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Tytuł artykułu

Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A theorem of Lusin states that every Borel function onRis equal almost everywhere to the derivative of a continuous function. This result was later generalized to Rn in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of differentiation by a famous theorem of Cheeger.
Wydawca
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2015-02-05
zaakceptowano
2015-09-15
online
2015-10-15
Twórcy
autor
  • Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012
Bibliografia
  • [1] G. Alberti, A Lusin type theorem for gradients, J. Funct. Anal., 100 (1991), 110-118.
  • [2] Z.M. Balogh, Size of characteristic sets and functions with prescribed gradient, J. Reine Angew. Math., 564 (2003), 63-83.
  • [3] D. Bate. Structure of measures in Lipschitz differentiability spaces. J. Amer. Math. Soc. 28 (2015), no. 2, 421-482.
  • [4] D. Bate, G. Speight, Differentiability, porosity and doubling in metric measure spaces, Proc. Amer. Math. Soc., 141 (2013), 971-985.
  • [5] M. Bourdon, H. Pajot, Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings, Proc. Amer. Math. Soc., 127 (1999), 2315-2324.
  • [6] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517.
  • [7] J. Cheeger, B. Kleiner, Inverse limit spaces satisfying a Poincaré inequality, Anal. Geom. Metr. Spaces 3 (2015), 15–39
  • [8] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61 (1990), 601-628.
  • [9] E. Durand-Cartagena, J.A. Jaramillo, N. Shanmugalingam, The 1-Poincaré inequality in metric measure spaces, Michigan Math. J., 61 (2012), 63-85. [WoS]
  • [10] G. Francos, The Luzin theorem for higher-order derivatives, Michigan Math. J., 61 (2012), 507-516.
  • [11] P. Hajłasz, J. Mirra, The Lusin theorem and horizontal graphs in the Heisenberg group. Anal. Geom. Metr. Spaces, 1 (2013), 295-301.
  • [12] J. Heinonen, P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181 (1998), 1-61..
  • [13] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson. Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. New Mathematical Monographs: 27. Cambridge University Press, Cambridge. 2015.
  • [14] S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z., 245 (2003), 255-292. [WoS]
  • [15] S. Keith, A differentiable structure for metric measure spaces. Adv. Math., 183 (2004), 271-315.
  • [16] B. Kleiner, J. Mackay, Differentiable structures on metric measure spaces: A primer, preprint (2011). To appear, Annali SNS. arXiv:1108.1324.
  • [17] B. Kleiner, A. Schioppa, PI spaces with analytic dimension 1 and arbitrary topological dimension, preprint (2015). arXiv:1504.06646.
  • [18] T. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality, Geom. Funct. Anal., 10 (2000), 111-123.
  • [19] N. Lusin, Sur la notion de l’intégrale, Ann. Mat. Pura Appl., 26 (1917), 77-129.
  • [20] L. Moonens, W.F. Pfeffer, The multidimensional Luzin theorem, J. Math. Anal. Appl., 339 (2008), 746-752.
  • [21] S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. (N.S.), 2(2) (1996), 155-295. [Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_agms-2015-0017
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