I am studying a about O'Neil's convolution inequality. It is stated that for  $\Phi_1$ and $\Phi_2$ be $N$-functions, with $\Phi_i(2t)\approx \Phi_i(t), \quad i=1,2$ with $t\gg 1$ and $k \in M_+(R^n)$ is the kernel of a convolution operator.

The $\rho$ is an r.i. norm on $M_+(R^n)$ given in terms of the r.i norm $\bar \rho$ on $M_+(R_+)$ by
$$
\rho(f)=\bar \rho(f^*), \quad f \in  M_+(R_+)
$$

Denote Orlicz gauge norms, $\rho_{\Phi}$, for which
$$
(\bar \rho_{\Phi})_d\approx \bar \rho_{\Phi}\left(\int_0^t h/t\right).
$$

It is stated that
$$
\rho_{\Phi_1}(k+f)\leq C \rho_{\Phi_2}(f)
$$
if
$$
(i) \quad \bar \rho_{\Phi_1}\left(\frac 1t \int_0^t k^*(s)\int_0^sf^*\right)\leq C \bar \rho_{\Phi_2}(f^*)
$$
$$
(ii) \quad \bar \rho_{\Phi_1}\left (\frac 1t\int_0^t f^*(s)\int_0^sk^*\right)\leq  C \bar \rho_{\Phi_2}(f^*)
$$
$$
(iii) \quad \bar \rho_{\Phi_1}\left(\int_t^{\infty}k^*f^*\right)\leq C \bar \rho_{\Phi_2}(f^*).
$$

I cannot understand under which conditions on kernel those inequalities (i),(ii) and (iii) would hold.