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

Let ψ:[0,1]→[0,∞)ψ:[0,1]→[0,∞), s:[0,1]→Rs:[0,1]→R be measurable functions and Γ   be a parameter curve in RnRn given by (t,x)∈[0,1]×Rn→s(t,x)=s(t)x(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 RnRn. Under certain conditions on s(t)s(t) and on an absolutely homogeneous weight function ω, we characterize the weight function ψ   such that Uψ,sUψ,s is bounded on weighted Morrey spaces Lp,λ(ω)Lp,λ(ω) and then compute the corresponding operator norm of Uψ,sUψ,s. We also give a necessary and sufficient condition on the function ψ  , which ensures the boundedness of the commutator of the operator Uψ,sUψ,s on Lp,λ(ω)Lp,λ(ω) with symbols in BMO(ω)BMO(ω).