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Boundedness of weighted Hardy operator and its adjoint on Triebel-Lizorkin-type spaces. (English) Zbl 1242.46046
Summary: Let $p \in [1, \infty]$, $q \in [1, \infty)$, $\tau \in (0, \infty)$, and $\alpha \in (0, 1)$ such that $\tau > 1/p - 1/q$ and $\alpha \leq n(1/p - \tau)$, let $U_\psi$ be the weighted Hardy operator and $V_\psi$ its adjoint operator with respect to the weight function $\psi$. In this paper, the authors establish a sufficient and necessary condition on the weight function $\psi$ to ensure the boundedness of $U_\psi$ and $V_\psi$ on the Triebel-Lizorkin-type spaces $\dot{F}^{\alpha,\tau}_{p,q}(\Bbb R^n)$ and their predual spaces, Triebel-Lizorkin-Hausdorff spaces, which unify and generalize the known results on $Q$-type spaces.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47B38 Operators on function spaces (general)
##### Keywords:
weighted Hardy operator; Triebel-Lizorkin-type spaces
Full Text:
##### References:
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