Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| author | Eriksson-Bique, Sylvester Soultanis, Elefterios |
| author_facet | Eriksson-Bique, Sylvester Soultanis, Elefterios |
| contents | We represent minimal upper gradients of Newtonian functions, in the range $1\le p<\infty$, 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. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_08097 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential Eriksson-Bique, Sylvester Soultanis, Elefterios Metric Geometry Functional Analysis Primary: 46E36, 49J52. Secondary: 53C23, 30L99, 26B05 We represent minimal upper gradients of Newtonian functions, in the range $1\le p<\infty$, 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. 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. |
| title | Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential |
| topic | Metric Geometry Functional Analysis Primary: 46E36, 49J52. Secondary: 53C23, 30L99, 26B05 |
| url | https://arxiv.org/abs/2102.08097 |