Pointwise multipliers of Orlicz spaces

Abstract

We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ₁, Φ₂ and Φ, generating corresponding Orlicz spaces, satisfy the estimate ${\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)}$ for all u > 0, we prove that if the pointwise product xy belongs to L ^Φ(μ) for all ${y \in L^{\Phi_1}(\mu)}$, then ${x \in L^{\Phi_2}(\mu)}$. The result with some restrictions either on Young functions or on the measure μ was proved by Maligranda and Persson (Indag. Math. 51 (1989), 323–338). Our result holds for any collection of three Young functions satisfying the above estimate and for an arbitrary complete σ-finite measure μ.