Peetre’s Theorem and Generalized Functions

Abstract

Sheaf morphisms are considered in sheaves of generalized functions. It is proved that for (ultra)distributions they must be continuous outside discrete points. Contrary to Peetre’s original theorem, which applies to sheaves of test functions, an example makes clear that these points can really be points of discontinuity. Finally, it is shown that in the sheaf of hyper-functions there are more general discontinuous sheaf morphisms.

Peetre’s theorem says that any sheaf morphism in the sheaf of C^∞-functions is a differential operator. We shall investigate sheaf morphisms in sheaves of generalized functions, in particular distributions, ultradistributions of the Beurling and of the Roumieu type, and hyperfunctions. All these sheaves are soft so that their sections with a compact support form flabby cosheaves which are the duals, with respect to a certain topology, of the sheaves of their associated test functions. The main point is to investigate the continuity of a cosheaf morphism P (= local operator) in one of these cosheaves. At places where P is continuous its transposed ^tP is a continuous sheaf morphism in the sheaf of test functions and it follows that ^tP, and hence P itself, are appropriate differential operators there. In this paper we shall only briefly mention these results, as well as the generalization of Peetre’s theorem to the soft sheaves of test functions. Our main attention will be on the continuity of a local operator in a space of generalized functions and we shall indicate what possibilities there are for a discontinuous sheaf morphism.