Landau type theorem for Orlicz spaces. (English) Zbl 0738.46013

This paper contains a short and elementary proof of the following fact: If \(L^ m\) is an Orlicz space generated by a convex (not necessarily finite-valued) function \(M\) and \(g\) is a measurable function such that \(fg\in L^ 1\) for all \(f\in L^ M\), then \(g\in L^{M*}\), where \(M^*\) is the conjugate (=complementary) function of \(M\).
This proposition implies analogous theorems for some classes of nonlocally convex Orlicz spaces (over atomless or counting measures). Moreover, this paper contains an example of an Orlicz space \(L^ M\), whose Köthe dual is not isomorphic to any space of the form \(L^{M*}(\nu)\).


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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