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Elliptic Operator Theory CHAPTER IV ELLIPTIC OPERATOR THEORY 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 only C∞ 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. 1. Sobolev Spaces In this section we shall restrict ourselves to compact differentiable mani- folds, for simplicity, although many of the topics that we shall discuss are certainly more general. Let X be a compact differentiable manifold with a strictly positive smooth measure µ. We mean by this that dµ is a volume element (or density) which can be expressed in local coordinates (x1,...,xn) by dµ ρ(x)dx ρ(x)dx dx = = 1 ··· n 108 Sec. 1 Sobolev Spaces 109 where the coefficients transform by ∂y(x) ρ(x)dx ρ(y(x)) det dx, =˜ ∂x where ρ(y)dy is the representation with respect to the coordinates y ˜ = (y1,...,yn), 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) det g (x) 1/2, =| ij | where ds2 g (x)dx dx is a Riemannian metric for X expressed = ij i j in terms of the local coordinates (x1,...,xn).† 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 E be a Hermitian (differentiable) vector bundle over X. Let Ek(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‡∞ by= D(X, E) E(X, E) and the compactly supported functions by D(X) E(X). Define⊂ an inner product(,)onE(X, E) by setting ⊂ (ξ, η) ξ(x),η(x) dµ, = E $X where , is the Hermitian metric on E. Let E ξ (ξ, ξ)1/2 0 = 2 0 be the L -norm and let W (X, E) be the completion of E(X, E). Let Uα,ϕα be a finite trivializing cover, where, in the diagram { } ϕ α m E Uα / U C | ˜α × ϕα Uα ¯ / U , ˜α ϕ is a bundle map isomorphism and ϕ U U Rn are local coordinate α α α ˜α systems for the manifold X. Then let¯ : → ⊂ m ϕ∗ E(U ,E) E(U ) α: α −→ [ ˜ α ] be the induced map. Let ρα be a partition of unity subordinate to Uα , and define, for ξ E(X, E){ , } { } ∈ ξ ϕ∗ρ ξ n , s,E = α α s,R α where s,Rn is the Sobolev norm for a compactly supported differentiable function f Rn Cm, : −→ †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. { ∈ : = } 110 Elliptic Operator Theory Chap. IV defined (for a scalar-valued function) by 2 2 2 s (1.1) f n f(y) (1 y ) dy, s,R = | ˆ | +| | $ where n i x,y f(y) (2π)− e− f(x)dx ˆ = 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 is defined for all s R, but we shall deal only with integral s ∈ values in our applications. Intuitively, ξ s < ,fors a positive integer, means that ξ has s derivatives in L2. This follows∞ from the fact that in Rn, n the norm n is equivalent [on D(R )] to the norm s,R 1/2 Dαf 2dx ,f D(Rn) n | | ∈ - α s R . ||≤ $ (see, e.g., Hörmander [1], Chap. 1). This follows essentially from the basic facts about Fourier transforms that Dαf(y) yαf(y), = ˆ α α1 αn α α α1 αn where y y y ,D ( i)| | D D , D ∂/∂x , and f = 1 ··· n = − 1 ··· n j = j 0 = f . / ˆ 0 The norm s defined on E depends on the choice of partition of unity and the local trivialization. We let W s (X, E) be the completion of E(X, E) with respect to the norm s . Then it is a fact, which we shall not verify here, that the topology on W s (X, E) is independent of the choices made; i.e., any two such norms are equivalent. Note that for s 0 we have made two dif- ferent choices of norms, one using the local trivializations= and one using the Hermitian structure on E, and that these two L2-norms are also equivalent. We have a sequence of inclusions of the Hilbert spaces W s (X, E), s s 1 s j W W + W + . ···⊃ ⊃ ⊃···⊃ ⊃··· 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 s s (W )∗ W − (s 0). =∼ ≥ s In fact, we could have defined W − in this manner, using the definition involving the norms s for the nonnegative values of s. Locally this is s n s n easy to see, since we have for f W (R ), g W − (R ) the duality (ignoring the conjugation problem by assuming∈ that ∈f and g are real-valued) f, g f(x)• g(x)dx f(ξ)• g(ξ)dξ, = = ˆ ˆ $ $ and this exists, since 2 s/2 2 s/2 f, g f(ξ)(1 ξ ) g(ξ) (1 ξ )− dξ f s g s < . | | ≤ | ˆ | +| | |ˆ | +| | ≤ − ∞ $ The growth is the important thing here, and the patching process (being a Sec. 1 Sobolev Spaces 111 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 (X, E). ⊂ k Proposition 1.2 (Rellich): The natural inclusion j W s (E) W t (E) : ⊂ for t<sis 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 k n f s < ,fors> n/2 k 1, a nonnegative integer. Then f C (R ) (after a∞ possible change[ ]+ on a+ set of measure zero). ∈ Proof: Our assumption f < means that s ∞ f(ξ) 2(1 ξ 2)s dξ < . n | ˆ | +| | ∞ $R Let i x,ξ f(x) e f(ξ)dξ ˜ = n ˆ $R be the inverse Fourier transform, if it exists. We know that if the inverse Fourier transform exists, then f(x)agrees with f(x)almost everywhere, and ˜ weagreetosaythatf C0(Rn) if this integral exists, making the appropriate change on a set of measure∈ zero. Similarly, for some constant c, α i x,ξ α D f(x) c e ξ f(ξ)dξ = ˆ $ will be continuous derivatives of f if the integral converges. Therefore we need to show that for α k, the integrals | |≤ i x,ξ α e ξ f(ξ)dξ ˆ $ 112 Elliptic Operator Theory Chap. IV converge, and it will follow that f Ck(Rn). But, indeed, we have ∈ α α 2 s/2 ξ | | f(ξ) ξ | | dξ f(ξ)(1 ξ ) | | dξ | ˆ || | = | ˆ | +| | (1 ξ 2)s/2 $ $ +| | 2 α 1/2 ξ | | f | | dξ . ≤ s (1 ξ 2)s $ +| | Now s has been chosen so that this last integral exists (which is easy to see by using polar coordinates), and so we have α f(ξ) ξ | | dξ < , | ˆ || | ∞ $ and the proposition is proved.
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