分形上的$p$-能量形式：最新进展
$p$-Energy forms on fractals: recent progress
AI总结：
本文综述自相似分形上$p$-能量形式的研究进展，梳理了$p≠2$情形的研究起源，介绍了近年研究者在多类自相似集上证明该形式存在性及相关性质的成果，并以标准Sierpiński gasket为例说明。
摘要：
本文综述了自相似分形上自相似$p$-能量形式的最新研究进展，其中$p∈(1,∞)$。当$p=2$时，这类形式的概念与自相似Dirichlet形式一致，自20世纪80年代末以来已有大量相关研究；而$p∈(1,∞)\setminus\{2\}$的情形研究起步晚得多，由Herman、Peirone与Strichartz[《Potential Anal.》20(2004)，125--148]以及Strichartz与Wong[《Nonlinearity》17(2004)，595--616]于2004年首次提出，此后直到数年前都未取得实质性进展。Kigami、Shimizu、Cao--Gu--Qiu以及Murugan--Shimizu的近期研究，证明了这类$p$-能量形式在一般后临界有限（p.-c.f.）自相似集以及大类低维无限分支自相似集上的存在性；作者们还主要针对p.-c.f.自相似集，证明了这些形式及相关$p$-调和函数的更多细致性质。本文致力于综述这些成果，重点介绍作者们的最新研究进展，并以二维标准Sierpiński gasket这一最简单的非平凡情形为例进行说明。
英文摘要：
In this article, we survey recent progress on self-similar $p$-energy forms on self-similar fractals, where $p\in(1,\infty)$. While for $p=2$ the notion of such forms coincides with that of self-similar Dirichlet forms and there have been plenty of studies on them since the late 1980s, studies on the case of $p\in(1,\infty)\setminus\{2\}$ was initiated much later in 2004 by Herman, Peirone and Strichartz [Potential Anal. 20 (2004), 125--148] and Strichartz and Wong [Nonlinearity 17 (2004), 595--616] and no essential progress on this case had been made since then until a few years ago. The recent progress by Kigami, Shimizu, Cao--Gu--Qiu and Murugan--Shimizu has established the existence of such $p$-energy forms on general post-critically finite (p.-c.f.) self-similar sets and on large classes of low-dimensional infinitely ramified self-similar sets, and the authors have proved further detailed properties of these forms and associated $p$-harmonic functions, mainly for p.-c.f. self-similar sets. This article is devoted to a review of these results, focusing on the most recent developments by the authors and illustrating them in the simplest non-trivial setting of the two-dimensional standard Sierpiński gasket.
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