thusRis the pseudodifferential operator associated with the symbolσ(x, ξ) =Kby(x,−ξ) e^−2πix·ξ. The properness condition for K implies thatKb_y(x, ξ)is rapidly decreasing inξ, thusσ is in the class R:=S^−∞(M;E, F). Such an operator is called a regularizing operator.

Regularizing operators play very nicely in regularity theory, in the sense that adding such operators does not interfere with the regularity of distributions at stake. They can somehow be considered as “negligible” in pseudodifferential calculus ; it is thus very frequent to make calculations only modulo R.

3.14. Composition of pseudodifferential operators associated to standard symbols.

For any σ ∈ S^δ(M;E, F) and σ⁰ ∈ S^δ0(M;F, G), δ, δ⁰ ∈ R, there exists a “composed symbol”

σ⁰σ ∈S^δ+δ0(M;E, G) such that Op_σ0◦Op_σ = Op_σ0σ. Moreover σ⁰σ−σ⁰·σ ∈S^δ+δ0−1(M;E, G).

More precisely, we have an asymptotic expansion σ⁰σ(x, ξ)∼X

`

X

|α|=`

1

(2πi)^|α|α!D_ξ^ασ⁰(x, ξ)D_x^ασ(x, ξ) modR where the `-th term is in S<sup>δ+δ0−`</sup>(M;E, G).

Proof. When Op_σ0 and Op_σ are differential operators (i.e. σ⁰(x, ξ) and σ(x, ξ) are polynomials in ξ), the asymptotic expansion is an exact formula, and is just a finite sum ; this follows easily from the Leibniz formula. The general manifold case requires to compute locally finite sums of the form

X

j,k

ψ_kOp_σ⁰

k ψ_kψ_jOp_σ

j(ψ_ju)

whereu ∈W_loc^s (M). The(j, k)-term is non zero only in caseSupp(ψj)∩Supp(ψk)6=∅. One can arrange the covering (Ω_j) and the partition of unity in such a way that whenever this happens Supp(ψj)∪Supp(ψk) is contained in Ωj or Ωk (e.g. by taking geodesic ballsΩj ' B(0, rj), so that B(0, r_j/4) still covers M, and Supp(ψ_j) ⊂ B(0, r_j/4)). Then, by using a diffeomorphism Ωj 'R^m, it is sufficient to consider the case where M = R^m, and to consider the composition of pseudodifferential operators associated with the symbols ψ_k(x)σ⁰_k(x, ξ) and ψ_kψ_j(x)σ_j(x, ξ).

Hence, we can also assume that σ⁰(x, ξ) andσ(x, ξ)have compact support in x, being non zero only when x ∈ K ⊂ R^m. In that case, for u ∈ D(Rⁿ) we have absolutely convergent integrals of rapidly decreasing functions

Op_σ⁰(u)(x) = Z

R^m

σ⁰(x, η)u(η) eb ^2πix·ηdλ(η), Op_σ(u)^∧(η) =

Z

R^m

bσx(η−ξ, ξ)bu(ξ)dλ(ξ),

3. Pseudodifferential operators 11

the first one being equivalent to definition (3.3) withξreplaced byη, and the second one already stated in 3.10. By substituting the second formula in the first, we find

Op_σ⁰◦Op_σ(u)(x) = Z

R^m

σ⁰(x, η) Z

R^m

σbx(η−ξ, ξ)u(ξ)b dλ(ξ)

e^2πix·ηdλ(η)

= Z

R^m

Z

R^m

σ⁰(x, η)σbx(η−ξ, ξ) e^2πix·ηdλ(η)

bu(ξ)dλ(ξ)

= Z

R^m

Z

R^m

σ⁰(x, η+ξ)bσx(η, ξ) e^2πix·ηdλ(η)

bu(ξ) e^2πix·ξdλ(ξ), where, in the last line, we have performed a change of variable η 7→ η +ξ. This shows that Op_σ◦Op_σ is associated to the symbol

(3.15) σ⁰σ(x, ξ) =

Z

R^m

σ⁰(x, η+ξ)σb_x(η, ξ) e^2πix·ηdλ(η), and we have to prove that σ⁰σ ∈S^δ+δ0(M;E, G). For this, we compute

D^α_xD_ξ^βσ⁰σ(x, ξ) =X

µ,ν

α µ

β ν

Z

R^m

D^µ_xD^ν_ξσ⁰(x, η+ξ)D_ξ^β−νbσ_x(η, ξ) (2πiη)^α−µe^2πix·ηdλ(η).

As we have already shown (cf. (3.11)), we have estimates

|D^β−ν_ξ bσx(η, ξ)|6C_β−ν,N(1 +|η|²)^−N/2(1 +|ξ|²)(δ−|β|+|ν|)/2, and also

|D^µ_xD^ν_ξσ⁰(x, η+ξ)|6C_µ,ν⁰ (1 +|η+ξ|²)^{(δ0−|ν|)/2}

6C_µ,ν⁰ 2^{|δ0−|ν||/2}(1 +|ξ|²)^{(δ0−|ν|)/2}(1 +|η|²)^{|δ0−|ν||/2} by the Peetre inequality. Therefore

D^α_xD_ξ^βσ⁰σ(x, ξ)

6C_α,β,N⁰⁰ Z

R^m

(1 +|ξ|²)^{(δ+δ0−|β|)/2}(1 +|η|²)^(|δ0−|ν||+|α|−|µ|−N)/2dλ(η).

For N large enough to ensure the convergence of the integral, we get the expected upper bound D^α_xD_ξ^βσ⁰σ(x, ξ)

6C_α,β⁰⁰⁰ (1 +|ξ|)^{δ+δ0−|β|}. Furthermore, we have a Taylor expansion

σ⁰(x, η+ξ) = X

|α|6p

1

α!D^α_ξσ⁰(x, ξ)η^α+ Z 1

0

X

|α|=p+1

1

α!D^α_ξσ⁰(x, ξ+tη)η^α(p+ 1)(1−t)^pdt The integral term is bounded by

C_p⁽⁴⁾(1 +|ξ+tη|²)^{(δ0−p−1)/2} 6C_p⁽⁵⁾(1 +|ξ|²)^{(δ0−p−1)/2}(1 +|η|²)^{|δ0−p−1|/2}. If we plug the Taylor expansion in (3.15), this remainder term contributes for

C_p,N⁽⁶⁾ Z

R^m

(1 +|ξ|²)^{(δ+δ0−p−1)/2}(1 +|η|²)^{(|δ0−p−1|−N)/2}dλ(η)6C_p⁽⁷⁾(1 +|ξ|)^{δ+δ0−p−1},

while the main terms yield Z

R^m

X

|α|6p

1

α!D^α_ξσ⁰(x, ξ)η^αbσ_x(η, ξ) e^2πix·ηdλ(η) = X

|α|6p

1

(2πi)^|α|α!D_ξ^ασ⁰(x, ξ)D_x^ασ(x, ξ) by the Fourier inversion formula. It is easy to check that the terms with |α| = ` are in S<sup>δ+δ0−`</sup>(M;E, F). Statement 3.14 is proved.

3.16. Adjoint of a pseudodifferential operator. Assume that E, F are equipped with smooth Hermitian metrics hE, hF. Given σ ∈ S^δ(M;E, F), there exists an adjoint symbol σ^† ∈ S^δ(M;F, E) such that Op_σ et Op_σ† are (formally) adjoint, i.e. for every u ∈ D(M, E), v ∈D(M, F)

Z

M

hOp_σ(u), viF dV = Z

M

hu,Op_σ^†(v)iEdV.

Proof. We first compute σ^† in the trivial case M = R^m, E = F = K with the trivial metric, when x7→σ(x, ξ) has compact support. Then

Z

M

hOp_σ(u), vidV = Z

R^m

Op_σ(u)(x)v(x)dλ(x) = Z

R^m×R^m

σ(x, ξ)bu(ξ) e^2πix·ξv(x)dλ(x)dλ(ξ).

A partial Fourier transform with respect to x gives Z

M

hOp_σ(u), vidV = Z

R^m×R^m

σb_x(η−ξ, ξ)u(ξ)b bv(η)dλ(ξ)dλ(η).

Similarly, hhu,Op_σ^†(v)ii=hhOp_σ^†(v), uii, and by switching u and v, ξ and η, σ and σ^†, we get Z

M

hOp_σ^†(v), uidV = Z

R^m×R^m

σc^†x(ξ−η, η)bv(η)u(ξ)b dλ(ξ)dλ(η).

The required relation between σ and σ^† is σcx^†(ξ −η, η) = σbx(η−ξ, ξ), or equivalently, after a one-to-one substitution (η, ξ)7→(ξ, ξ+η),

cσx^†(η, ξ) =σbx(−η, ξ+η).

In the higher rank case E =K^r, F =K^r

0 we find

(3.17) σcx^†(η, ξ) =σbx(−η, ξ+η)^∗

where A^∗ is the complex adjoint of a matrix A. We leave the reader check that this defines σ^† ∈ S^δ(M;F, E), again by means of the Peetre inequality and by the fast decay of bσx(η, ξ) in η. The general case follows from the fact that (ψ_jOp_σ

jψ_j)^∗ = ψ_j(Op_σ

j)^∗ψ_j = ψ_jOp_σ^†

j

ψ_j, when orthormal trivializations of E, F are taken in each chart.

5. Finiteness theorem 13

4. Fundamental results on elliptic operators

The concept of ellipticity is easily extended to pseudodifferential operators by expressing that the symbol σ(x, ξ) is “uniformly” injective when ξ → ∞.

4.1. Definition. A pseudodifferential operator Op_σ of degree δ is called elliptic if it can be defined by a symbol σ ∈S^δ(M, E, F) such that

|σ(x, ξ)·u|>c|ξ|^δ|u|, ∀(x, ξ)∈T_M^? , ∀u∈Ex

for |ξ| large enough, the estimation being uniform for x∈M.

If E and F have the same rank, the ellipticity condition implies that σ(x, ξ) is invertible for ξ large. By taking a suitable C^∞ truncating function θ(ξ) >0 equal to 0 for |ξ| 6R large and to 1 for |ξ| > 2R, one sees that the function σ⁰(x, ξ) = θ(ξ)σ(x, ξ)⁻¹ defines a symbol in the space S^−δ(M;F, E). Also, since

1−σ⁰(x, ξ)σ(x, ξ) = 1−θ(ξ)∈D(R^m)⊂S(R^m),

this difference is a regularizing operator. According to 3.14, we have Id−Op_σ0◦Op_σ = Op_ρ, where ρ ∈ S⁻¹(M;E, E), and thus ρ^j ∈ S^−j(M;E, E). Choose a symbol τ equivalent to the asymptotic expansion Id +ρ+ρ² +· · ·+ρ^j +· · · (this is possible by 3.7). Then Op_τ is an inverse of Op_σ⁰◦Op_σ = Id−Op_ρ modulo R. It is then clear that one obtains an inverseOp_τσ⁰ of Op_σ modulo R. An easy consequence of this observation is the following :

4.2. Gårding inequality. Assume given P :C^∞(M, E) → C^∞(M, F) an elliptic differential operator of degree δ, where rankE = rankF = r, and let Pe be an extension of P with distributional coefficient sections. For all u ∈ W⁰(M, E) such that P ue ∈ W^s(M, F), one then has u ∈W^s+δ(M, E) and

kuk_s+δ 6C_s(kP uke _s+kuk₀), where C_s is a positive constant depending only on s.

Proof. Since P is elliptic of degree δ, there exists a symbol σ ∈ S^−δ(M;F, E) such that Op_σ◦Pe = Id +R, R ∈ R. Then kOp_σ(v)ks+δ 6 Ckvks by applying 3.10. Consequently, in setting v =P u, we see thate u= Op_σ(P u)e −Ru satisfies the desired estimate.

5. Finiteness theorem

We conclude these notes with the proof of the following fundamental finiteness theorem, which is the starting point of L² Hodge theory. We assume throughout this section thatM is compact, in general finiteness cannot hold on non compact manifolds.

5.1. Finiteness theorem. Assume given E, F Hermitian vector bundles on a compact manifold M, such that rankE = rankF = r, and let P : C^∞(M, E) → C^∞(M, F) be an elliptic differential operator of degree δ. Then :

i) KerP is finite dimensional.

ii) P C^∞(M, E)

is closed and of finite codimension in C^∞(M, F) ; moreover, if P^? is the formal adjoint of P, there exists a decomposition

C^∞(M, F) =P C^∞(M, E)

⊕KerP^? as an orthogonal direct sum in W⁰(M, F) =L²(M, F).

Proof. (i) The Gårding inequality shows thatkuk_s+δ6C_skuk₀ for allu∈KerP. By the Sobolev lemma, this implies that KerP is closed in W⁰(M, E). Moreover, the k k0-closed unit ball of KerP is contained in the k k_δ-boule of radius C₀, therefore it is compact according to the Rellich lemma. The Riesz Theorem implies dim KerP <+∞.

(ii) We first show that the extension

Pe:W^s+δ(M, E)→W^s(M, F)

has a closed image all s. For any ε >0, there exists a finite number of elements v1, . . . , vN ∈ W^s+δ(M, F), N =N(ε), such that

(5.2) kuk0 6εkuks+δ+

N

X

j=1

|hhu, vjii0|.

Indeed, the set

K_(vj)=n

u∈W^s+δ(M, F) ; εkuk_s+δ+

N

X

j=1

|hhu, v_jii₀|61o is relatively compact in W⁰(M, F) and T

(v_j)K_(vj) = {0}. It follows that there are elements (vj) such that the K_(vj) are contained in the unit ball of W⁰(M, E), as required. Substituting the kuk₀ by its upper bound (5.2) in the Gårding inquality, we obtain

(1−C_sε)kuk_s+δ 6C_s

kP uke _s+

N

X

j=1

|hhu, v_jii₀| .

Define T =

u ∈W^s+δ(M, E) ; u⊥vj, 16j 6n} and put ε= 1/2Cs. It follows that kuks+δ 62CskP uke s, ∀u∈T.

This implies that Pe(T) is closed. As a consequence P We ^s+δ(M, E)

=Pe(T) + Span Pe(v₁), . . . ,Pe(v_N)

is closed in W^s(M, E). Consider now the case s = 0. Since C^∞(M, E) is dense in W^δ(M, E), we see that in W⁰(M, E) =L²(M, E), one has

P We ^δ(M, E)⊥

=

P C^∞(M, E)⊥

= KerPf^?. We have thus proven that

(5.3) W⁰(M, E) =P We ^δ(M, E)

⊕KerPf^?.

SinceP^? is also elliptic, it follows that KerPf^? is finite dimensional and thatKerPf^? = KerP^? is contained in C^∞(M, F). By applying the Gårding inequality, the decomposition formula (5.3) gives

W^s(M, E) =P We ^s+δ(M, E)

⊕KerP^?, (5.4)

C^∞(M, E) =P C^∞(M, E)

⊕KerP^?. (5.5)

5. Finiteness theorem 15

We finish this section by the construction of the Green operator associated with a self- adjoint elliptic operator.

5.6. Theorem. Assume given E a Hermitian vector bundle of rank r over a compact mani- fold M, and P :C^∞(M, E)→C^∞(M, E) a self-adjoint elliptic differential operator of degreeδ.

Then if H denotes the orthogonal projection H : C^∞(M, E) → KerP, there exists a unique operator G on C^∞(M, E) such that

P G+H =GP +H = Id.

Moreover G is a pseudodifferential operator of degree−δ, called Green operator associated to P. Proof. According to Theorem 5.1, KerP = KerP^? is finite dimensional, and ImP = (KerP)^⊥. It then follows that the restriction of P to (KerP)^⊥ is a bijective operator. One defines G toe be 0⊕P⁻¹ relative to the orthogonal decompositionC^∞(M, E) = KerP ⊕(KerP)^⊥. The relations P G+H =GP +H = Id are then obvious, as well as the uniqueness of G. Moreover, G is continuous in the Fréchet space topology ofC^∞(M, E), according to the Banach theorem.

One also uses the fact that there exists a pseudodifferential operator Qof order −δ which is an inverse of P modulo R, i.e., P Q= Id +R, R∈R. It then follows that

Q= (GP +H)Q=G(Id +R) +HQ=G+GR+HQ,

whereGRandHQare regularizing operators (H is a regularizing operator of finite rank defined by the kernel P

ϕs(x)⊗ϕ^?_s(y), where (ϕs) is a base of eigenfunctions of KerP ⊂C^∞(M, E)).

Consequently G=Q mod R and G is a pseudodifferential operator of degree −δ.

5.7. Corollary.Under the hypotheses of Theorem5.6, the eigenvalues ofP form a real sequence (λk) such that lim_k→+∞|λk| = +∞, the eigenspaces Vλ_k of P are finite dimensional, and one has a Hilbert space (completed) orthogonal direct sum

L²(M, E) =Md

kVλk. For any integer m ∈ N, an element u = P

kuk ∈ L²(M, E) is in W^mδ(X, E) if and only if P|λ_k|^2mku_kk² <+∞.

Proof. The Green operator extends to a self-adjoint operator

Ge:L²(M, E)→L²(M, E)

which factors through W^δ(M, E), and is therefore compact. This operator defines an inverse to Pe:W^δ(M, E)→L²(M, E)on(KerP)^⊥. The spectral theory of compact self-adjoint operators shows that the eigenvaluesµk ofGe form a real sequenceµk tending to0and thatL²(M, E)is a Hilbert orthogonal direct sum of eigenspaces. The corresponding eigenvalues of Pe areλk =µ⁻¹_k if µ_k6= 0, and according to the ellipticity ofP−λ_kId, the eigenspacesV_λk = Ker(P−λ_kId)are finite dimensional and contained in C^∞(M, E). Finally, if u=P

kuk ∈L²(M, E), the Gårding inequality shows that u ∈ W^mδ(M, E) if and only if Pe^mu ∈ L²(M, E) = W⁰(M, E), which easily gives the condition P

|λk|^2mkukk² <+∞.