AI 中文总结
研究将阿尔贝蒂关于博雷尔向量场与\(C^1\)函数梯度关系的结果扩展到分数阶梯度等非局部梯度情形，核心方法是平移方法，主要贡献是建立了相关结果并展示该方法在几何测度论中的作用。
AI 中文摘要
我们扩展了阿尔贝蒂的著名结果，即博雷尔向量场在任意小测度集之外与\(C^1\)函数的梯度一致。我们证明在分数阶梯度和\(C^{0,s}\)函数的背景下类似的陈述成立。此外，我们为具有适当函数空间的一般非局部梯度建立了类似结果。两个陈述均通过平移方法证明，展示了其在几何测度论中的用途。
英文摘要
We extend the celebrated result of Alberti, stating that Borel vector fields coincide with gradients of $C^1$-functions outside of a set of arbitrary small measure. We prove that a similar statement holds true in the setting of fractional gradients and $C^{0,s}-functions. Furthermore, we establish analogous results for general nonlocal gradients with appropriate function spaces. Both statements are proven by means of the translation method, demonstrating its use for the purposes of geometric measure theory. In particular, this article illustrates how the translation method transfers rigidity phenomena from the classical gradient to a broad class of nonlocal gradients.
