Abstract
We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a \(\sigma \)-porous set. The second result states that irregular points of a Lipschitz function form a \(\sigma \)-porous set. We use these observations to give a new proof of Pansu’s theorem for Lipschitz maps from a general Carnot group to a Euclidean space.
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References
Agrachev, A., Barilari, D., Boscain, U.: Introduction to Riemannian and Sub-Riemannian Geometry (from Hamiltonian Viewpoint). http://webusers.imj-prg.fr/davide.barilari/
Bellaiche, A.: The tangent space in sub-Riemannian geometry. J. Math. Sci. 83(4), 461–476 (1994)
Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for their Sub-Laplacians. Springer Monographs in Mathematics, vol. 26. Springer, New York (2007)
Citti, G., Manfredini, M., Pinamonti, A., Serra Cassano, F.: Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group. Calc. Var. Partial Differ. Equ. 49(3–4), 1279–1308 (2014)
Citti, G., Manfredini, M., Pinamonti, A., Serra Cassano, F.: Poincaré-type inequality for Lipschitz continuous vector fields. J. Math. Pures Appl. (9) 105(3), 265–292 (2016)
Sarti, A., Citti, G., Petitot, J.: The symplectic structure of the primary visual cortex. Biol. Cybern. 98(1), 33–48 (2008)
Capogna, L., Danielli, D., Pauls, S., Tyson, J.: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Progress in Mathematics, vol. 259. Birkhauser, Basel (2007)
Franchi, B., Serapioni, R.: Intrinsic Lipschitz graphs within Carnot groups. J. Geom. Anal. 26(3), 1946–1994 (2016)
Franchi, B., Serapioni, R., Serra Cassano, F.: Differentiability of intrinsic Lipschitz functions within Heisenberg groups. J. Geom. Anal. 21(4), 1044–1084 (2011)
Franchi, B., Serapioni, R., Serra Cassano, F.: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321, 479–531 (2001)
Gromov, M.: Carnot-Caratheodory spaces seen from within. Prog. Math. 144, 79–323 (1996)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001)
Lindenstrauss, J., Preiss, D.: On Fréchet differentiability of Lipschitz maps between Banach spaces. Ann. Math. 157, 257–288 (2003)
Lindenstrauss, J., Preiss, D., Tiser, J.: Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces. Annals of Mathematics Studies, vol. 179. Princeton University Press, Princeton (2012)
Magnani, V.: Differentiability and area formula on stratified Lie groups. Houst. J. Math. 27(2), 297–323 (2001)
Magnani, V.: Towards differential calculus in stratified groups. J. Aust. Math. Soc. 95(1), 76–128 (2013)
Montgomery, R.: A Tour of Sub-Riemannian Geometries, their Geodesics and Applications. Mathematical Surveys and Monographs, vol. 91. American Mathematical Society, Providence (2006)
Pansu, P.: Metriques de Carnot-Carathéodory et quasiisometries des espaces symetriques de rang un. Ann. Math. 129(1), 1–60 (1989)
Preiss, D., Speight, G.: Differentiability of Lipschitz functions in Lebesgue null sets. Invent. Math. 199(2), 517–559 (2015)
Pinamonti, A., Speight, G.: A measure zero universal differentiability set in the Heisenberg group. Math. Ann. (2016). doi:10.1007/s00208-016-1434-x
Preiss, D., Zajicek, L.: Directional derivatives of Lipschitz functions. Israel J. Math. 125, 1–27 (2001)
Semmes, S.: On the nonexistence of bi-Lipschitz parameterizations and geometric problems about \(A_{\infty }\)-weights. Revista Matematica Iberoamericana 12(2), 337–410 (1996)
Serra Cassano, F.: Some Topics of Geometric Measure Theory in Carnot Groups. EMS Series of Lectures in Mathematics, vol. I. European Mathematical Society (EMS), Zürich. To appear
Serra Cassano, F., Vittone, D.: Graphs of bounded variation, existence and local boundedness of non-parametric minimal surfaces in the Heisenberg group. Adv. Calc. Var. 7(4), 409–492 (2014)
Vittone, D.: The Regularity Problem for Sub-Riemannian Geodesics. CRM Series, vol. 17, pp. 193–226. Ed. Norm., Pisa (2014). http://cvgmt.sns.it/paper/2416/
Zajicek, L.: Porosity and \(\sigma \)-porosity. Real Anal. Exch. 13(2), 314–350 (1987/1988)
Zajicek, L.: On \(\sigma \)-porous sets in abstract spaces. Abstr. Appl. Anal. 5, 509–534 (2005)
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The authors thank the referee for his/her useful comments and suggestions. The authors also thank Enrico Le Donne and Valentino Magnani for interesting discussions and suggestions.
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Pinamonti, A., Speight, G. Porosity, Differentiability and Pansu’s Theorem. J Geom Anal 27, 2055–2080 (2017). https://doi.org/10.1007/s12220-016-9751-6
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DOI: https://doi.org/10.1007/s12220-016-9751-6