# Condition on kernel convolution operator

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.