Abstract
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.
Intrinsic Square Function Characterizations of Several Hardy--Type Spaces--a Survey
