发表机构
Universidade de Santiago de Compostela; Galician Center for Mathematical Research and Technology (CITMAga)（圣地亚哥德孔波斯特拉大学; 加利西亚数学研究与技术中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对 Stieltjes--Bochner 演化图建立 Aubin--Lions 紧性定理，结合原子求值与局部平均，推广经典结果至混合测度情形。
AI 中文摘要
设 $g$ 为非降左连续函数，$\u03bc_g$ 为其 Lebesgue--Stieltjes 测度。我们建立了 Banach 值微积分基本定理和由 $\u03bc_g$ 度量的演化的 Aubin--Lions 紧性定理，允许绝对连续、奇异连续和原子分量。若值域空间具有 Radon--Nikodým 性质，则曲线 $g$-绝对连续当且仅当它是无界 Bochner 积分；其强 $g$-导数为 Bochner 密度，且变差测度的密度等于其范数。末端原子可能使导数从 Bochner 状态类中不可见，因此自然演化空间是状态--导数对构成的图。对于 $B_0\Subset B\hookrightarrow B_1$，其中 $B_0$ 和 $B_1$ 自反且 $1<p_0,p_1<\infty$，状态投影是到 $L_g^{p_0}([a,b);B)$ 的紧线性算子。证明结合了原子处的有界求值、非原子点的局部 $\u03bc_g$-平均值以及 Ehrling 不等式。该结果恢复了经典定理，并为有界时间尺度、加权序列和混合 Stieltjes 测度提供了紧性原理。
英文摘要
Let $g$ be a nondecreasing left-continuous function and let $μ_g$ be its Lebesgue--Stieltjes measure. We establish a Banach-valued fundamental theorem of calculus and an Aubin--Lions compactness theorem for evolution measured by $μ_g$, allowing absolutely continuous, singular continuous and atomic components. If the range space has the Radon--Nikodým property, a curve is $g$-absolutely continuous if and only if it is an indefinite Bochner integral; its strong $g$-derivative is the Bochner density and the variation measure has density equal to its norm. A terminal atom may make the derivative invisible from the Bochner state class, so the natural evolution space is a graph of state--derivative pairs. For $B_0\Subset B\hookrightarrow B_1$, with $B_0$ and $B_1$ reflexive and $1<p_0,p_1<\infty$, the state projection is a compact linear operator into $L_g^{p_0}([a,b);B)$. The proof combines bounded evaluation at atoms, local $μ_g$-averages at nonatomic points and Ehrling's inequality. The result recovers the classical theorem and yields compactness principles for bounded time scales, weighted sequences and mixed Stieltjes measures.
