Unbounded Translation Invariant Operators and the Derivation Property

We are going to investigate translation invariant derivations on L p spaces of locally compact abelian groups, 1⩽ p <∞. By these we mean densely defined closed linear operators which commute with translations and obey a Leibniz rule, that is, T ( fg )=( Tf )· g + f ·( Tg ); see Definition 1 for details. The original motivation for studying these operators was to find an abstract description of constant coefficient partial differential operators as a link to perturbation theory which is usually formulated in terms of abstract operator theoretic notions. This fits, for example, Schrödinger operator theory (as in [ 1 ]). Here, in view of the applications to perturbation theory, the point is that this identification is not just a formal one but includes assertions about domains (Theorem 1). In this paper, however, we shall concentrate on groups other than ℝ n .