Differences of Vector-Valued Functions on Topological Groups
Let G be a locally compact group equipped with right Haar measure. The right differences Δhφof functions φ on G are defined by Δhφ(t) = φ(th) - φ(t) for h, t ∈ G. Let φ ∈ L∞(G) and suppose Δhφ∈ Lp(G) for some $1 \leq p < \infty$ and all h ∈ G. We prove that |Δhφ|pis a right uniformly continuous f...
| Veröffentlicht in: | Proceedings of the American Mathematical Society. - American Mathematical Society, 1950. - 124(1996), 7, Seite 1969-1975 |
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| Format: | Online-Aufsatz |
| Sprache: | English |
| Veröffentlicht: |
1996
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| Zugriff auf das übergeordnete Werk: | Proceedings of the American Mathematical Society |
| Schlagworte: | Differences weight functions spectrum right uniform continuity G-modules weak continuity absolutely continuous elements Mathematics |
| Zusammenfassung: | Let G be a locally compact group equipped with right Haar measure. The right differences Δhφof functions φ on G are defined by Δhφ(t) = φ(th) - φ(t) for h, t ∈ G. Let φ ∈ L∞(G) and suppose Δhφ∈ Lp(G) for some $1 \leq p < \infty$ and all h ∈ G. We prove that |Δhφ|pis a right uniformly continuous function of h. If G is abelian and the Beurling spectrum sp(φ) does not contain the unit of the dual group Ĝ, then we show φ ∈ Lp(G). These results have analogues for functions φ: G → 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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| ISSN: | 10886826 |