Differentiability of maps of Carnot groups of Sobolev classes

The $\mathscr P$-differentiability in the topology of the Sobolev space of weakly contact maps of Carnot groups is proved. The $\mathscr P$-differentiability in the sense of Pansu of contact maps in the class $W_p^1$, $p>\nu$, and other results are established as consequences. The method of proof is new even in the case of a Euclidean space and yields, for instance, a new proof of well-known results of Reshetnyak and Calderon–Zygmund on the differentiability of functions of Sobolev classes. In addition, a new proof of Lusin's condition $\mathscr N$ is given for quasimonotone maps in the class $W_\nu^1$. As a consequence, change-of-variables formulae are obtained for maps of Carnot groups.