Sobolev Spaces

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

1. Sobolev Spaces

E L L I P T I C

O P E R AT O R T H E O RY

In this chapter we shall describe the general theory of elliptic differential operators on compact differentiable manifolds, leading up to a presentation of a general Hodge theory. In Sec. 1 we shall develop the relevant theory of the function spaces on which we shall do analysis, namely the Sobolev spaces of sections of vector bundles, with proofs of the fundamental Sobolev and Rellich lemmas. In Sec. 2 we shall discuss the basic structure of differential operators and their symbols, and in Sec. 3 this same structure is generalized to the context of pseudodifferential operators. Using the results in the first three sections, we shall present in Sec. 4 the fundamental theorems concerning homogeneous solutions of elliptic differential equations on a manifold. The pseudodifferential operators in Sec. 3 are used to construct a parametrix (pseudoinverse) for a given operator L. Using the parametrix we shall show that the kernel (null space) of L is finite dimensional and contains onlyC^∞ sections (regularity). In the case of self-adjoint operators, we shall obtain the decomposition theorem of Hodge, which asserts that the vector space of sections of a bundle is the (orthogonal) direct sum of the (finite dimensional) kernel and the range of the operator. In Sec. 5 we shall introduce elliptic complexes (a generalization of the basic model, the de Rham complex) and show that the Hodge decomposition in Sec. 4 carries over to this context, thus obtaining as a corollary Hodge’s representation of de Rham cohomology by harmonic forms.

where the coefficients transform by ρ(x) dx = ˜ρ(y(x))

det∂y(x)

∂x dx,

where ρ(y) dy˜ is the representation with respect to the coordinates y = (y₁, . . . , y_n), where x → y(x) and ∂y/∂x is the corresponding Jacobian matrix of the change of coordinates. Such measures always exist; take, for instance,

ρ(x)= |detgij(x)|^1/2, where ds² =

g_ij(x)dx_i,

dx_j is a Riemannian metric for X expressed in terms of the local coordinates (x₁, . . . , x_n).† If X is orientable, then the volume element dµ can be chosen to be a positive differential form of degree n (which can be taken as a definition of orientability).

Let Ebe a Hermitian (differentiable) vector bundle overX. LetE_k(X, E) be the kth order differentiable sections of E over X,0 ≤ k ≤ ∞, where E_∞(X, E) =E(X, E). As usual, we shall denote the compactly supported sections‡ byD(X, E)⊂E(X, E)and the compactly supported functions by D(X)⊂E(X). Define an inner product ( , ) on E(X, E) by setting

(ξ,η)=

$

X

ξ(x),η(x)E dµ, where ,^E is the Hermitian metric onE. Let

ξ0=(ξ, ξ )^1/2

be theL²-norm and letW⁰(X, E)be the completion ofE(X, E). Let{U_α, ϕ_α} be a finite trivializing cover, where, in the diagram

E|Uα ϕα

//

˜ U_α×C^m

U_α ^ϕ¯α // U˜_α,

ϕαis a bundle map isomorphism andϕ¯α:Uα→ ˜Uα⊂Rⁿare local coordinate systems for the manifold X. Then let

ϕ_α^∗: E(U_α, E)−→ [E(U˜_α)]^m

be the induced map. Let {ρ_α}be a partition of unity subordinate to {U_α}, and define, for ξ ∈E(X, E),

ξs,E =

α

ϕ_α^∗ρ_αξs,Rⁿ,

where s,Rⁿ is the Sobolev norm for a compactly supported differentiable function

f: Rⁿ−→C^m,

†See any elementary text dealing with calculus on manifolds, e.g., Lang [1].

‡A section ξ∈E(X, E) has compact support on a (not necessarily compact) manifold X if {x∈X: ξ(x)=0} is relatively compact in X.

defined (for a scalar-valued function) by (1.1) f²s,Rⁿ =

$

| ˆf (y)|²(1+ |y|²)^sdy, where

ˆ

f (y)=(2π )⁻ⁿ

$

e^−ix,yf (x) dx

is the Fourier transform in Rⁿ. We extend this to a vector-valued function by taking the s-norm of the Euclidean norm of the vector, for instance.

Note that sis defined for alls∈R, but we shall deal only with integral values in our applications. Intuitively, ξ^s <∞, for s a positive integer, means thatξ has s derivatives inL². This follows from the fact that in Rⁿ, the norm s,Rⁿ is equivalent [on D(Rⁿ)] to the norm

-

|α|≤s

$

Rⁿ

|D^αf|²dx .1/2

, f ∈D(Rⁿ)

(see, e.g., Hörmander [1], Chap. 1). This follows essentially from the basic facts about Fourier transforms that

D/^αf (y)=y^αf (y),ˆ

where y^α=y₁^α1· · ·y_n^αn, D^α=(−i)^|α| D^α₁¹· · ·D_n^αn, D_j =∂/∂x_j, and f⁰ = ˆf⁰.

The norm sdefined onEdepends on the choice of partition of unity and the local trivialization. We let W^s(X, E) be the completion ofE(X, E) with respect to the norm s. Then it is a fact, which we shall not verify here, that thetopologyonW^s(X, E)is independent of the choices made; i.e., any two such norms are equivalent. Note that fors=0 we have made two dif- ferent choices of norms, one using the local trivializations and one using the Hermitian structure onE, and that these twoL²-norms are also equivalent.

We have a sequence of inclusions of the Hilbert spaces W^s(X, E),

· · · ⊃W^s⊃W^s+1⊃ · · · ⊃W^s+j ⊃ · · ·.

If we let H^∗ denote the antidual of a topological vector space over C (the conjugate-linear continuous functionals), then it can be shown that

(W^s)^∗∼=W^−s (s≥0).

In fact, we could have defined W^−s in this manner, using the definition involving the norms s for the nonnegative values of s. Locally this is easy to see, since we have forf ∈W^s(Rⁿ),g∈W^−s(Rⁿ)the duality (ignoring the conjugation problem by assuming that f andg are real-valued)

f, g =

$

f (x)•g(x)dx=

$ ˆ

f (ξ )•g(ξ ) dξ,ˆ and this exists, since

|f, g| ≤

$

| ˆf (ξ )|(1+ |ξ|²)^s/2| ˆg(ξ )|(1+ |ξ|²)^−s/2dξ ≤ fsg−s<∞. The growth is the important thing here, and the patching process (being a

C^∞ process with compact supports) does not affect the growth conditions and hence the existence of the integrals. Thus the global result stated above is easily obtained. We have the following two important results concerning this sequence of Hilbert spaces.

Proposition 1.1 (Sobolev): Let n=dimRX, and suppose that s >[n/2] + k+1. Then

W^s(X, E)⊂E_k(X, E).

Proposition 1.2 (Rellich): The natural inclusion j:W^s(E)⊂W^t(E) for t < s is a completely continuous linear map.

Recall that completely continuous means that the image of a closed ball is relatively compact, i.e., j is a compact operator. In Proposition 1.2 the compactness of X is strongly used, whereas it is inessential for Pro- position 1.1.

To give the reader some appreciation of these propositions, we shall give proofs of them in special cases to show what is involved. The general results for vector bundles are essentially formalism and the piecing together of these special cases.

Proposition 1.3 (Sobolev): Let f be a measurable L² function in Rⁿ with f^s<∞, for s >[n/2] +k+1, a nonnegative integer. Then f ∈C^k(Rⁿ) (after a possible change on a set of measure zero).

Proof: Our assumptionfs <∞ means that

$

Rⁿ

| ˆf (ξ )|²(1+ |ξ|²)^sdξ <∞. Let

˜ f (x)=

$

Rⁿ

e^ix,ξf (ξ ) dξˆ

be the inverse Fourier transform, if it exists. We know that if the inverse Fourier transform exists, thenf (x)˜ agrees withf (x)almost everywhere, and we agree to say thatf ∈C⁰(Rⁿ)if this integral exists, making the appropriate change on a set of measure zero. Similarly, for some constant c,

D^αf (x)=c

$

e^ix,ξξ^αf (ξ ) dξˆ

will be continuous derivatives of f if the integral converges. Therefore we need to show that for |α| ≤k, the integrals

$

e^ix,ξξ^αf (ξ ) dξˆ

converge, and it will follow that f ∈C^k(Rⁿ). But, indeed, we have

$

| ˆf (ξ )||ξ|^|α|dξ =

$

| ˆf (ξ )|(1+ |ξ|²)^s/2 |ξ|^|α|

(1+ |ξ|²)^s/2dξ

≤ f^s

$ |ξ|^2|α| (1+ |ξ|²)^sdξ

1/2

.

Now s has been chosen so that this last integral exists (which is easy to see by using polar coordinates), and so we have

$

| ˆf (ξ )||ξ|^|α|dξ <∞, and the proposition is proved.

Q.E.D.

Similarly, we can prove a simple version of Rellich’s lemma.

Proposition 1.4 (Rellich): Suppose that f_v ∈W^s(Rⁿ) and that all f_v have compact support in K⊂⊂Rⁿ. Assume thatf_vs≤1. Then for anyt < s there exists a subsequence f_vj which converges in t.

Proof: We observe first that forξ,η∈Rⁿ, s∈Z⁺, (1.2) (1+ |ξ|²)^s/2≤2^s/2(1+ |ξ−η|²)^s/2(1+ |η|²)^s/2. To see this we write, using the Schwarz inequality,

1+ |ζ+η|²≤1+(|ζ| + |η|)²≤1+2(|ζ|²+ |η|²)

≤2(1+ |ζ|²)(1+ |η|²).

Now let ξ =ζ+η, and we obtain (1.2) easily.

Let ϕ ∈D(Rⁿ) be chosen so that ϕ≡1 nearK. Then from a standard relation between the Fourier transform and convolution we have that

f_v=ϕf_v implies

(1.3) fˆ_v(ξ )=

$ ˆ

ϕ(ξ−η)fˆ_v(η) dη.

Therefore we obtain from (1.2) and (1.3) that (1+ |ξ|²)^s/2| ˆf_v(ξ )|

≤2^s/2

$

(1+ |ξ −η|²)^s/2| ˆϕ(ξ−η)|(1+ |η|²)^s/2| ˆf_v(η)|dη

≤K_s,ϕf_v^s≤K_s,ϕ,

whereK_s,ϕis a constant depending onsandϕ. Therefore| ˆf_v(ξ )|is uniformly bounded on compact subsets of Rⁿ. Similarly, by differentiating (1.3) we obtain that all derivatives offˆv are uniformly bounded on compact subsets in the same manner. Therefore, there is, by Ascoli’s theorem, a subsequence f_vj such that fˆ_vj converges in the C^∞ topology to a C^∞ function on Rⁿ. Let us call {f_v}this new sequence.

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.