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

Abstract

Given a quasi-subaditive operator T:L₀(μ)→ L₀(v), we want to study mapping properties of interpolation type for which the following modular inequality holds \[∈t_{\mathscr{N}}P(|Tf(x)|)dv(x)≤∈t_{\mathscr{M}}Q(|f(x)|)dμ(x) \] where P and Q are modular functions. These results generalize the classical Marcinkiewicz interpolation theorem.