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

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 Rf(x)=\sup_{x\leq\tau < b}u(\tau)\int_0^\tau k(\tau,y)\sigma (y)f(y)dy, x > 0 in Orlicz-Lorentz spaces are investigated. We obtain sufficient conditions of boundedness of R: \Lambda_{u_0}^{G_0}(w_0)\\ \rightarrow \Lambda_{u_1}^{G_1}(w_1) and R: \Lambda_{u_0}^{G_0}(w_0)\rightarrow \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)\rightarrow\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.