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We show that every $mathbb{R}^d$-valued Sobolev path with regularity $alpha$ and integrability $p$ can be lifted to a Sobolev rough path in the sense of T. Lyons provided $alpha >1/p>0$. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. to a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.
We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that the solut
In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of the classical quasidisks. After that, we also find some applications of their Sobolev extension property.
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 $alpha < 1/p<1/3$. The novelty of our approach is its use of ideas underlying Hairers reconstruction the
We provide an extension operator for weighted Sobolev spaces on bounded polyhedral cones $K$ involving a mixture of weights, which measure the distance to the vertex and the edges of the cone, respectively. Our results are based on Steins extension operator for Sobolev spaces.
We develop some characterizations for Sobolev spaces on the setting of graded Lie groups. A key role is played by several mean value inequalities that may be of independent interest.