A Measure Zero UDS in the Heisenberg Group


  • Andrea Pinamonti Università degli Studi di Trento
  • Gareth Speight University of Cincinnati




Universal differentiability sets, Heisenberg group, Pansu differentiability


We show that the Heisenberg group contains a measure zero set N such that every real-valued Lipschitz function is Pansu differentiable at a point of N.


Alberti, G., Csornyei, M., Preiss, D.: Differentiability of Lipschitz functions, structure of null sets, and other problems, Proc. Int. Congress Math. III (2010), 1379-1394.

Bate, D.: Structure of measures in Lipschitz differentiability spaces, J. Amer. Math. Soc. 28 (2015), 421-482.

Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics (2007).

Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9(3) (1999), 428-517.

Capogna, L., Danielli, D., Pauls, S., Tyson, J.: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Birkhauser Progress in Mathematics 259 (2007).

Csornyei, M., Jones, P.: Product formulas for measures and applications to analysis and geometry, announced result: http://www.math.sunysb.edu/Videos/dfest/PDFs/38-Jones.pdf.

De Philippis, G., Rindler, F.: On the structure of A-free measures and applications, Annals of Math. ArXiv: 1601.06543.

Doré M., Maleva O.: A compact null set containing a Differentiability point of every Lipschitz function, Math. Ann. 351(3) (2011), 633-663.

Doré, M., Maleva, O.: A compact universal differentiability set with Hausdorff dimension one, Israel J. Math. 191(2) (2012), 889-900.

Dymond, M., Maleva, O.: Differentiability inside sets with upper Minkowski dimension one, preprint available at arXiv:1305.3154.

Fitzpatrick, S.: Differentiation of real-valued functions and continuity of metric projections, Proc. Amer. Math. Soc. 91(4) (1984), 544-548.

Gromov, M.: Carnot-Carath_eodory spaces seen from within, Progress in Mathematics 144 (1996), 79-323.

Lindenstrauss, J., Preiss, D., Tiser, J.: Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces, Annals of Mathematics Studies 179, Princeton University Press (2012).

Magnani, V.: Towards differential calculus in stratified groups, J. Aust. Math. Soc. 95(1) (2013), 76-128.

Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications, American Mathematical Society, Mathematical Surveys and Monographs 91 (2006).

Pansu, P.: Metriques de Carnot-Carathéodory et quasiisometries des espaces symetriques de rang un, Annals of Mathematics 129(1) (1989), 1-60.

Preiss, D.: Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91(2) (1990), 312-345.

Preiss, D., Speight, G.: Differentiability of Lipschitz functions in Lebesgue null sets, Inven. Math. 199(2) (2015), 517-559.

Pinamonti, A., Speight, G.: A measure zero universal differentiability set in the Heisenberg group, to appear in Mathematische Annalen, DOI: 10.1007/s00208-016-1434-x, preprint available at arXiv:1505.07986.

Zahorski, Z.: Sur l'ensemble des points de non-derivabilite d'une fonction continue, Bull. Soc. Math. France 74 (1946), 147-178.




How to Cite

Pinamonti, A., & Speight, G. (2017). A Measure Zero UDS in the Heisenberg Group. Bruno Pini Mathematical Analysis Seminar, 7(1), 85–96. https://doi.org/10.6092/issn.2240-2829/6692