A Non-Differentiability Result for the Inversion Operator between Sobolev Spaces

Abstract

The order of differentiability of the inversion operator $\mathcal{J}$ between certain spaces or manifolds of distributionally differentiable functions is shown to be sharp in the following sense. Up to a certain order $k$ guaranted by inverse function arguments, the operator $\mathcal{J}$ is everywhere differentiable and${\mathcal{J}}^{(k)}$ is continuous. On the other hand, $\mathcal{J}$ is nowhere $k+1$ times differentiable.