Characterization of Lipschitz Functions via the Commutators of Singular and Fractional Integral Operators in Variable Lebesgue Spaces

We obtain characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of Caldern-Zygmund and fractional type operators in the context of the variable exponent Lebesgue spaces L (p(ai...)), where the symbols of the commutators belong to the Lipschitz spaces. A useful tool is a pointwise estimate involving the sharp maximal operator of the commutator and certain associated maximal operators, which is new even in the classical context. Some boundedness properties of the commutators between Lebesgue and Lipschitz spaces in the variable context are also proved.