VARIATION INEQUALITIES FOR THE COMMUTATORS OF APPROXIMATE IDENTITIES WITH LIPSCHITZ FUNCTIONS

This paper is devoted to establishing the boundedness of the variation operators for commutators generated by approximate identities with Lipschitz functions in the weighted Lebesgue spaces and the endpoint spaces. As applications, we obtain the corresponding boundedness results for $ \lambda $-jump operator, the number of up-crossing, heat semigroups, Poisson semigroups and maximal operator.