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 R n given by ( t , x ) ∈ [ 0 , 1 ] × R n → s ( t , x ) = s ( t ) x . In this paper, we study a new weighted Hardy–Cesàro operator defined by U ψ , s f ( x ) = ∫ 0 1 f ( s ( t ) x ) ψ ( t ) d t , for measurable complex-valued functions f on R n . 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 ) .