A measure zero UDS in the Heisenberg group
DOI:
https://doi.org/10.6092/issn.2240-2829/6692Parole chiave:
Universal differentiability sets, Heisenberg group, Pansu differentiabilityAbstract
Proveremo che ogni gruppo di Heisenberg contiene un insieme di misura nulla tale che ogni funzione lipschitziana ammette almeno un punto di Pansu differenziabilità al suo interno.
Riferimenti bibliografici
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.
Downloads
Pubblicato
Come citare
Fascicolo
Sezione
Licenza
Copyright (c) 2016 Andrea Pinamonti, Gareth Speight
I diritti d'autore e di pubblicazione di tutti i testi nella rivista appartengono ai rispettivi autori senza restrizioni.
La rivista è distribuita sotto una Creative Commons Attribution 4.0 International License (licenza completa).
Vedere inoltre la nostra Open Access Policy.