Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces

Given measurable functions ϕ, ψ on {\mathbb{R}^{+}} and a kernel function {k(x,y)\geq 0} satisfying the Oinarov condition, we study the Hardy operator Kf(x)=\psi(x)\int_{0}^{x}k(x,y)\phi(y)f(y)\,dy,\quad x>0, between Orlicz–Lorentz spaces {\Lambda_{X}^{G}(w)} , where f is a measurable function on {\mathbb{R}^{+}} . We obtain sufficient conditions of boundedness of {K:\Lambda_{u_{0}}^{G_{0}}(w_{0})\rightarrow\Lambda_{u_{1}}^{G_{1}}(w_{1})} and {K:\Lambda_{u_{0}}^{G_{0}}(w_{0})\rightarrow\Lambda_{u_{1}}^{G_{1},\infty}(w_{% 1})} . We also look into boundedness and compactness of {K:\Lambda_{u_{0}}^{p_{0}}(w_{0})\rightarrow\Lambda_{u_{1}}^{p_{1},q_{1}}(w_{1% })} between weighted Lorentz spaces. The function spaces considered here are quasi-Banach spaces rather than Banach spaces. Specializing the weights and the Orlicz functions, we restore the existing results as well as we achieve new results in the new and old settings.