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

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.