Hilbert space methods in the theory of harmonic integrals

Abstract

The theory of harmonic integrals was created by Hodge [15], and the theorem which bears his name is the central result of the subject. Kodaira [17] and-independently-de Rham and Bidal [1] used the generalized harmonic operator A in their treatments of the theory. A was also used by Milgram and Rosenbloom [19] in their study of harmonic integrals with the heat equation. It is our purpose to develop the properties of A from the point of view of Hilbert space theory, thus arriving at Hodge's theorem without the use of a generalized integral equation theory. In addition, in ?2 we study A on a class of open manifolds-those with negligible boundary; these include all complete manifolds. ?3 contains a proof of the fundamental differentiability lemma. We wish to express our appreciation to Professor M. H. Stone, who suggested this topic to us. He provided us with the proof (in ?2) that I+ 5*+dd* is self-adjoint, and he suggested the application of Rellich's results to the proof of the complete continuity of the Green's operator. 1. Compact manifolds; Hodge's theorem. We assume familiarity with the basic concepts of differential forms on Riemannian manifolds. Expositions of