Integro-Differential Operators

Abstract

Introduction. This article considers a fairly general class of operators on sections of a vector bundle over a compact manifold, including the differential operators and singular integral operators. The members of this class share many of the properties of differential operators, particularly the elliptic ones. Two general advantages have motivated the development. First, it leads to transparent proofs of the familiar results for elliptic equations, on regularity, the Fredholm alternative, and eigenfunction expansions; and for a larger class than the differential operators. These proofs are not new; rather some of the techniques used in the case of differential operators appear here as general properties of the class of integro-differential operators considered. A second advantage of the larger system, not extensively exploited in this article, is topological. Homotopies (in the class of smooth functions) of the characteristic polynomial of a differential operator can be lifted to homotopies of the operator itself in the class of integro-differential operators considered, but not (generally) in the class of differential operators. This is an important help in treating some questions raised by Gelfand [6]; some of the questions concerning the index have now been answered by Atiyah and Singer [1]. To find the notation and main results, one can read ?1-?3 (except for proofs), ?6, and the definitions and statements of theorems and corollaries from the remaining sections. The paper is organized as follows. ?1 describes the well-known function spaces on Rn that are involved, as well as certain operators on them. ?2 describes the singular integral operators and their symbols. ?3 extends this collection to one that contains the differential operators on R, as well as the inverses of the invertible elliptic operators. The symbol u(A) of an operator A is defined, and the behavior of a under composition of operators is discussed. ?4 considers the behavior of a under coordinate changes. ?5 gives some necessary lemmas from functional analysis. ?6 establishes the notation for vector bundles, and the analogs for bundles of the function spaces of ?1. ?7 defines the singular integral operators on sections of a vector bundle E over a compact manifold X, and their symbols. If A is a singular integral operator from sections of one bundle E