广义1/2变差下路的二阶可微性
Second order differentiability of paths via a generalized 1/2-variation
摘要：
给出了连续向量值路与二次可微函数的Lebesgue等价的一个等价条件。为此，我们引入了$VBG_{1/2}$函数的概念，它在二阶可微性中起着类似于一阶可微性的$VBG_*$函数的经典概念的作用。事实上，对于一个函数$f:[a,b]\to X$，是勒贝格等价于一个二次可微函数与是勒贝格等价于一个具有逐点Lipschitz导数的可微函数是一样的。我们还考虑了一阶导数几乎处处为非零的情况。
英文摘要：
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.
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