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arXiv · 2404.10422

Lipshitzian Vector Fields, Upper Gradients And Distributional Derivatives

Sergio Venturini
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Abstract

We prove that given a locally integrable function $f$ on an open set of an Euclidean space the distributional derivative $Xf$ with respect to a locally Lipshitzian vector field $X$ is locally integrable if, and only if, the function $f$ admits a locally integrable upper gradient along the vector field $X$; in this case $Xf$ coincides with the Lie derivative $L_X f$ and $|Xf|$ is the least upper gradient of the function $f$. Applications to systems of locally Lipshitzian vector fields are given.

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Sergio Venturini. 2024-04-16. Lipshitzian Vector Fields, Upper Gradients And Distributional Derivatives. https://arxiv.org/abs/2404.10422

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