度量测度空间上$p$-能量的截断函数的能量不等式
Energy inequalities for cutoff functions of $p$-energies on metric measure spaces
AI总结：
本文研究度量测度空间上$p$-能量的截断Sobolev不等式的成立条件，通过椭圆Harnack不等式、容量上下界等几何与函数条件，在不假设Poincaré不等式的情况下证明其成立，并应用于Sierpiński地毯上$p$-能量测度的奇异性。
摘要：
对于$p>1$，在体积倍增的度量测度空间上，我们给出了$p$-能量的截断Sobolev不等式成立的若干几何与函数条件。特别地，我们证明了椭圆Harnack不等式、双侧容量界以及一些额外的几何与分析假设蕴含截断Sobolev不等式，而不需要假设Poincaré不等式。反之，我们表明Poincaré不等式与截断Sobolev不等式一起可以恢复第一个蕴含中使用的分析输入。此外，在低维情形下，Poincaré不等式和容量上界蕴含截断Sobolev不等式。作为应用，我们证明了对于任意$p>1$，Sierpiński地毯上的$p$-能量测度关于Hausdorff测度是奇异的，从而解决了Murugan和Shimizu提出的问题[Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608]。
英文摘要：
For $p>1$, and for a $p$-energy on a volume doubling metric measure space, we provide several geometric and functional conditions for the validity of the cutoff Sobolev inequality. In particular, we prove that the elliptic Harnack inequality, two-sided capacity bounds, and some additional geometric and analytic assumptions imply the cutoff Sobolev inequality, without assuming the Poincaré inequality. Conversely, we show that the Poincaré inequality together with the cutoff Sobolev inequality recovers the analytic inputs used in the first implication. Moreover, in a lower-dimensional regime, the Poincaré inequality and the capacity upper bound imply the cutoff Sobolev inequality. As an application, we prove that the $p$-energy measure is singular with respect to the Hausdorff measure on the Sierpiński carpet for any $p>1$, thereby resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608].
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