度量空间上有界p-调和函数与拟调和函数在局部假设下的刘维尔定理及可去集
Liouville theorems and removable sets for bounded $p$-harmonic and quasiharmonic functions on metric spaces under local assumptions
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中文总结 AI 辅助
本文针对满足局部p-庞加莱不等式等条件的度量空间，刻画了有界p-调和函数与拟调和函数的可去紧集，揭示可去性与刘维尔型定理等价，确定了相关关键性质并给出刘维尔定理的初等证明，结果适用于流形等空间。
中文摘要 AI 辅助
对于连通的真度量空间X，其配备了局部加倍测度且满足局部p-庞加莱不等式，我们完全刻画了具有正容量的哪些紧集K是有界p-调和函数（p>1）的可去集，同时也对有界拟调和函数证明了类似结果。该刻画兼具几何与分析表述，特别地，可去性等价于X\backslash K上刘维尔型定理的有效性。局部连通性、序列环形拟凸性、容量集中性及p-抛物性等性质被确定为可去性的关键因素。在此过程中，我们给出了p-抛物空间中拟超调和函数刘维尔定理的一个相当初等的证明。我们的结果尤其适用于配备（局部）p-容许权重的流形和Rⁿ。
英文摘要
For connected proper metric spaces $X$, equipped with a locally doubling measure supporting a local $p$-Poincaré inequality, we completely characterize which compact sets $K$ with positive capacity are removable for bounded $p$-harmonic functions, $p>1$. Similar results are proved also for bounded quasiharmonic functions. The characterization is both in geometric and analytic terms. In particular, removability is shown to be equivalent to the validity of a Liouville type theorem in $X\setminus K$. Properties such as local connectedness, sequential annular quasiconvexity, concentration of capacity and $p$-parabolicity are identified as crucial for removability. Along the way, we give a rather elementary proof of the Liouville theorem for quasisuperharmonic functions in $p$-parabolic spaces. Our results apply in particular to manifolds and $\mathbf{R}^n$ equipped with (locally) $p$-admissible weights.
