  We prove that if $\Omega\subset \mathbb R^n$ is a bounded open set and
$n\alpha> {\rm dim}_b (\partial \Omega) = d$, then the Brouwer degree
deg$(v,\Omega,\cdot)$ of any H\"older function $v\in C^{0,\alpha}\left (\Omega,
\mathbb R^{n}\right)$ belongs to the Sobolev space $W^{\beta, p} (\mathbb R^n)$
for every $0\leq \beta < \frac{n}{p} - \frac{d}{\alpha}$. This extends a
summability result of Olbermann and in fact we get, as a byproduct, a more
elementary proof of it. Moreover we show the optimality of the range of
exponents in the following sense: for every $\beta\geq 0$ and $p\geq 1$ with
$\beta > \frac{n}{p} - \frac{n-1}{\alpha}$ there is a vector field $v\in C^{0,
\alpha} (B_1, \mathbb R^n)$ with $\mbox{deg}\, (v, \Omega, \cdot)\notin
W^{\beta, p}$, where $B_1 \subset \mathbb R^n$ is the unit ball.
