L1/w(<i>G</i>) -invariant weighted Orlicz spaces

Abstract

In this paper, we prove that σ-compactness of a locally compact group G is a necessary condition for the weighted Orlicz space LΦ/w (G)   to be a convolution algebra, and for a class of Orlicz spaces we give an equivalent condition. Also, we show that if LΦ/w (G)  is  L1/w-invariant, then there exists a constant c > 0 such that Φ(ω(st)) ≤ c ω(s)ω_Φ (t) for locally a.e. s, t ∈ G, where ω_Φ is a locally integrable function satisfying an inclusion property.