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

Abstract

Let $f\in M_{+}({\mathbb{R}}_{+})$, the class of nonnegative, Lebesgure-measurable functions on ${\mathbb{R}}_{+}=(0,\infty )$. We deal with integral operators of the form \[ (T_Kf)(x)=\int_{\R_+}K(x,y)f(y)\, dy, \quad x \in \R_+, \] with $K\in M_{+}({\mathbb{R}}_{+}^2)$. We are interested in inequalities \[ ρ_{1}((T_Kf)^*)\leq Cρ_2(f^*), \] in which ${\rho }_1$ and ${\rho }_2$ are functionals on functions $h\in M_{+}({\mathbb{R}}_{+})$, and \[ h^*(t)=μ_h^{-1}(t), \quad t \in \R_+, \] where \[ μ_h(λ)=|\{x \in \R_+: \, h(x)> λ\}|, λ\in \R_+. \] Specifically, ${\rho }_1$ and ${\rho }_2$ are so-called Orlicz-Lorentz gauge functionals of the type \[ ρ(h)=ρ_{Φ, u}(h)=\inf\left\{λ>0:\, \int_{\R_+}Φ\left(\frac{h(x)}λ\right)u(x)\, dx \leq 1\right\}, \quad h \in M_+(\R_+); \] here $\Phi (x)={\int }_0^x\varphi (y)dy$, $\varphi$ an increasing function mapping ${\mathbb{R}}_{+}$ onto itself and $u\in M_{+}({\mathbb{R}}_{+})$.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2102.11431