The supremum-involving Hardy-type operators on Lorentz-type spaces

Abstract

Given measurable functions $u,\sigma$ on an interval $(0,b)$ and a kernel function $k(x,y)$ on $(0,b{)}^2$ satisfying Oinarov condition, the supremum-involving Hardy-type operators

in Orlicz-Lorentz spaces are investigated. We obtain sufficient conditions of boundedness of $$\begin{gathered} R:{\Lambda }_{u_0}^{G_0}(w_0) \\ \to {\Lambda }_{u_1}^{G_1}(w_1) \end{gathered}$$ and $R:{\Lambda }_{u_0}^{G_0}(w_0)\to {\Lambda }_{u_1}^{G_1,\infty }(w_1)$. Furthermore, in the case of weighted Lorentz spaces, two characterizations of the boundedness of the operator $R:{\Lambda }_{u_0}^{p_0}(w_0)\to {\Lambda }_{u_1}^{p_1,q_1}(w_1)$ are achieved as well as the compactness of the operator $R$ is characterized. It is notable that in the present paper the spaces are only required to be quasi-Banach spaces other than Banach spaces.