On optimal parameters involved with two-weighted estimates of commutators of singular and fractional operators with Lipschitz symbols

Abstract

In this paper we prove two-weighted norm estimates for higher order commutatorof singular integral and fractional type operators between weighted Lp and certainspaces that include Lipschitz, BMO and Morrey spaces. We also give the optimalparameters involved with these results, where the optimality is understood in thesense that the parameters dening the corresponding spaces belong to certain regionout of which the classes of weights are satised by trivial weights. We also exhibitpairs of non-trivial weights in the optimal region satisfying the conditions required.