Reverse-Holder classes in the Orlicz spaces setting

In connection with the ${\displaystyle A_{\varphi }}$ classes of weights (see [K-T] and [B-K]), we study, in the context of Orlicz spaces, the corresponding reverse-Hölder classes ${\displaystyle R{H}_{\varphi }}$. We prove that when ϕ is ${\displaystyle {\Delta }_2}$ and has lower index greater than one, the class ${\displaystyle R{H}_{\varphi }}$ coincides with some reverse-Hölder class ${\displaystyle R{H}_q,q>1}$. For more general ϕ we still get ${\displaystyle R{H}_{\varphi }\subset A_{\infty }={\bigcup }_{q>1}R{H}_q}$ although the intersection of all these ${\displaystyle R{H}_{\varphi }}$ gives a proper subset of ${\displaystyle {\bigcap }_{q>1}R{H}_q}$.