向量值里斯位势：Cartan型估计和相关容量
Vector-valued Riesz potentials: Cartan type estimates and related capacities
摘要：
关于多个电荷的静电势有许多有趣的问题。我们考虑其中一个与场的强度有关，换句话说，与这个势的梯度的大小有关。我们想给出一个尖锐的估计的大小的点的集合，这个梯度是大的。当然，我们希望估计在$N$的收费数量上是尖锐的。尺寸将通过具有各种度规函数的Hausdorff含量来测量。这样的设置使我们能够考虑广泛的措施（不一定有许多费用）。主要技术将是Calderón-Zygmund容量和非齐次Calderón-Zygmund算子。在这里，我们建立了各种类型的能力与奇异核（e。G.解析容量，lipschitz谐波容量等）和非线性容量从理论的潜力阿拉亚当斯，Hedberg，Havin，Maz'ya，沃尔夫。本文的“电容”部分推广了Mateu，Prat和Verdera [J. reine und angew.数学、578（2005），201- 223]。“尺寸估计”部分推广了M. Anderson和V. Eiderman [Annals of Math.，163（2005），1057- 1076]。困难在于我们不能再使用门格尔曲率，因为我们是在大于2维的空间中工作的。
英文摘要：
There are many interesting problems about the electrostatic potential of finitely many charges. We consider one of them concerning the intensity of the field, in other words, about the magnitude of the gradient of this potential. We want to give a sharp estimate of the size of the set of points where this gradient is large. Of course we want the estimate to be sharp in number $N$ of charges. The size will be measured by the Hausdorff content with various gauge functions. Such a setting allows us to consider a wide class of measures (not necessarily with finitely many charges). The main technique will be Calderón-Zygmund capacities and nonhomogeneous Calderón-Zygmund operators. Here we establish a relationship between various types of capacities with singular kernels (e. g. analytic capacity, lipschitz harmonic capacity, etc) and non-linear capacity from the theory of potential á la Adams, Hedberg, Havin, Maz'ya, Wolff. "Capacitary" part of the paper extends the theorem of Mateu, Prat and Verdera [J. reine und angew. Math., 578 (2005), 201--223]. "Size estimates" part of the paper extends the theorem of M. Anderson and V. Eiderman [Annals of Math., 163 (2005), 1057--1076]. The difficulty lies in the fact that we cannot use Menger's curvature anymore because we are working in spaces of dimension bigger than two.
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