On the basis of fractional calculus, we introduce an integral of weakly controlled paths, which is a generalization of integrals in the context of rough path analysis. As an application, we provide an alternative proof of Lyons' extension theorem for geometric Hölder rough paths together with an explicit expression of the extension map.
"Extension theorem for rough paths via fractional calculus." J. Math. Soc. Japan 69 (3) 893 - 912, July, 2017. https://doi.org/10.2969/jmsj/06930893