NONCONSERVATIVE PRODUCTS IN BOUNDED VARIATION FUNCTIONS

There exist two definitions of products of a bounded variation function by a derivative of another bounded variation function. One of them follows from a concept of generalized functions in which arbitrary products of distributions make sense: one has only one product but its understanding involves a nonclassical concept contained in each generalized function- Another one has been recently introduced by Dal Maso, Le Floch, and Murat as a generalization of a definition of Volpert; one has a family of different products indexed by a "path-phi"; each phi-product is well defined as a measure, and the scenario takes place in the familiar framework of functions of bounded variation and measures. In spite of their apparent great difference, these products are closely related: the purpose of this paper is to prove a clear correspondence between the two approaches: the path-phi is the nonclassical ingredient inherent in the concept of generalized functions.