fa.functional analysis – Condition on kernel convolution operator


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 $tgg 1$ 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
$$
(bar rho_{Phi})_dapprox bar rho_{Phi}left(int_0^t h/tright).
$$

It is stated that
$$
rho_{Phi_1}(k+f)leq C rho_{Phi_2}(f)
$$

if
$$
(i) quad bar rho_{Phi_1}left(frac 1t int_0^t k^*(s)int_0^sf^*right)leq C bar rho_{Phi_2}(f^*)
$$

$$
(ii) quad bar rho_{Phi_1}left (frac 1tint_0^t f^*(s)int_0^sk^*right)leq C bar rho_{Phi_2}(f^*)
$$

$$
(iii) quad bar rho_{Phi_1}left(int_t^{infty}k^*f^*right)leq C bar rho_{Phi_2}(f^*).
$$

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