  Let $f \in M_+(\R_+)$, the class of nonnegative, Lebesgure-measurable
functions on $\R_+=(0, \infty)$. We deal with integral operators of the form \[
(T_Kf)(x)=\int_{\R_+}K(x,y)f(y)\, dy, \quad x \in \R_+, \] with $K \in
M_+(\R_+^2)$.
  We are interested in inequalities \[ \rho_{1}((T_Kf)^*)\leq C\rho_2(f^*), \]
in which $\rho_1$ and $\rho_2$ are functionals on functions $h \in M_+(\R_+)$,
and \[ h^*(t)=\mu_h^{-1}(t), \quad t \in \R_+, \] where \[ \mu_h(\lambda)=|\{x
\in \R_+: \, h(x)> \lambda\}|, \lambda \in \R_+. \] Specifically, $\rho_1$ and
$\rho_2$ are so-called Orlicz-Lorentz gauge functionals of the type \[
\rho(h)=\rho_{\Phi, u}(h)=\inf\left\{\lambda>0:\,
\int_{\R_+}\Phi\left(\frac{h(x)}{\lambda}\right)u(x)\, dx \leq 1\right\}, \quad
h \in M_+(\R_+); \] here $\Phi(x)=\int_0^x\phi(y)\, dy$, $\phi$ an increasing
function mapping $\R_+$ onto itself and $u\in M_+(\R_+)$.
