Curves between Lipschitz and C¹ and their relation to geometric knot theory

In this article we investigate regular curves whose derivatives have vanishing mean oscillations. We show that smoothing these curves using a standard mollifier one gets regular curves again. We apply this result to solve a couple of open problems. We show that curves with finite M\"obius energy can be approximated by smooth curves in the energy space $W^{\frac 32,2}$ such that the energy converges which answers a question of He. Furthermore, we extend the result of Scholtes on the $\Gamma$-convergence of the discrete M\"obius energies towards the M\"obius energy and prove conjectures of Ishizeki and Nagasawa on certain parts of a decomposition of the M\"obius energy. Finally, we extend a theorem of Wu on inscribed polygons to curves with derivatives with vanishing mean oscillation