最小上梯度的逐曲线刻画与Sobolev微分的构造
Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential
AI总结：
本文通过一般曲线上的极大方向导数刻画Newton函数的最小上梯度，引入p-弱图构造Sobolev微分，建立适配有限Hausdorff维数空间的p-弱可微结构，给出相关经典结论的新证明与几何解释。
摘要：
我们运用平面模对偶与分解技术，通过给定点处“一般”曲线上的极大方向导数，刻画了1≤p<∞范围内Newton函数的最小上梯度。作为应用，我们引入p-弱图的概念，证明每个Newton函数都存在关于这类图的微分，能在p-几乎每条曲线上给出线性逼近。该微分可逐曲线计算，具有线性性，且满足通常的莱布尼茨法则与链式法则。\n 所得到的p-弱可微结构适用于有限Hausdorff维数的空间，在Poincaré不等式成立的条件下与Cheeger结构一致。此外，当Gigli的抽象可微结构存在时，该p-弱可微结构与之兼容，并为其提供了几何解释。p-弱图可生成有限维p-弱余切丛与逐点范数，该范数能恢复Newton函数的最小上梯度，且可通过在一般曲线上取极大值的过程计算得到。由此，我们得到了Newton空间中Lipschitz函数的自反性与稠密性的新证明，以及用逐点范数刻画无穷小Hilbert性的结论。
英文摘要：
We represent minimal upper gradients of Newtonian functions, in the range $1\le p<\infty$, by maximal directional derivatives along "generic" curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of $p$-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along $p$-almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules. The arising $p$-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger's structure in the presence of a Poincaré inequality. It is moreover compatible with, and gives a geometric interpretation of, Gigli's abstract differentiable structure, whenever it exists. The $p$-weak charts give rise to a finite dimensional $p$-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.
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