A variation embedding theorem and applications

Fractional Sobolev spaces, also known as Besov or Slobodetski spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces. Applications include a new route to a regularity result by Kusuoka for stochastic differential equations, integration against Besov-paths, a regularity criterion for rough paths and a new regularity result for Cameron–Martin paths associated to fractional Brownian motion.