Hardy-Type Operators in Lorentz-Type Spaces Defined on Measure Spaces

Abstract

Weight criteria for the boundedness and compactness of generalized Hardy-type operators (0.1) $$Tf(x) = {u_1}(x)\int_{\left\{ {\phi (y) \le \psi (x)} \right\}} {f(y){u_2}(y){v_0}(y)d\mu (y),\;\;\;\;\;x \in X},$$ in Orlicz-Lorentz spaces defined on measure spaces is investigated where the functions ϕ, ψ, u1, u2, v0 are positive measurable functions. Some sufficient conditions of boundedness of $$T:\;\Lambda _{{v_0}}^{{G_0}}({w_0}) \to \Lambda _{{v_1}}^{{G_1}}({w_1})$$ and $$T:\;\Lambda _{{v_0}}^{{G_0}}({w_0}) \to \Lambda _{{v_1}}^{{G_1},\infty }({w_1})$$ are obtained on Orlicz-Lorentz spaces. Furthermore, we achieve sufficient and necessary conditions for T to be bounded and compact from a weighted Lorentz space $$\Lambda _{{v_0}}^{{p_0}}({w_0})$$ to another $$\Lambda _{{v_1}}^{{p_1},{q_1}}({w_1})$$ . It is notable that the function spaces concerned here are quasi-Banach spaces instead of Banach spaces.