On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

By Giovanni S. Alberti, Andrea Marchese. Geometric and Functional Analysis, 2014

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For every finite measure μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document} on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{R}}^n}$$\end{document} we define a decomposability bundle V(μ,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V(\mu,\,\cdot)}$$\end{document} related to the decompositions of μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document} in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{R}}^n}$$\end{document} is differentiable at μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document}-a.e. x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x}$$\end{document} with respect to the subspace V(μ,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V(\mu,\,x)}$$\end{document}, and prove that this differentiability result is optimal, in the sense that, following (Alberti et al., Structure of null sets, differentiability of Lipschitz functions, and other problems, 2016), we can construct Lipschitz functions which are not differentiable at μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu}$$\end{document}-a.e. x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x}$$\end{document} in any direction which is not in V(μ,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V(\mu,\,x)}$$\end{document}. As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k}$$\end{document}-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C^1}$$\end{document} to Lipschitz maps.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/caeb16079fab3f4ee846c1c777a86cf621f1ab7d

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Citations in the corpus

Behaviour of the reference measure on $\sf RCD$ spaces under charts (2016, background): “1, without the need of using also the deep results in [5], [4] and [1].”
Dimensional estimates and rectifiability for measures satisfying linear PDE constraints (2018, background): “1 with the result of [2], one obtains the weak converse of Rademacher’s theorem (see [14, 19, 21] for other consequences in metric geometry).”
On the volume measure of non-smooth spaces with Ricci curvature bounded below (2016, background): “Let us mention that the aforementioned statement will also follow by a stronger result announced by Csörnyei and Jones [17], namely that for every Lebesgue null set E ⊂ R there exists a Lipschitz map f : R → R which is no-where differentiable; see also the discussion in the introduction of [2] for a”
On a conjecture of Cheeger (2016, result): “When combined with the main result of [AM16], this proves the weak converse of Rademacher’s theorem in any dimension, see [DR16, Theorem 1.”
On the structure of A-free measures and applications (2016, methodology): “By relying on the results of Alberti and Marchese [4] and of Schioppa [40], the latter result immediately implies that the Rademacher theorem can hold only for absolutely continuous measures and that every top-dimensional Ambrosio–Kirchheim metric current in R d is a Federer–Fleming ﬂat chain (a par”