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Curvewise
characterizations of minimal upper gradients and the
construction of a Sobolev differential
Sylvester Eriksson-Bique and Elefterios Soultanis |
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Vol. 17 (2024), No. 2, 455–498
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Abstract |
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We represent minimal upper gradients of Newtonian functions, in the range , 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 -weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along -almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules. The arising -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 -weak charts give rise to a finite-dimensional -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
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Mathematical Subject Classification
Primary: 46E36, 49J52
Secondary: 26B05, 30L99, 53C23
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Milestones
Received: 2 June 2021
Revised: 11 May 2022
Accepted: 11 July 2022
Published: 6 March 2024
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Authors |
| Sylvester Eriksson-Bique | |
| Research Unit of Mathematical
Sciences University of Oulu Oulu Finland |
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| Elefterios Soultanis | |
| Department of Mathematics and
Statistics University of Jyväskylä Jyväskylä Finland |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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