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

TL;DR

This paper investigates inequalities involving positive integral operators and Orlicz-Lorentz gauge functionals, establishing bounds that relate the rearranged outputs of the operators to the inputs within this functional framework.

Contribution

It introduces new inequalities for positive integral operators using Orlicz-Lorentz gauge functionals, expanding the understanding of their boundedness properties.

Findings

Derived bounds for integral operators with Orlicz-Lorentz functionals.

Established conditions for inequalities involving rearranged functions.

Extended classical inequalities to a broader functional setting.

Abstract

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