Generalized weighted Hardy-Cesàro operators and their commutators on weighted Morrey spaces

Let ψ:[0, 1]→[0, ∞), s:[0,1]→R be measurable functions and Γ be a parameter curve in Rn given by (t,x)∈[0,1]×Rn→s(t,x)=s(t)x. In this paper, we study a new weighted Hardy-Cesàro operator defined by Uψ,sf(x)=∫01f(s(t)x)ψ(t)dt, for measurable complex-valued functions f on Rn. Under certain conditions on s(t) and on an absolutely homogeneous weight function ω, we characterize the weight function ψ such that U _ψ,s is bounded on weighted Morrey spaces L ^p,λ(ω) and then compute the corresponding operator norm of U _ψ,s. We also give a necessary and sufficient condition on the function ψ, which ensures the boundedness of the commutator of the operator U _ψ,s on L ^p,λ(ω) with symbols in BMO(ω).