# 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.