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Second order differentiability of paths via a generalized $$\frac 1 2$$-variation. (English) Zbl 1135.46021
Summary: 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 $$f:[a,b]\rightarrow X$$, being Lebesgue equivalent to a twice differentiable function is the same as being Lebesgue equivalent to a differentiable function $$g$$ with a pointwise Lipschitz derivative such that $$g^{\prime\prime }(x)$$ exists whenever $$g^{\prime }(x)\neq 0$$. We also consider the case when the first derivative can be taken non-zero almost everywhere.

##### MSC:
 46G05 Derivatives of functions in infinite-dimensional spaces 26E20 Calculus of functions taking values in infinite-dimensional spaces
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##### References:
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