On the Assouad spectrum of Hölder and Sobolev graphs
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911039964053504 |
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| author | Garitsis, Efstathios Konstantinos Chrontsios Tyson, Jeremy T. |
| author_facet | Garitsis, Efstathios Konstantinos Chrontsios Tyson, Jeremy T. |
| contents | We provide upper bounds for the Assouad spectrum $\dim_A^θ(\text{Gr}(f))$ of the graph of a real-valued Hölder or Sobolev function $f$ defined on an interval $I \subset \mathbb{R}$. We demonstrate via examples that all of our bounds are sharp. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $α$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $α$ and returns the graph of a new $α$-Hölder continuous function for which the Assouad $θ$-spectrum realizes the stated upper bound for all $θ\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07783 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Assouad spectrum of Hölder and Sobolev graphs Garitsis, Efstathios Konstantinos Chrontsios Tyson, Jeremy T. Classical Analysis and ODEs Dynamical Systems We provide upper bounds for the Assouad spectrum $\dim_A^θ(\text{Gr}(f))$ of the graph of a real-valued Hölder or Sobolev function $f$ defined on an interval $I \subset \mathbb{R}$. We demonstrate via examples that all of our bounds are sharp. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $α$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $α$ and returns the graph of a new $α$-Hölder continuous function for which the Assouad $θ$-spectrum realizes the stated upper bound for all $θ\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi. |
| title | On the Assouad spectrum of Hölder and Sobolev graphs |
| topic | Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2309.07783 |