Integrability of the Brouwer degree for irregular arguments

Abstract: We prove that the Brouwer degree $\mathrm{deg}(u,U,\cdot)$ for a function $u\in C^{0,\alpha}( U;\mathbb{R}<sup>n)$ is in $L<sup>p(\mathbb{R}<sup>n)$ if $1\leq p&lt;\frac{n\alpha}d$, where $U\subset \mathbb{R}<sup>n$ is open and bounded and $d$ is the box dimension of $\partial U$. This is supplemented by a theorem showing that $u_j\to u$ in $C<sup>{0,\alpha}(U;\mathbb{R}<sup>n)$ implies $$\mathrm{deg}(u_j,U,\cdot)\to \mathrm{deg}(u,U,\cdot)$$ in $L<sup>p(\mathbb{R}<sup>n)$ for the parameter regime $1\leq p&lt;\frac{n\alpha}d$, while there exist convergent sequences $u_j\to u$ in $C<sup>{0,\alpha}(U;\mathbb{R}<sup>n)$ such that $$|\mathrm{deg}(u_j,U,\cdot)|_{L^p}\to \infty$$ for the opposite regime $p&gt;\frac{n\alpha}d$.