Local spaces of distributions

Abstract

A space of distributions E is local if, roughly, a distribution T belongs to E whenever T belongs to E in the neighborhood of every point. A space E, in whose definition growth conditions enter, is not local but one can associate with E a local space $E_{loc}$. This is classical for the spaces $L^p$ [6], and was done for the Sobolev spaces $\mathcal H⁵$ by Laurent Schwartz in his 1956 Bogotà lectures [8], where he presented an expository account of B. Malgrange's doctoral dissertation. In the present paper I establish some simple properties of the space $E_{loc}$ attached to a space of distributions E. To a distribution space E we can also attach the space E, consisting of those elements of E which have compact support. At the end of the paper I make some remarks concerning the duality between local spaces and spaces of distributions with compact support.