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