Universal family of translations on weighted Orlicz spaces

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis 2022-02, Vol.26 (1), Article 1
• Main Authors: Akbarbaglu, Ibrahim, Azimi, Mohammad R.
• Format: Article
• Language: English
• Subjects: Calculus of Variations and Optimal Control; Optimization; Econometrics; Fourier Analysis; Mathematics; Mathematics and Statistics; Operator Theory; Orlicz space; Potential Theory; Translations
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-022-00869-2

Let G be a locally compact group, Φ be a Young function and ω be a weight on G . An associated weighted Orlicz space is denoted by L Φ ( G , ω ) . For any S ⊆ G , a family of left translations { L s } s ∈ S on L Φ ( G , ω ) , defined by L s f ( t ) : = f ( s - 1 t ) for all t ∈ G , is said S-universal if there exists a function f ∈ L Φ ( G , ω ) , called an S - universal vector , such that its S -orbit, namely, O r b S ( f ) = { L s ( f ) : s ∈ S } is dense in L Φ ( G , ω ) . In this paper, first it is shown that any compact group G , does not admit an S -universal vector in L Φ ( G , ω ) and only the infinite dimensional weighted Orlicz spaces L Φ ( G , ω ) may contain it. In the sequel, under natural restrictions, the existence of S -universal vector leads to find out that G must be second countable. Moreover, we give a necessary and sufficient condition for { L s } s ∈ S to be S -universal on L Φ ( G , ω ) . At last, some other specific situations which guarantee an S -universal vector for { L s } s ∈ S , are also studied.