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.