It is known that $ \alpha$ -Hölderness impose restrictions on the bad behavior of a graph in the sense of dimension. To put it precisely, let $ I=[0,1]$ and $ f:[0,1] \to \mathbb{R}$ . Then, if $ f$ is $ \alpha$ -Hölder, its graph $ \Gamma_f = \{(x,f(x)):x\in I\}$ has Hausdorff dimension at most $ 2-\alpha$ . Equivalently, if $ \operatorname{dim}_H(\Gamma_f) > 2 – \alpha$ , then $ f$ can’t be $ \alpha$ -Hölder.

The previous example is an instance of the following principle: if the graph of $ f$ is bad enough in the sense of dimension, then the regularity of $ f$ is bounded above. By “bouded above” I mean “less than $ C^\alpha(I)$ ” as in the previous example.

Are there reverse implications in the spirit of “upper bound on dimension of $ \Gamma_f$ implies some regularity of $ f$ “? I’m not necessarily talking about being $ C^\alpha$ , it can be some weaker notion of regularity: being in some Sobolev space, agreeing with a $ C^\alpha$ function on some big set, being approximable in a particularly good/efficient way, etc…