Multiplicativity Factors for Orlicz Space Function Norms

Let ρφ be a function norm defined by a Young function φ with respect to a measure space (T, Ω, m), and let Lφ be the Orlicz space determined by ρφ. If Lφ is an algebra, then a constant μ > 0 is called a multiplicativity factor for ρφ, if ρφ,(fg) ≤ μρφ(f) ρφ(g) for all f, g ∈ Lφ. The main objective of this paper is to give conditions under which Lφ is indeed an algebra, and to obtain in this case the best (least) multiplicativity factor for ρφ. The first of our principal results is that Lφ is an algebra if and only if minf ≡ inf{m(A) > 0 : A ∈ Ω} > 0 or x∞(φ) ≡ sup{x ≥ 0 : φ(x) < ∞} < ∞ Our second main result states that if Lφ is an algebra and (T, Ω, m) is free of infinite atoms, then the best multiplicativity factor for ρφ is φ−1(1/minf if minf > 0, and x∞(φ) if minf = 0.