Second order differentiability of paths via a generalized 1/2-variation

Abstract

We find an equivalent condition for a continuous vector-valued path to be Lebesgue equivalent to a twice differentiable function. For that purpose, we introduce the notion of a $VBG_{1/2}$ function, which plays an analogous role for the second order differentiability as the classical notion of a $VBG_*$ function for the first order differentiability. In fact, for a function $f:[a,b]\to X$, being Lebesgue equivalent to a twice differentiable function is the same as being Lebesgue equivalent to a differentiable function with a pointwise Lipschitz derivative. We also consider the case when the first derivative can be taken non-zero almost everywhere.