A Sobolev rough path extension theorem via regularity structures

We show that every ℝ d -valued Sobolev path with regularity a and integrability p can be lifted to a Sobolev rough path provided 1/2 > α > 1/ p > ⋁ 1/3. 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.