Title: Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces Show Chinese title

Title: 自相似能量形式的构造与Laakso型分形空间上Sobolev空间的奇异性

Abstract: We construct self-similar $p$-energy forms as normalized limits of discretized $p$-energies on a rich class of Laakso-type fractal spaces. Collectively, we refer to them as IGS-fractals, where IGS stands for (edge-)iterated graph systems. We propose this framework as a rich source of "toy models" that can be consulted for tackling challenging questions that are not well understood on most other fractal spaces. Supporting this, our framework uncovers a novel analytic phenomenon, which we term as singularity of Sobolev spaces. This means that the associated Sobolev spaces $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ for distinct $p_1,p_2 \in (1,\infty)$ intersect only at constant functions. We provide the first example of a self-similar fractal on which this singularity phenomenon occurs for all pairs of distinct exponents. In particular, we show that the Laakso diamond space is one such example. Show Chinese abstract

Abstract: 我们构造了自相似的 $p$-能量形式，作为在一类丰富的Laakso型分形空间上离散化 $p$-能量的归一化极限。我们将这些统称为IGS-分形，其中IGS代表（边）迭代图系统。我们认为这一框架可以作为“玩具模型”的丰富来源，用于解决大多数其他分形空间上尚未充分理解的难题。支持这一点，我们的框架揭示了一种新的分析现象，我们称之为Sobolev空间的奇异性。这意味着对于不同的 $p_1,p_2 \in (1,\infty)$，对应的Sobolev空间 $\mathscr{F}_{p_1}$ 和 $\mathscr{F}_{p_2}$ 仅相交于常数函数。我们提供了第一个例子，即在该自相似分形上，所有不同指数对都表现出这种奇异性现象。特别是，我们证明了Laakso钻石空间就是一个这样的例子。