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Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential

Sylvester Eriksson-Bique and Elefterios Soultanis

Vol. 17 (2024), No. 2, 455–498
Abstract

We represent minimal upper gradients of Newtonian functions, in the range 1 p < , by maximal directional derivatives along “generic” curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of p-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along p-almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules.

The arising p-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger’s structure in the presence of a Poincaré inequality. In particular, it exists whenever the space is metrically doubling. It is moreover compatible with, and gives a geometric interpretation of, Gigli’s abstract differentiable structure, whenever it exists. The p-weak charts give rise to a finite-dimensional p-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.

Keywords
Sobolev, test plan, minimal upper gradient, differential structure, differential, chart
Mathematical Subject Classification
Primary: 46E36, 49J52
Secondary: 26B05, 30L99, 53C23
Milestones
Received: 2 June 2021
Revised: 11 May 2022
Accepted: 11 July 2022
Published: 6 March 2024
Authors
Sylvester Eriksson-Bique
Research Unit of Mathematical Sciences
University of Oulu
Oulu
Finland
Elefterios Soultanis
Department of Mathematics and Statistics
University of Jyväskylä
Jyväskylä
Finland

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