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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.

46G05 Derivatives of functions in infinite-dimensional spaces
26E20 Calculus of functions taking values in infinite-dimensional spaces
Full Text: DOI arXiv
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