We are going to investigate translation invariant derivations on Lp spaces of locally compact abelian groups, 1[les ]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 ℝn.