Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces

Details Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces

2012-03-20

arxiv.org/abs/1203.4337v1

Mathematics

Functional Analysis

12 pages; Classical Analysis and ODEs (math.CA), Functional Analysis (math.FA)

Scientific paper

We prove that $b$ is in $Lip_{\bz}(\bz)$ if and only if the commutator $[b,L^{-\alpha/2}]$ of the multiplication operator by $b$ and the general fractional integral operator $L^{-\alpha/2}$ is bounded from the weighed Morrey space $L^{p,k}(\omega)$ to $L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$, where $0<\beta<1$, $0<\alpha+\beta \frac{1-k}{p/q-k},$ and here $r_\omega$ denotes the critical index of $\omega$ for the reverse H\"{o}lder condition.

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