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Characterization of CMO via compactness of the commutators of bilinear fractional integral operators

Characterization of CMO via compactness of the commutators of bilinear fractional integral operators 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Characterization of CMO via compactness of the commutators of bilinear fractional integral operators

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Nature Switzerland AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-018-0264-2
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Nov 1, 2018

References