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Zbl 1206.42027
Fu, Zunwei; Lu, Shanzhen
Weighted Hardy operators and commutators on Morrey spaces.
(English)
[J] Front. Math. China 5, No. 3, 531-539 (2010). ISSN 1673-3452; ISSN 1673-3576/e

The authors discuss the Morrey space boundedness of the weighted Hardy operators $U_\psi$ 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 $1<q<\infty$ and $-1/q<\lambda<0$, $U_\psi$ is bounded on the Morrey space $L^{q,\lambda}(\Bbb R^n)$ if and only if $\int_{0}^{1}t^{n\lambda }\psi(t)\,dt<\infty$, and $\|U_\psi\|_{\text {op}}=\int_{0}^{1}t^{n\lambda }\psi(t)\,dt$. They also characterize those $\psi$ for which the commutators of $U_\psi$ and the function multipliers $M_b$ are bounded on $L^{q,\lambda}(\Bbb R^n)$ for all $\text {BMO}(\Bbb R^n)$ functions $b$. They give the same results for the Cesàro operators which are adjoint to $U_\psi$, too. Their results generalize the corresponding ones in $L^q(\Bbb R^n)$ spaces.
[Kôzô Yabuta (Nishinomiya)]
MSC 2000:
*42B99 Fourier analysis in several variables
26D15 Inequalities for sums, series and integrals of real functions
42B25 Maximal functions

Keywords: weighted Hardy operators; Morrey spaces; commutators; BMO

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