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Differences of vector-valued functions on topological groups

Authors:	Bolis Basit A J Pryde

Institution:	Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia ; Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia

Abstract:	Let be a locally compact group equipped with right Haar measure. The right differences $\triangle _{h} \varphi$ of functions $\varphi$ on 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 . If 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 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 -modules.

Keywords:	Differences weight functions spectrum right uniform continuity $G$-modules weak continuity absolutely continuous elements

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