Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]