Contribution aux espaces d'Orlicz (Q788245)

Let \((T,\Sigma\),\(\mu)\) be a \(\sigma\)-finite measure space and let \(\phi\) :\(T\times [0,\infty)\to [0,\infty)\) be a function with certain additional properties. The author considers the generalized Orlicz class \(L_ o^{\phi}\) of (\(\mu\)-equivalence classes of) complex-valued measurable functions x on T such that \(\int_{T}\phi(t,| x(t)|)d\mu(t)<\infty\) and the corresponding Orlicz space \(L^{\phi}\) consisting of those functions x such that \(\alpha x\in L_ 0^{\phi}\) for some \(\alpha>0\). The main results of the paper give necessary and sufficient conditions for containment relations between such classes \(L_ o^{\phi}\) or spaces \(L^{\phi}\) for different functions \(\phi\).