The Commutators of Fractional Integrals on Besov Spaces

In this paper we consider the commutators of fractional integrals Rnb(x)b(y)|xy|nf(y)dy on the Besov spaces \ifmmode\expandafter\else\expandafter\.\fiBs,qp , where b is a locally integrable function and 0 > > n. We first establish the equivalence between the boundedness of the commutators and the paraproduct of J. M. Bony. Then we obtain two conditions on the boundedness of the commutators. One of these conditions is necessary and the other is sufficient.