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.
@article{bwmeta1.element.doi-10_1515_agms-2015-0017,
author = {Guy C. David},
title = {Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces},
journal = {Analysis and Geometry in Metric Spaces},
volume = {3},
year = {2015},
zbl = {1325.26030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_agms-2015-0017}
}
Guy C. David. Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces. Analysis and Geometry in Metric Spaces, Tome 3 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_agms-2015-0017/
[1] G. Alberti, A Lusin type theorem for gradients, J. Funct. Anal., 100 (1991), 110-118. | Zbl 0752.46025
[2] Z.M. Balogh, Size of characteristic sets and functions with prescribed gradient, J. Reine Angew. Math., 564 (2003), 63-83. | Zbl 1051.53024
[3] D. Bate. Structure of measures in Lipschitz differentiability spaces. J. Amer. Math. Soc. 28 (2015), no. 2, 421-482. | Zbl 1307.30097
[4] D. Bate, G. Speight, Differentiability, porosity and doubling in metric measure spaces, Proc. Amer. Math. Soc., 141 (2013), 971-985. | Zbl 1272.30083
[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. | Zbl 0924.30030
[6] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517. | Zbl 0942.58018
[7] J. Cheeger, B. Kleiner, Inverse limit spaces satisfying a Poincaré inequality, Anal. Geom. Metr. Spaces 3 (2015), 15–39 | Zbl 1331.46016
[8] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61 (1990), 601-628. | Zbl 0758.42009
[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] | Zbl 1275.46018
[10] G. Francos, The Luzin theorem for higher-order derivatives, Michigan Math. J., 61 (2012), 507-516. | Zbl 1256.28006
[11] P. Hajłasz, J. Mirra, The Lusin theorem and horizontal graphs in the Heisenberg group. Anal. Geom. Metr. Spaces, 1 (2013), 295-301. | Zbl 1286.46036
[12] J. Heinonen, P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181 (1998), 1-61.. | Zbl 0915.30018
[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. | Zbl 1332.46001
[14] S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z., 245 (2003), 255-292. [WoS] | Zbl 1037.31009
[15] S. Keith, A differentiable structure for metric measure spaces. Adv. Math., 183 (2004), 271-315. | Zbl 1077.46027
[16] B. Kleiner, J. Mackay, Differentiable structures on metric measure spaces: A primer, preprint (2011). To appear, Annali SNS. arXiv:1108.1324. | Zbl 1339.30001
[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. | Zbl 0962.30006
[19] N. Lusin, Sur la notion de l’intégrale, Ann. Mat. Pura Appl., 26 (1917), 77-129. | Zbl 46.0391.01
[20] L. Moonens, W.F. Pfeffer, The multidimensional Luzin theorem, J. Math. Anal. Appl., 339 (2008), 746-752. | Zbl 1141.28008
[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] | Zbl 0870.54031