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Let p∈[1,∞], q∈[1,∞), τ∈(0,∞), and α∈(0,1) such that τ>1/p-1/q and α≤n(1/p-τ), let Uψ be the weighted Hardy operator and Vψ its adjoint operator with respect to the weight function ψ. In this paper, the authors establish a sufficient and necessary condition on weight function ψ to ensure the boundedness of Uψ and Vψ on the Triebel-Lizorkin-type spaces Ḟp,qα,τ(ℝn) and their predual spaces, Triebel-Lizorkin-Hausdorff spaces, which unify and generalize the known results on Q-type spaces

Boundedness of Weighted Hardy Operator and Its Adjoint on Triebel-Lizorkin-Type Spaces