Let $f$ be a $\alpha$-Hölder function in $\mathbb{R}^n$. Question : does it have distributional derivatives in a $L^p$ space ? (modulo a suitable relationship between $\alpha$ and $n$).
I know the other way around : if we look at elements of Sobolev spaces, then we get a Hölder regularity depending on the dimension $n$ and exponent $p$. Is there a converse there ? When I tried googling the question, I only found an article with an example of nowhere differentiability in the classical sense.
If you want an easy example, take the Cantor function, which is Hölder for exponents small enough.
If you want a more elaborate example, take this or this function, which are not Lipschitz, but are Hölder for every exponent less than $1$ for careful choices of the parameters (they are is actually quasi-Lipschitz). However they are not BV, so their distributional derivative cannot be in any Sobolev space.