Some extensions of the Marcinkiewicz interpolation theorem in terms of modular inequalities

Abstract

Given a quasi-subaditive operator $T:L_0\left(\mu \right)$$\to L_0\left(v\right)$, we want to study mapping properties of interpolation type for which the following modular inequality holds

${\int }_{N}P\left(\right|Tf\left(x\right)\left|\right)dv\left(x\right)\le {\int }_{M}Q\left(\right|f\left(x\right)\left|\right)d\mu \left(x\right)$

where $P$ and $Q$ are modular functions. These results generalize the Marcinkiewicz interpolation theorem.