Fragment-wise differentiable structures
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913236301905920 |
|---|---|
| author | Bate, David Eriksson-Bique, Sylvester Soultanis, Elefterios |
| author_facet | Bate, David Eriksson-Bique, Sylvester Soultanis, Elefterios |
| contents | The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this paper is to give an analogous geometric and ``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where $\infty$-modulus of curve fragments, $\ast$-upper gradients and Alberti representations play a central role. We give a new definition of fragment-wise charts and prove that they exists for spaces with finite Hausdorff dimension. We give a replacement for $p$-duality in terms of Alberti representations and $\infty$-modulus and present the theory of $\ast$-upper gradients. Further, we give new and sharper results for approximations of Lipschitz functions, which yields the density of directions. Our results are applicable to all complete and separable metric measure spaces. In the process, we show that there are strong parallels between the Sobolev and Lipschitz worlds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11284 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fragment-wise differentiable structures Bate, David Eriksson-Bique, Sylvester Soultanis, Elefterios Classical Analysis and ODEs Functional Analysis Metric Geometry 30L99, 51F30, 28A75 The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this paper is to give an analogous geometric and ``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where $\infty$-modulus of curve fragments, $\ast$-upper gradients and Alberti representations play a central role. We give a new definition of fragment-wise charts and prove that they exists for spaces with finite Hausdorff dimension. We give a replacement for $p$-duality in terms of Alberti representations and $\infty$-modulus and present the theory of $\ast$-upper gradients. Further, we give new and sharper results for approximations of Lipschitz functions, which yields the density of directions. Our results are applicable to all complete and separable metric measure spaces. In the process, we show that there are strong parallels between the Sobolev and Lipschitz worlds. |
| title | Fragment-wise differentiable structures |
| topic | Classical Analysis and ODEs Functional Analysis Metric Geometry 30L99, 51F30, 28A75 |
| url | https://arxiv.org/abs/2402.11284 |