Towards Differential Calculus in stratified groups

Abstract

In this paper we establish the basic tools to develop the "Calculus" associated with group-valued continuously Pansu differentiable mappings. We develop the technical machinery on which all of our results rely. In particular, the linearization of addends appearing in the Baker-Campbell-Hausdorff formula is one of the major tools in several proofs. This allows us to characterize graded group-valued continuously Pansu differentiable mappings through a system of nonlinear first order PDEs, namely, the "contact equations". Through this approach we establish a quantitative estimate for the Pansu difference quotient. This key result is crucial to get the mean value inequality, the inverse mapping theorem, the rank theorem and the implicit function theorem for graded group-valued continuously Pansu differentiable mappings. All of these results represent the backbone to develop a reasonable "Calculus" in this context. On the other hand, these theorems rise the question of possible factorizations of stratified groups into inner semidirect products. In fact, we consider this algebraic issue to show that image-sets and level-sets of suitable continuously Pansu differentiable mappings are intrinsic graphs. These sets represent the intrinsic regular submanifolds modeled on a couple of fixed groups. Notice that the metric properties of these sets are completely different from the ones of classical submanifolds.