Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions

in this paper we show that b epsilon Lip(beta,mu) if and only if the commutator [b, T] of the multiplication operator by b and the singular integral operator T is bounded from L-p (mu) to L-q (mu(1-q)), where 1 < p < q < infinity, 0 < beta < 1 and 1/q = 1/p = 1/p - beta/n. Also we will obtain that b epsilon Lip(beta,mu) if and only if the commutator [b, I-alpha] of the multiplication operator by b and the fractional integral operator I-alpha is bounded from L-p (mu) to L-r (mu(1-(1-alpha/n)r)), where 1 < p < infinity, 0 < beta < 1 and 1/r = 1/p - (beta + alpha)/n with 1/p > (beta + alpha)/n. (c) 2007 Elsevier Inc. All rights reserved.