Let G be a Hausdorff locally
compact abelian group. In this paper we characterise completely those continuous
linear operators T from Cc(G) (the space of continuous functions with compact
supports endowed with the inductive limit topology) into M(G) (the space
of measures with the vague topology of measures) which commute with
convolution: T(f ∗ g) = (Tf) ∗ g. They are represented by convolution with a
“quasimeasure”. As a corollary of this theorem, we have the result that the space of
multipliers from Lp(G)(p≠∞) to Lq(G) is isomorphic to a subspace of the space of
quasimeasures.
The quasimeasures are defined as the elements of the dual of a certain inductive
limit of Banach spaces. We develop some of the theory of pseudomeasures and of
quasimeasures and establish the structural relationship of quasimeasures to
pseudomeasures.
