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.
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L. Maligranda’s research partially supported by the Swedish Research Council (VR) grant 621-2008-5058. E. Nakai’s research partially supported by Grant-in-Aid for Scientific Research (C), No. 20540167, Japan Society for the Promotion of Science.
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Maligranda, L., Nakai, E. Pointwise multipliers of Orlicz spaces. Arch. Math. 95, 251–256 (2010). https://doi.org/10.1007/s00013-010-0160-y
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DOI: https://doi.org/10.1007/s00013-010-0160-y