Abstract
In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure \(\mu \). We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at \(\mu \)-almost every point x in any direction which is not contained in the decomposability bundle \(V(\mu ,x)\), recently introduced by Alberti and the first author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for a closed set of points of arbitrarily small measure. Here a function is an admissible blowup at a point x if it is null at the origin and it is the sum of a linear function on \(V(\mu ,x)\) and a Lipschitz function on \(V(\mu ,x)^{\perp }\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alberti, G.: A Lusin type theorem for gradients. J. Funct. Anal. 100(1), 110–118 (1991)
Alberti, G., Marchese, A.: On the differentiability of Lipschitz functions with respect to measures in the Euclidean space. Geom. Funct. Anal. 26(1), 1–66 (2016)
David, G.C.: Lusin-type theorems for Cheeger derivatives on metric measure spaces. Anal. Geom. Metr. Spaces 3, 296–312 (2015)
De Philippis, G., Rindler, F.: On the structure of \({\mathscr {A}}\)-free measures and applications. Ann. Math. 184, 1017–1039 (2016)
Garnett, J., Killip, R., Schul, R.: A doubling measure on \({\mathbb{R}}^d\) can charge a rectifiable curve. Proc. Amer. Math. Soc. 138(5), 1673–1679 (2010)
Marchese, A.: Lusin type theorems for Radon measures. Rend. Semin. Mat. Univ. Padova 138, 193–207 (2017)
Moonens, L., Pfeffer, W.F.: The multidimensional Luzin theorem. J. Math. Anal. Appl. 339(1), 746–752 (2008)
Preiss, D.: Geometry of measures in \({\bf R}^n\): distribution, rectifiability, and densities. Ann. Math. 125(3), 537–643 (1987)
Schioppa, A.: Derivations and Alberti representations. Adv. Math. 293, 436–528 (2016)
Schioppa, A.: The Lip-lip equality is stable under blow-up. Calc. Var. Partial Differ. Equ. 55(1), 22–30 (2016)
Acknowledgements
The authors would like to thank Giovanni Alberti for several discussions. A. M. was supported by the ERC-grant “Regularity of area-minimizing currents” (306247). A.S. was supported by the “ETH Zurich Postdoctoral Fellowship Program and the Marie Curie Actions for People COFUND Program”.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
