Differences of vector-valued functions on topological groups

Let $G$ be a locally compact group equipped with right Haar measure. The right differences $\triangle _{h} \varphi$ of functions $\varphi$ on $G$ are defined by $\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)$ for $h,t \in G$. Let $\varphi \in L^{\infty }(G)$ and suppose $\triangle _{h} \varphi \in L^{p} (G)$ for some $1 \leq p < \infty$ and all $h \in G$. We prove that $\Vert \triangle _{h} \varphi \Vert _{p}$ is a right uniformly continuous function of $h$. If $G$ is abelian and the Beurling spectrum $sp(\varphi )$ does not contain the unit of the dual group $\hat {G}$, then we show $\varphi \in L^{p} (G)$. These results have analogues for functions $\varphi : G\to X$, where $X$ is a separable or reflexive Banach space. Finally, we apply our methods to vector-valued right uniformly continuous differences and to absolutely continuous elements of left Banach $G$-modules.