A Sobolev rough path extension theorem via regularity structures

We show that every ${\mathbb{R}}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path provided $\alpha <1\mathrm{/}p<1\mathrm{/}3$. The novelty of our approach is its use of ideas underlying Hairer’s reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.