Differential Operators

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

2. Differential Operators

Let ǫ >0 be given. Suppose that t < s. Then there is a ballBǫ such that 1

(1+ |ξ|²)^s−t < ǫ for ξ outside the ball B_ǫ. Then consider

fv−fµ²t =

$

Rⁿ

|(fˆ_v− ˆf_µ)(ξ )|²

(1+ |ξ|²)^s−t (1+ |ξ|²)^sdξ

≤

$

B_E

|(fˆ_v− ˆf_µ)(ξ )|²(1+ |ξ|²)^tdξ +ǫ

$

Rⁿ−BE

|(fˆ_v− ˆf_µ)(ξ )|²(1+ |ξ|²)^sdξ

≤

$

BE

|(fˆv− ˆfµ)(ξ )|²(1+ |ξ|²)^tdξ+2ǫ,

where we have used the fact thatf_vs≤1. Since we know thatfˆ_vconverges on compact sets, we can choose v, µlarge enough so that the first integral is < ǫ, and thus fv is a Cauchy sequence in the t norm.

Q.E.D.

We now need to discuss briefly the concept of a formal adjoint operator in this setting.

Definition 1.5: Let

L: E(X, E)−→E(X, F ) be a C-linear map. Then a C-linear map

S: E(X, F )−→E(X, E) is called an adjoint of L if

(1.4) (Lf, g)=(f, Sg)

for all f ∈E(X, E), g∈E(X, F ).

It is an easy exercise, using the density of E(X, E) in W⁰(X, E), to see that an adjoint of an operator L is unique, if it exists. We denote this transpose by L^∗. In later sections we shall discuss adjoints of various types of operators. This definition extends to Hilbert spaces over noncompact manifolds (e.g., Rⁿ) by using (1.4) as the defining relation for sections with compact support. This is then the formal adjoint in that context.

be a C-linear map. We say thatL is adifferential operatorif for any choice of local coordinates and local trivializations there exists a linear partial differential operator L˜ such that the diagram (for such a trivialization)

[E(U )]^{p L˜} // [E(U )]^q

≀ ≀

E(U, U×C^p) // E(U, U×C^q)

∪ ∪

E(X, E)|U

L // E(X, F )|U

commutes. That is, for f =(f₁, . . . , f_p)∈ [E(U )]^p

˜ L(f )_i =

p j=1

|α|≤k

a^ij_αD^αf_j, i=1, . . . , q.

A differential operator is said to be oforder k if there are no derivatives of order≥k+1 appearing in a local representation. (For an intrinsic definition involving jet-bundles, see Palais [1], Chap. IV.) Let Diffk(E, F ) denote the vector space of all differential operators of order k mapping E(X, E) to E(X, F ).

SupposeX is a compact differentiable manifold. We define OPk(E, F )as the vector space of C-linear mappings

T: E(X, E)−→E(X, F ) such that there is a continuous extension of T

¯

T_s: W^s(X, E)−→W^s−k(X, F )

for all s. These are the operators of order k mapping E to F.

Proposition 2.1: Let L∈ OP_k(E, F ). Then L^∗ exists, and moreover L^∗∈ OPk(F, E), and the extension

(L^∗)_s: W^s(X, F )−→W^s−k(X, E) is given by the adjoint map

(L_k−s)^∗:W^s(X, F )−→W^s−k(X, E).

This proposition is easy to prove since one has a candidate (L_k−s)^∗ (for each s) which gives the desired adjoint when restricted to E(X, F ) in a suitable manner. One uses the uniqueness of adjoints and Proposition 1.1.

Proposition 2.2: Diffk(E, F )⊂OPk(E, F ).

The proof of this proposition is not hard. Locally it involves, again, D/^αf (ξ )=ξ^αf (ξ ), and the definition of theˆ s-norm.

We now want to define the symbolof a differential operator. The symbol will be used for the classification of differential operators into various types.

First we have to define the set of all admissible symbols. Let T^∗(X) be the real cotangent bundle to a differentiable manifoldX, letT^′(X)denoteT^∗(X)

with the zero cross section deleted (the bundle of nonzero cotangent vectors), and let T^′(X)−→^π X denote the projection mapping. Then π^∗E and π^∗F denote the pullbacks of E and F over T^′(X). We set, for any k∈Z,

Smblk(E, F ):= {σ ∈ Hom(π^∗E, π^∗F ): σ (x, ρv)

=ρ^kσ (x, v), (x, v)∈T^′(X), ρ >0}. We now define a linear map

(2.1) σ_k: Diffk(E, F )−→Smblk(E, F ),

where σ_k(L)is called thek-symbol of the differential operatorL. To define σ_k(L), we first note thatσ_k(L)(x, v)is to be a linear mapping fromE_x toF_x, where(x, v)∈T^′(X). Therefore let(x, v)∈T^′(X)ande∈E_xbe given. Find g∈E(X)andf ∈E(X, E)such thatdg_x=v, andf (x)=e. Then we define†

σ_k(L)(x, v)e=L i^k

k!(g−g(x))^kf

(x)∈F_x. This defines a linear mapping

σ_k(L)(x, v): E_x−→F_x,

which then defines an element of Smbl_k(E, F ), as is easily checked. It is also easy to see that the σk(L), so defined, is independent of the choices made. We call σk(L) thek-symbol of L.

Proposition 2.3: The symbol map σ_k gives rise to an exact sequence (2.2) 0−→Diff_k−1(E, F )−→^j Diff_k(E, F )−→^σk Symbl_k(E, F ), where j is the natural inclusion.

Proof: One must show that the k-symbol of a differential operator of order k has a certain form in local coordinates. Let L be a linear partial differential operator

L: [E(U )]^p−→ [E(U )]^q where U is open in Rⁿ. Then it is easy to see that if

L=

|ν|≤k

A_νD^ν,

where {A_ν}are q×p matrices of C^∞ functions onU, then

(2.3) σk(L)(x, v)=

|ν|=k

Aν(x)ξ^ν,

where v =ξ₁dx₁+ · · · +ξ_ndx_n. For each fixed (x, v), σ_k(L)(x, v) is a linear mapping from x×C^p → x×C^q, given by the usual multiplication of a vector in C^p by the matrix

|ν|=k

Aν(x)ξ^ν.

†We include the factor i^k so that the symbol of a differential operator is compatible with the symbol of a pseudodifferential operator defined in Sec. 3 by means of the Fourier transform.

What one observes is that if σk(L) =0, then the differential operator L has kth order terms equal to zero, and thus L is a differential operator of order k−1. Let us show that (2.3) is true. Choose g∈E(U ) such that v=dg=

ξ_jdx_j; i.e., D_jg(x)=ξ_j. Let e∈C^p. Then we have σk(L)(x, v)e=

|ν|≤k

AνD^ν i^k

k!(g−g(x))^ke

(x).

Clearly, the evaluation at x of derivatives of order ≤k−1 will give zero, since there will be a factor of[g−g(x)]|x=0 remaining. The only nonzero term is the one of the form (recalling that D^ν=(−i)^νD₁^ν1· · ·D^ν_nⁿ)

|ν|=k

A_ν(x)k!

k!(D1g(x))^ν1· · ·(D_ng(x))^νn

=

|ν|=k

A_ν(x)ξ₁^ν1· · ·ξ_n^νn =

|ν|=k

A_ν(x)ξ^ν,

which gives us (2.3). The mapping σ_k in (2.2) is well defined, and to see that the kernel is contained in Diffk−1(E, F ), it suffices to see that this is true for a local representation of the operator. This then follows from the local representation for the symbol given by (2.3).

Q.E.D.

We observe that the following property is true: If L1∈Diff_k(E, F ) and L2∈Diff_m(F, G), then L2L1=L2◦L1∈Diff_k+m(E, G), and, moreover, (2.4) σ_k+m(L2L1)=σ_m(L2)•σ_k(L1),

where the right-hand product is the product of the linear mappings involved.

The relation (2.4) is easily checked for local differential operators on trivial bundles (the chain rule for composition) and the general case is reduced to this one in a straightforward manner.

We now look at some examples.

Example 2.4: IfL: [E(Rⁿ)]^p→ [E(Rⁿ)]^q is an element of Diffk(Rⁿ×C^p, Rⁿ×C^q), then

σ_k(L)(x, v)=

|ν|=k

A_v(x)ξ^ν, where

L=

|ν|≤k

AνD^ν, v= n j=1

ξjdxj,

the{A_ν}beingq×pmatrices of differentiable functions inRⁿ (cf. the proof of Proposition 2.3).

Example 2.5: Consider the de Rham complex E⁰(X) −→^d E¹(X) −→ · · ·^d −→^d Eⁿ(X),

given by exterior differentiation of differential forms. Written somewhat differently, we have, for T^∗=T^∗(X)⊗C,

E(X,∧⁰T^∗)−→^d E(X,∧¹T^∗)−→ · · ·^d ,

and we want to compute the associated 1-symbol mappings, (2.5) ∧⁰T_x^∗

σ₁(d)(x,υ)

// ∧¹T^∗_x

σ₁(d)(x,υ)

// ∧²T^∗_x // · · ·. We claim that for e∈ ∧^pT_x^∗, we have

σ₁(d)(x, υ)e=iυ∧e.

Moreover, the sequence of linear mappings in (2.5) is an exact sequence of vector spaces. These are easy computations and will be omitted.

Example 2.6: Consider the Dolbeault complex on a complex manifoldX, E^p,0(X)−→^∂¯ E^p,1(X)−→ · · ·^∂¯ −→^∂¯ E^p,n(X)−→0.

Then this has an associated symbol sequence // ∧^p,q−1T^∗_x(X)

σ₁(¯∂)(x,υ)

// ∧^p,qT^∗_x(X)

σ₁(¯∂)(x,υ)

// ∧^p,q+1T^∗_x(X) //, where the vector bundles ∧^p,qT^∗(X) are defined in Chap. I, Sec. 3. We have that ν∈T_x^∗(X), considered as a real cotangent bundle. Consequently, ν =ν^1,0+υ^0,1, given by the injection

0−→T_x^∗(X)−→T_x^∗(X)⊗RC=T^∗(X)^1,0⊕T^∗(X)^0,1

= ∧^1,0T^∗(X)⊕ ∧^0,1T^∗(X).

Then we claim that

σ1(∂)(x, υ)e¯ =iυ^0,1∧e,

and the above symbol sequence is exact. Once again we omit the simple computations.

Example 2.7: Let E −→ X be a holomorphic vector bundle over a complex manifold X. Then consider the differentiable (p, q)-forms with coefficients in E,E^p,q(X, E), defined in (II.3.9), and we have the complex (II.3.10)

−→E^p,q(X, E)−→^∂¯E E^p,q+1(X, E)−→, which gives rise to the symbol sequence

// ∧^p,qT^∗_x ⊗E_x

σ₁(¯∂_E)(x,υ)

// ∧^p,q+1T^∗_x ⊗E_x //.

We let υ=υ^1,0+υ^0,1, as before, and we have for f ⊗e∈ ∧^p,qT_x^∗⊗E σ₁(¯∂)(x, υ)f ⊗e=(iυ^0,1∧f )⊗e,

and the symbol sequence is again exact.

We shall introduce the concept ofelliptic complexin Sec. 5, which gener- alizes these four examples.

The last basic property of differential operators which we shall need is the existence of a formal adjoint.

Proposition 2.8: LetL∈Diffk(E, F ). ThenL^∗exists andL^∗∈Diffk(F, E).

Moreover, σk(L^∗)=σk(L)^∗, where σk(L)^∗ is the adjoint of the linear map σ_k(L)(x, υ): E_x−→F_x.

Proof: Let L ∈ Diff_k(E, F ), and suppose that µ is a strictly positive smooth measure on X and that h_E and h_E are Hermitian metrics on E and F. Then the inner product for any ξ,η∈D(X, E) is given by

(ξ,η)=

$

X

ξ,ηEdµ,

and if ξ,η have compact support in a neighborhood where E admits a local frame f, we have

(ξ,η)=

$

Rⁿ

tη(x)h¯ _E(x)ξ(x)ρ(x) dx, where ρ(x)is a density,

η(x)=η(f )(x)=

⎡

⎢⎢

⎢⎢

⎣

η¹(f )(x)

·

·

· η^r(f )(x)

⎤

⎥⎥

⎥⎥

⎦ ,

etc. Similarly, for σ, τ ∈D(X, F ), we have (σ, τ )=

$

Rⁿ

tτ (x)h¯ F(x)σ (x)ρ(x) dx.

Suppose thatL:D(X, E)→D(X, F )is a linear differential operator of order k, and assume that the sections have support in a trivializing neighborhood U which gives local coordinates forX near some point. Then we may write

(Lξ, τ )=

$

Rⁿ

tτ (x)h¯ _F(x)(M(x, D)ξ(x))ρ(x) dx, where

M(x, D)=

|α|≤k

C_α(x)D^α

is an s×r matrix of partial differential operators; i.e., C_α(x) is an s×r matrix of C^∞ functions in Rⁿ. Note that ξ and τ have compact support here. We can then write

(Lξ, τ )=

$

Rⁿ

|α|≤k

tτ (x)ρ(x)h¯ _F(x)C_α(x)D^αξ(x) dx, and we can integrate by parts, obtaining

(Lξ, τ )=

$

Rⁿ

|α|≤k

(−1)^|α|D^α(^tτ (x)ρ(x)h¯ _F(x)C_α(x))ξ(x) dx

=

$

Rⁿ t(

|α|≤k

˜

C_α(x)D^ατ (x))h_E(x)ξ(x)ρ(x) dx,

where C˜_α(x)arer×s matrices of smooth functions defined by the formula

(2.6) ^t(

|α|≤k

˜

C_αD^ατ )=

|α|≤k

(−1)^|α|D^α(^tτ ρh¯ _FC_α)h⁻_E¹ρ⁻¹,

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.