Note on coarea formulae in the Heisenberg group
Let \({\mathcal H}^3\) be the 3-dimensional Heisenberg group. The author proves the following coarea formula. Let \(A\to \mathbb{R}^2\) be a Lipschitz map, where \(A\subset{\mathcal H}^3\) is a measurable subset. Then for every measurable function \(u: A\to [0,+\infty]\) the formula \[ \int_A u(x) J_Hf(x)\,dx= \int_{\mathbb{R}^2} \Biggl(\int_{f^{-1}(t)\cap A} u(y)\,d{\mathcal S}^2_{{\mathcal H}^3}(y)\Biggr)\,dt \] holds, where \(J_Hf\) is the horizontal Jacobian of \(f\), i.e., the Jacobian of the matrix representing the Pansu differential [cf. \textit{P. Pansu}, Ann. Math. (2) 129, No. 1, 1--60 (1989; Zbl 0678.53042)], and where \({\mathcal S}^2_{{\mathcal H}^3}\) is the spherical Hausdorff measure with respect to the Carnot-Carathéodory distance [cf. \textit{H. Federer}, Geometric measure theory (Die Grundlagen der mathematischen Wissenschaften in Einzeldarstellungen 153, Berlin: Springer-Verlag) (1969; Zbl 0176.00801)].
scientific article; zbMATH DE number 2145403