Elliptic Operator Theory

Abstract

In this chapter we shall describe the general theory of elliptic differential operators on compact differentiate 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 cohomo-logy by harmonic forms.