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arXiv · 2309.07783

On the Assouad spectrum of H\"older and Sobolev graphs

Efstathios Konstantinos Chrontsios Garitsis·Jeremy T. Tyson
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Abstract

We provide upper bounds for the Assouad spectrum $\dim_A^\theta(\text{Gr}(f))$ of the graph of a real-valued H\"older 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 H\"older graphs, we further provide a geometric algorithm which takes as input the graph of an $\alpha$-H\"older continuous function satisfying a matching lower oscillation condition with exponent $\alpha$ and returns the graph of a new $\alpha$-H\"older continuous function for which the Assouad $\theta$-spectrum realizes the stated upper bound for all $\theta\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.

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Efstathios Konstantinos Chrontsios Garitsis, Jeremy T. Tyson. 2023-09-14. On the Assouad spectrum of H\"older and Sobolev graphs. https://arxiv.org/abs/2309.07783

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