Note on coarea formulae in the Heisenberg group
type: Item
instance of: scholarly article
title: Note on coarea formulae in the Heisenberg group
zbMATH Open document ID: 1062.22020
DOI: 10.5565/PUBLMAT_48204_07
author: Valentino Magnani
published in: Publicacions Matemàtiques
publication date: 2005-03-14T00:00:00Z
full work available at URL: https://eudml.org/doc/41509
review text: 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)].
reviewed by: Hidenori Fujiwara
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 2145403
zbMATH Keywords: coarea formula
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Carnot-Carathéodory space
MaRDI profile type: Publication
OpenAlex ID: W2060124736
description: scientific article; zbMATH DE number 2145403