I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with
$$
\Phi_i(2t)\approx \Phi_i(t), \quad i=1,2
$$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the kernel of a convolution operator.
Furthermore let $\rho$ be an r.i. norm on $M_+(\mathbf R^n)$ given in terms of the r.i norm $\bar \rho$ on $M_+(\mathbf R_+)$ by
$$
\rho(f)=\bar \rho(f^*), \quad f \in M_+(\mathbf R_+)
$$
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
$$
\begin{align}
\bar \rho_{\Phi_1}\left(\frac 1t \int_0^t k^*(s)\int_0^sf^*\right) &\leq C \bar \rho_{\Phi_2}(f^*)\label{1}\tag{i}\\
\bar \rho_{\Phi_1}\left (\frac 1t\int_0^t f^*(s)\int_0^sk^*\right) &\leq C \bar \rho_{\Phi_2}(f^*)\label{2}\tag{ii}\\
\quad \bar \rho_{\Phi_1}\left(\int_t^{\infty}k^*f^*\right)& \leq C \bar \rho_{\Phi_2}(f^*)\label{3}\tag{iii}.
\end{align}
$$
I cannot understand under which conditions on the kernel those inequalities \eqref{1},\eqref{2} and \eqref{3} would hold.
