Differentiability and area formula on stratified Lie groups (Q2744631)

The Area Formula NEWLINE\[NEWLINE\int_AJ_Q(d_xf) d{\mathcal H}_d^Q(x) = \int_{\mathbb P}N(f,A,y) d{\mathcal H}_\rho^Q(y)NEWLINE\]NEWLINE for Lipschitz maps \(f : A \subseteq {\mathbb G} \to {\mathbb P}\) is proved, where \({\mathbb G}\) and \({\mathbb P}\) are stratified nilpotent Lie groups and \(A\) is a measurable subset of \({\mathbb G}\). The symbols \({\mathcal H}_d^Q\) and \({\mathcal H}_\rho^Q\) denote the \(Q\)-Hausdorff measures defined on the metric spaces \(({\mathbb G},d)\) and \(({\mathbb P}, \rho)\), respectively, where \(Q=\sum_{i=1}^ni\dim V_i\) and the Lie algebra \(\mathcal G\) of \(\mathbb G\) of nilpotency degree \(n\) is the direct sum of the vector spaces \(V_i\) satisfying \([V_i,V_1]=V_{i+1}\). The symbol \(J_Q(d_xf)\) denotes the Jacobian of the differential \(d_xf\) and is defined as \(J_Q(d_xf)={\mathcal H}_\rho^Q(d_xf(B_1)) / {\mathcal H}_d^Q(B_1)\), where \(B_1\) denotes the unit metric ball of \({\mathbb G}\). The multiplicity function \(N\) of \(f\) relative to \(A\) is given by \(N(f,A,y)=\#(f^{-1}(y)\cap A) \in {\mathbb N} \cup \{\infty\}\). In order to prove the Area Formula the author extends a result of \textit{P. Pansu} [Ann. Math. (2) 129, 1-60 (1989; Zbl 0678.53042)] about the differentiability of Lipschitz maps to the case of measurable domains (covered in Section 3 of the paper). The quite elaborate proof of this extension is the key part of the paper, with Theorem 3.9 being the main result: Any Lipschitz map \(f : A \to {\mathbb P}\), defined on any measurable subset \(A \subseteq {\mathbb G}\) is differentiable \({\mathcal H}_d^Q\)-almost everywhere. Using this theorem the proof of the Area Formula is straightforward.