On a variant of Hardy inequality between weighted Orlicz spaces

Let M be an N-function satisfying the ${\mathrm{\Delta }}_2$- condition, let $\omega, \vp$betwootherfunctions,$\omega\ge 0$. We study Hardy-type inequalities $${\int }_{\text{rp}}M(\omega (x)\mathrm{\mid }u(x)\mathrm{\mid })\mathrm{e}\mathrm{x}\mathrm{p}(-\text{vp}(x))dx\le C{\int }_{\text{rp}}M(\mathrm{\mid }{u}^{\mathrm{\prime }}(x)\mathrm{\mid })\mathrm{e}\mathrm{x}\mathrm{p}(-\text{vp}(x))dx,$$ where $u$ belongs to some dilation invariant set $\mathcal{R}$ contained in the space of locally absolutely continuous functions. We give sufficient conditions the triple $(\omega ,\text{vp},M)$ must satisfy in order to have such inequalities valid for $u$ from a given set $\mathcal{R}$. The set $\mathcal{R}$ can be smaller than the set of Hardy transforms. Bounds for constants, retrieving classical Hardy inequalities with best constants, are also given.