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We give a partial positive answer to the open problem proposed in Wang et al. (Acta Math Sin Ser A 35:1106–1114, 2015), that is, we characterize the BMO space via the boundedness of iterated commutator of bilinear fractional integral operator $$[\Pi \vec {b},I_{\alpha }]$$ [ Π b → , I α ] . Moreover, it is showed that the symbol b belongs to CMO, the closure in $$\mathrm{BMO}$$ BMO of the space of $$C^{\infty }$$ C ∞ functions with compact support, if and only if the commutator $$[\Pi \vec {b},I_{\alpha }]$$ [ Π b → , I α ] is a compact operator with $$\vec {b}=(b,b)$$ b → = ( b , b ) . On the other hand, Bényi et al. (Math Z 208:569–582, 2015) obtained the separate compactness for commutators of the class $$B_{\alpha }$$ B α , when $$b\in \mathrm{CMO}$$ b ∈ CMO . In this paper, it is proved that $$b\in \mathrm{CMO}$$ b ∈ CMO is necessary for $$[b,B_{\alpha }]_{i}(i=1,2)$$ [ b , B α ] i ( i = 1 , 2 ) is a compact operator.
Analysis and Mathematical Physics – Springer Journals
Published: Nov 1, 2018
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