Two-weight weak type maximal inequalities in Orlicz classes

Necessary and sufficient conditions are shown in order that the inequalities of the form ${\displaystyle \varrho \left(\{M_{\mu }f>\lambda \}\right)\Phi (\lambda )\le C{ʃ}_X\Psi \left(C\left|f\left(x\right)\right|\right)\sigma \left(x\right)d\mu }$, or ${\displaystyle \varrho \left(\{M_{\mu }f>\lambda \}\right)\le C{ʃ}_X\Phi \left(C{\lambda }^{-1}\left|f\left(x\right)\right|\right)\sigma \left(x\right)d\mu }$ hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, ${\displaystyle M_{\mu }}$ is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.