## 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.

### MSC:

 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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