Integrability of the Brouwer degree for irregular arguments

We prove that the Brouwer degree (a.k.a. mapping degree) for a Hölder continuous function is L^p integrable if the Hölder exponent is large enough in comparison to p times the box dimension of the boundary of the domain. This is supplemented by a theorem showing that convergence of a sequence of functions in an appropriate Hölder space implies convergence of the associated sequence of mapping degrees in L^p, again for a suitably chosen regime of the Hölder and Lebesgue exponents. In a different regime, we show the existence of counterexamples to the latter case.