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
$0<\beta<1$, $0<\alpha+\beta<n, 1<p<{n}/({\alpha+\beta})$,
${1}/{q}={1}/{p}-{(\alpha+\beta)}/{n},$ $0\leq k<{p}/{q},$ $\omega^{{q}/{p}}\in
A_1$ and $ r_\omega> \frac{1-k}{p/q-k},$ and here $r_\omega$ denotes the
critical index of $\omega$ for the reverse H\"{o}lder condition.Comment: 12 pages; Classical Analysis and ODEs (math.CA), Functional Analysis
  (math.FA

Si, Zengyan

Zhao, Fayou

English

arXiv

We prove thatbis inLipβ(ω)if and only if the commutator[b,L-α/2]of the multiplication operator byband the general fractional integral operatorL-α/2is bounded from the weighted Morrey spaceLp,k(ω)toLq,kq/p(ω1-(1-α/n)q,ω), where0&lt;β&lt;1,0&lt;α+β&lt;n,1&lt;p&lt;n/(α+β),1/q=1/p-(α+β)/n,0≤k&lt;p/q,ωq/p∈A1,andrω&gt;(1-k)/(p/(q-k)), and hererωdenotes the critical index ofωfor the reverse Hölder condition.</jats:p

Zengyan Si

Fayou Zhao

Crossref

Abstract and Applied Analysis

Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

We prove that b is in Lipβ(ω) if and only if the commutator [b,L-α/2] of the multiplication operator by b and the general fractional integral operator L-α/2 is bounded from the weighted Morrey space Lp,k(ω) to Lq,kq/p(ω1-(1-α/n)q,ω), where 0(1-k)/(p/(q-k)), and here rω denotes the critical index of ω for the reverse Hölder condition

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

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