The coarea formula for real-valued Lipschitz maps on stratified groups. (English) Zbl 1079.49030
Summary: We establish a coarea formula for real-valued Lipschitz maps on stratified groups when the domain is endowed with a homogeneous distance and level sets are measured by the \(Q - 1\) dimensional spherical Hausdorff measure. The number \(Q\) is the Hausdorff dimension of the group with respect to its Carnot-Carathéodory distance. We construct a Lipschitz function on the Heisenberg group which is not approximately differentiable on a set of positive measure, provided that the Euclidean notion of differentiability is adopted. The coarea formula for stratified groups also applies to this function, where the Euclidean one clearly fails. This phenomenon shows that the coarea formula holds for the natural class of Lipschitz functions which arises from the geometry of the group and that this class may be strictly larger than the usual one.
MSC:
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
| 28A75 | Length, area, volume, other geometric measure theory |
| 22E25 | Nilpotent and solvable Lie groups |
| 28A80 | Fractals |
| 49K05 | Optimality conditions for free problems in one independent variable |
References:
| [1] | Ambrosio, Adv. Math. 159 pp 51– (2001) |
| [2] | , and , Functions of Bounded Variation and Free Discontinuity Problems (Oxford University Press, New York, 2000). · Zbl 0957.49001 |
| [3] | Ambrosio, Math. Ann. 318 pp 527– (2000) |
| [4] | and , Weak differentiability of BV functions on stratified groups, Math. Z., to appear. · Zbl 1048.49030 |
| [5] | Size of characteristic sets and functions with prescribed gradients, J. Reine Angew. Math., to appear. |
| [6] | and (eds.), Sub-Riemannian Geometry, Progress in Mathematics Vol. 144 (Birkhäuser, Basel, 1996). |
| [7] | Calderón, Studia Mat. 20 pp 179– (1961) |
| [8] | Capogna, Comm. Anal. Geom. 2 pp 203– (1994) · Zbl 0864.46018 · doi:10.4310/CAG.1994.v2.n2.a2 |
| [9] | and , Representation of Nilpotent Lie Groups and their Applications. Part 1: Basic Theory and Examples (Cambridge University Press, Cambridge, 1990). |
| [10] | , and , Non-doubling Ahlfors measures, Perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces, Mem. Amer. Math. Soc., to appear. |
| [11] | The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1986). |
| [12] | Geometric Measure Theory (Springer, New York, 1969). · Zbl 0176.00801 |
| [13] | and , Hardy Spaces on Homogeneous Groups (Princeton University Press, 1982). · Zbl 0508.42025 |
| [14] | Franchi, Houston J. Math. 22 pp 859– (1996) |
| [15] | Franchi, Math. Ann. 321 pp 479– (2001) |
| [16] | , and , Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom., to appear. · Zbl 1077.22008 |
| [17] | , and , On the structure of finite perimeter sets in step 2 Carnot groups, in preparation. |
| [18] | Garofalo, Comm. Pure Appl. Math. 49 pp 1081– (1996) |
| [19] | Metric structures for Riemannian and non-Riemannian spaces, with appendices by M. Katz, P. Pansu, and S. Semmes, Progress in Mathematics, edited by Lafontaine and Pansu (Birkäuser, Boston, 1999). |
| [20] | and , Sobolev met Poincaré, Memoirs of the American Mathematical Society Vol. 145, No. 688 (Amer. Math. Soc., Providence, RI, 2000). |
| [21] | Heinonen, Ber. Univ. Jyväskylä Math. Inst. 68 pp 1– (1995) |
| [22] | and , Rectifiability and parametrization of intrinsic regular surfaces in the Heisenberg group, in preparation. |
| [23] | and , Isoperimetric sets on Carnot groups, Houston J. Math., to appear. · Zbl 1039.49037 |
| [24] | Magnani, Houston J. Math. 27 pp 297– (2001) |
| [25] | Magnani, Ann. Acad. Sci. Fenn. Math. 27 pp 121– (2002) |
| [26] | Magnani, Manuscripta Math. 110 pp 55– (2003) |
| [27] | An area formula in metric spaces, Scuola Normale Superiore di Pisa, preprint No. 19 (October, 2002). |
| [28] | Malý, Trans. Amer. Math. Soc. 355 pp 477– (2003) |
| [29] | Functions of bounded variation on ”good” metric measure spaces, in preparation. |
| [30] | A Tour of Subriemannian Geometries, their Geodesics and Applications (Amer. Math. Soc., Providence, RI, 2002). · Zbl 1044.53022 |
| [31] | Monti, Calc. Var. Partial Differential Equations 13 pp 339– (2001) |
| [32] | Pansu, C. R. Acad. Sci. Paris 295 pp 127– (1982) |
| [33] | Pansu, Ann. of Math. (2) 129 pp 1– (1989) |
| [34] | Pauls, Comm. Anal. Geom. 9 pp 951– (2001) · Zbl 1005.53033 · doi:10.4310/CAG.2001.v9.n5.a2 |
| [35] | A notion of rectifiability modelled on Carnot groups, in preparation. |
| [36] | Minimal surfaces in the Heisenberg group, Geom. Dedicata, to appear. · Zbl 1054.49029 |
| [37] | Semmes, Rev. Mat. Iberoamericana 12 pp 337– (1996) · Zbl 0858.46017 · doi:10.4171/RMI/201 |
| [38] | Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, N. J., 1970). · Zbl 0207.13501 |
| [39] | , and , Analysis and Geometry on Groups (Cambridge University Press, Cambridge, 1992). |
| [40] | P -Differentiability on Carnot groups in different topologies and related topics, in: Proc. Anal. Geom. (Sobolev Institute Press, Novosibirsk, 2000), pp. 603-670. · Zbl 0992.58005 |
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