Orlicz-Lorentz Gauge Functional Inequalities for Positive Integral Operators. Revised Version

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 $u\in M_+(\R_+)$.