Aubin--Lions Compactness for Stieltjes--Bochner Evolution Graphs
Abstract
Let g be a nondecreasing left-continuous function and let μg be its Lebesgue--Stieltjes measure. We establish a Banach-valued fundamental theorem of calculus and an Aubin--Lions compactness theorem for evolution measured by μg, 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 B0Subset B↪ B1, with B0 and B1 reflexive and 1<p0,p1<∞, the state projection is a compact linear operator into Lgp0([a,b);B). The proof combines bounded evaluation at atoms, local μg-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.
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BibTeX
@misc{fernandez2026aubinlions,
title = {{Aubin--Lions Compactness for Stieltjes--Bochner Evolution Graphs}},
author = {Francisco J. Fernández},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10064},
eprint = {2609.10064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10064},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Aubin--Lions Compactness for Stieltjes--Bochner Evolution Graphs
AU - Francisco J. Fernández
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10064
UR - https://arxiv.org/abs/2609.10064
AB - 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_0Subset B↪ B_1, with B_0 and B_1 reflexive and 1<p_0,p_1<∞, 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.
ER -