标题： 关于度量测度空间上$p$能量的截断函数的能量不等式 显示英文标题

标题： Energy inequalities for cutoff functions of $p$-energies on metric measures spaces

摘要： 对于$p\in(1,+\infty)$，以及在度量测度空间上的$p$能量，我们建立了Poincaré不等式和截断Sobolev不等式的联合的等价条件。 特别是，我们采用Trudinger和Wang [Amer. J. Math. 124 (2002), no. 2, 369--410] 的技术来推导超调和函数的Wolff势估计，并采用Holopainen [Contemp. Math. 338 (2003), 219--239] 的方法来证明调和函数的椭圆Harnack不等式。 作为应用，我们在Grigor'yan、Hu和Lau [Springer Proc. Math. Stat. 88 (2014), 147--207] 的容量猜想上取得了进展，并证明了对于所有$p>1$，$p$能量测度相对于Sierpiński地毯上的Hausdorff测度是奇异的，解决了Murugan和Shimizu [Comm. Pure Appl. Math., to appear] 提出的问题。 显示英文摘要

摘要： For $p\in(1,+\infty)$, and for a $p$-energy on a metric measure space, we establish equivalent conditions for the conjunction of the Poincar\'e inequality and the cutoff Sobolev inequality. In particular, we employ a technique of Trudinger and Wang [Amer. J. Math. 124 (2002), no. 2, 369--410] to derive a Wolff potential estimate for superharmonic functions, and a method of Holopainen [Contemp. Math. 338 (2003), 219--239] to prove the elliptic Harnack inequality for harmonic functions. As applications, we make progress toward the capacity conjecture of Grigor'yan, Hu, and Lau [Springer Proc. Math. Stat. 88 (2014), 147--207], and we prove that the $p$-energy measure is singular with respect to the Hausdorff measure on the Sierpi\'nski carpet for all $p>1$, resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math., to appear].