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Commutators of weighted Hardy operators on Herz-type spaces. (English) Zbl 1230.42023
The authors discuss the Morrey-Herz space boundedness of the commutator $U_\psi^b$ of the function multiplier $M_b$ and the weighted Hardy operator $U_\psi$ defined by $U_\psi f(x)=\int_{0}^{1}f(tx)\psi(t)\,dt$ $(x\in \Bbb R^n)$, where $\psi:[0,1)\to[0,\infty)$. When $\psi\equiv 1$ and $n=1$, this reduces to the classical Hardy operator $U: Uf(x)=x^{-1}\int_{0}^{x}f(t)\,dt$. They show that when $-\infty<\alpha<\infty$, $\lambda\ge0$, $0<p\le\infty$, $0<q_2\le q_1<\infty$, $b\in{\roman {Lip}}_\beta(\Bbb R^n)$ $(0<\beta<1)$ and $\int_{0}^{1}t^{-(\alpha+\beta+n/q_2-\lambda)}\psi(t) \,dt<\infty$, then the commutator $U_\psi^b$ is bounded from the homogeneous Morrey-Herz space $M\dot K_{p,q_1}^{\alpha+\beta+n/q_2-n/q_1,\lambda}(\Bbb R^n)$ to $M\dot K_{p,q_1}^{\alpha,\lambda}(\Bbb R^n)$. Here the homogeneous Morrey-Herz space $M\dot K_{p,q}^{\alpha,\lambda}(\Bbb R^n)$ is defined by $M\dot K_{p,q}^{\alpha,\lambda}(\Bbb R^n)=\{f\in L_{\roman {loc}}^q(\Bbb R^n \backslash \{0\}); \|f\|_{M\dot K_{p,q}^{\alpha,\lambda}(\Bbb R^n)}<\infty\}$, where $\|f\|_{M\dot K_{p,q}^{\alpha,\lambda}(\Bbb R^n)}=\sup_{k_0\in\Bbb Z} 2^{-k_0\lambda}\{\sum_{k=-\infty}^{k_0}2^{k\alpha p}\|f\chi_{\{2^{k-1}<|x|\le 2^{k}\}}\|_{L^q(\Bbb R^n)}^p\}^{1/p}$. This extends the $(L^p,L^q)$ result for the classical Hardy operator by {\it Z. W. Fu} [J. Beijing Norm. Univ., Nat. Sci. 42, No. 4, 342--345 (2006; Zbl 1135.42320)].

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 42B25 Maximal functions, Littlewood-Paley theory
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