AI 中文总结
本文研究图上Lipschitz函数空间的不变逐点闭子空间及其预对偶空间，证明ℤᵈ上Lipschitz空间的平移不变逐点闭子空间要么有限维要么非可分，对应预对偶空间要么有限维要么非自反。
AI 中文摘要
受Lipschitz自由空间和图上Lipschitz调和函数研究的双重驱动，我们研究图上Lipschitz函数空间的不变逐点闭子空间，特别关注作为图的有限生成群。这类空间构成一类自然的弱*闭子空间，在某些情形下可被完全描述，因此具有对应Lipschitz自由空间的典范商预对偶空间。我们证明这类空间由有限局部约束描述；在群情形下，它们是卷积方程组的Lipschitz解。我们通过泛性质描述并刻画其预对偶空间，证明只要这类空间包含一个具有c₀梯度的非零元素，就会包含ℓ_∞，因此其预对偶空间包含一个可补的ℓ₁副本。一个核心问题是这类Lipschitz空间及其预对偶空间是否包含无限维自反Banach空间。对此，本文的主要结果是利用抽象调和分析证明的二分性：ℤᵈ上Lipschitz空间的每个平移不变逐点闭子空间要么是有限维的，要么是非可分的——特别地，对应的预对偶空间要么是有限维的，要么是非自反的。
英文摘要
Motivated by both the research on Lipschitz-free spaces and on Lipschitz harmonic functions on graphs, we study invariant pointwise closed subspaces of spaces of Lipschitz functions over graphs, with special emphasis on finitely generated groups as graphs. Such spaces form a natural class of $\text{weak}^*$-closed subspaces, that can be fully described in some cases, and hence have canonical quotient preduals of the corresponding Lipschitz-free spaces. We show that these spaces are described by finite local constraints; in the group case as Lipschitz solutions of systems of convolution equations. We describe and characterize their preduals via a universal property and show that whenever they contain a non-zero element with a $c_0$-gradient, then they contain $\ell_\infty$, and consequently their preduals contain a complemented copy of $\ell_1$. A guiding question is whether this class of Lipschitz spaces and their preduals contains an infinite-dimensional reflexive Banach space. In this regard, the main result of the paper is the following dichotomy proved using abstract harmonic analysis. Every translation-invariant pointwise closed subspace of the Lipschitz space over $\mathbb{Z}^d$ is either finite-dimensional or it is non-separable --in particular, the corresponding predual is either finite-dimensional or non-reflexive.
