Some extensions of the Marcinkiewicz interpolation theorem in terms of modular inequalities. (Q1396420)

Let \(({\mathcal M},\mu)\) and \(({\mathcal N},\nu)\) be two \(\sigma\)-finite measure spaces and let \(P\) and \(Q\) two modular functions (i.e., \(Q: [0,\infty)\to [0,\infty)\) is a nondecreasing right-continuous function with \(Q(0^+)= 0\)). For a given subadditive operator \(T: L^0(\mu)\to L^0(\nu)\), several mapping properties of interpolation type for which the modular inequality \[ \int_{\mathcal N} P(| Tf(x)|\,d\nu(x)\leq \int_{\mathcal M}Q(| f(x)|)\,d\mu(x) \] holds are studied and applied. These results generalize the classical Marcinkiewicz interpolation theorem.