Local operators between spaces of ultradifferentiable functions and ultradistributions

Abstract

In this article we characterize certain ultradifferential operators by the condition of being local. First, we examine the continuity properties of local linear operators on spaces of ultradifferentiable functions in the sense of Beurling and of Roumieu. Next, a structure theorem for vector-valued ultradistributions with support at the origin is proved. This result leads to a representation theorem for continuous local operators from spaces of ultradifferentiable functions into various spaces of ultradistributions. In combination with the continuity results we thus obtain in many cases the desired characterization.