Some extensions of the Marcinkiewicz interpolation theorem in terms of modular inequalities. (English) Zbl 1042.46049

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.


46M35 Abstract interpolation of topological vector spaces
26D15 Inequalities for sums, series and integrals
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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