DIFFERENCES OF VECTOR-VALUED FUNCTIONS ON TOPOLOGICAL-GROUPS

Let G be a locally compact group equipped with right Haar measure. The right differences Delta(h) phi of functions phi on G are defined by D elta(h) phi(t) = phi(th) - phi(t) for h, t is an element of G. Let phi is an element of L(infinity)(G) and suppose Delta(h) phi is an elemen t of L(p)(G) for some 1 less than or equal to p < infinity and all h i s an element of G. We prove that parallel to Delta(h) phi parallel to( p) is a light uniformly continuous function of h. If G is abelian and the Beurling spectrum sp(phi) does not contain the unit of the dual gr oup (G) over cap, then we show phi is an element of L(p)(G). These res ults have analogues for functions phi : G --> X, where X is a separabl e or reflexive Banach space. Finally, we apply our methods to vector-v alued right uniformly continuous differences and to absolutely continu ous elements of left Banach G-modules.