Classification of metric measure spaces and their ends using p-harmonic functions
Classification of metric measure spaces and their ends using p-harmonic functions
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DOI：
10.54330/afm.120618
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发表时间：
2021-06
期刊：
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通讯作者：
Anders Bjorn;Jana Bjorn;N. Shanmugalingam
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作者：
Anders Bjorn;Jana Bjorn;N. Shanmugalingam
通过观察刘维尔型定理对于正的、有界的和/或有限的\(p\)-能量\(p\)-调和函数和\(p\)-准调和函数是否成立，我们对配备有支持局部\(p\)-庞加莱不等式的局部加倍测度的适当度量空间进行分类。黎曼曲面和黎曼流形先前已获得类似的分类。我们研究这些类型的度量测度空间之间的包含关系，以及它们与度量空间及其端点的\(p\)-双曲性的关系。特别是，我们将携带具有有限 p 能量的非常量 p 调和函数的空间表征为具有至少两个分离良好的趋于无穷大的集合的双曲序列的空间。我们还表明，每个这样的空间 \(X\) 都有一个具有有限 \(p\) 能量的函数 \(f \notin L^p(X) + \mathbf{R}\)。
By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite \(p\)-energy \(p\)-harmonic and \(p\)-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local \(p\)-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the \(p\)-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant \(p\)-harmonic functions with finite \(p\)-energy as spaces having at least two well-separated \(p\)-hyperbolic sequences of sets towards infinity. We also show that every such space \(X\) has a function \(f \notin L^p(X) + \mathbf{R}\) with finite \(p\)-energy.