Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913740960563200 |
|---|---|
| author | Anttila, Riku Eriksson-Bique, Sylvester Shimizu, Ryosuke |
| author_facet | Anttila, Riku Eriksson-Bique, Sylvester Shimizu, Ryosuke |
| contents | We construct self-similar $p$-energy forms as normalized limits of discretized $p$-energies on a rich class of Laakso-type fractal spaces. Collectively, we refer to them as IGS-fractals, where IGS stands for (edge-)iterated graph systems. We propose this framework as a rich source of "toy models" that can be consulted for tackling challenging questions that are not well understood on most other fractal spaces. Supporting this, our framework uncovers a novel analytic phenomenon, which we term as singularity of Sobolev spaces. This means that the associated Sobolev spaces $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ for distinct $p_1,p_2 \in (1,\infty)$ intersect only at constant functions. We provide the first example of a self-similar fractal on which this singularity phenomenon occurs for all pairs of distinct exponents. In particular, we show that the Laakso diamond space is one such example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13258 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces Anttila, Riku Eriksson-Bique, Sylvester Shimizu, Ryosuke Metric Geometry Analysis of PDEs Functional Analysis 28A80, 31E05, 31C45, 46E36 We construct self-similar $p$-energy forms as normalized limits of discretized $p$-energies on a rich class of Laakso-type fractal spaces. Collectively, we refer to them as IGS-fractals, where IGS stands for (edge-)iterated graph systems. We propose this framework as a rich source of "toy models" that can be consulted for tackling challenging questions that are not well understood on most other fractal spaces. Supporting this, our framework uncovers a novel analytic phenomenon, which we term as singularity of Sobolev spaces. This means that the associated Sobolev spaces $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ for distinct $p_1,p_2 \in (1,\infty)$ intersect only at constant functions. We provide the first example of a self-similar fractal on which this singularity phenomenon occurs for all pairs of distinct exponents. In particular, we show that the Laakso diamond space is one such example. |
| title | Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces |
| topic | Metric Geometry Analysis of PDEs Functional Analysis 28A80, 31E05, 31C45, 46E36 |
| url | https://arxiv.org/abs/2503.13258 |