关于电阻猜想
On the Resistance Conjecture
AI总结：
本文通过证明体积倍增、上容量界和Poincaré不等式可推出截止Sobolev不等式，从而解决电阻猜想，并统一了度量空间、分形、图和流形上所有p∈(1,∞)的Harnack不等式和稳定性现象的证明。
摘要：
我们给出了电阻猜想的肯定回答，即通过体积倍增、上容量界和Poincaré不等式来刻画抛物线Harnack不等式。关键步骤是证明这三个假设可推出所谓的截止Sobolev不等式，这是研究异常扩散、Dirichlet形式和分形中重新标定能量的重要不等式。此推论在p-Dirichlet空间的一般设定下成立，由作者和Murugan引入，从而可能统一证明度量空间和分形上，以及图和流形上所有p∈(1,∞)的Harnack不等式和稳定性现象。作为应用，我们还证明满足体积倍增、Poincaré和上容量界的Dirichlet空间具有有限鞅维数，并具有类似Cheeger工作的微分结构。在证明过程中，我们建立了通过Poincaré不等式扩展和刻画Sobolev函数的方法，并将Jones和Koskela的方法扩展到p-Dirichlet空间的一般设定中。
英文摘要：
We give an affirmative answer to the resistance conjecture on characterization of parabolic Harnack inequalities in terms of volume doubling, upper capacity bounds and a Poincaré inequalities. The key step is to show that these three assumptions imply the so called cutoff Sobolev inequality, an important inequality in the study of anomalous diffusions, Dirichlet forms and re-scaled energies in fractals. This implication is shown in the general setting of $p$-Dirichlet Spaces introduced by the author and Murugan, and thus a unified treatment becomes possible to proving Harnack inequalities and stability phenomena in both analysis on metric spaces and fractals and for graphs and manifolds for all exponents $p\in (1,\infty)$. As an application, we also show that a Dirichlet space satisfying volume doubling, Poincaré and upper capacity bounds has finite martingale dimension and admits a type of differential structure similar to the work of Cheeger. In the course of the proof, we establish methods of extension and characterizations of Sobolev functions by Poincaré-inequalities, and extend the methods of Jones and Koskela to the general setting of $p$-Dirichlet spaces.
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