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 <=tau <= b) u(tau) integral(tau)(0) 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(G0)(u0)(w(0)) -> Lambda(G1)(u1)(w(1)) and R : Lambda(G0)(u0)(w(0)) -> Lambda(G1, infinity)(u1)(w(1)). Furthermore, in the case of weighted Lorentz spaces, two characterizations of the boundedness of the operator R : Lambda(p0)(u0)(w(0)) -> Lambda(p1, q1)(u1)(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.