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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Differences of vector-valued
functions on topological groups


Authors: Bolis Basit and A. J. Pryde
Journal: Proc. Amer. Math. Soc. 124 (1996), 1969-1975
MSC (1991): Primary 43A15; Secondary 28B05, 39A05
MathSciNet review: 1317031
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Abstract: 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.


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Additional Information

Bolis Basit
Affiliation: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
Email: bbasit(ajpryde)@vaxc.cc.monash.edu.au

A. J. Pryde
Affiliation: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
Email: bbasit(ajpryde)@vaxc.cc.monash.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03258-3
PII: S 0002-9939(96)03258-3
Keywords: Differences, weight functions, spectrum, right uniform continuity, $G$-modules, weak continuity, absolutely continuous elements
Received by editor(s): September 21, 1994
Received by editor(s) in revised form: January 4, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society