NECESSARY AND SUFFICIENT CONDITIONS FOR BOUNDEDNESS OF COMMUTATORS OF STRONGLY SINGULAR INTEGRAL OPERATORS WITH WEIGHTED LIPSCHITZ FUNCTIONS

In this paper, we prove the commutator T-b generated by the strongly singular integral operator T and the function b is bounded from L-p(w) to L-q(w(1- q)) if and only if b is an element of Lip(beta)(w), where w is an element of A(1), 0 < beta < 1, 1 < p < n/beta and 1/q = 1/p - beta/n. To do this, we first show a maximal function estimate for the commutator.