Vector‐valued Riesz potentials: Cartan‐type estimates and related capacities
Vector‐valued Riesz potentials: Cartan‐type estimates and related capacities
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向量值 Riesz 势：Cartan 型估计和相关能力
DOI：
10.1112/plms/pdq003
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发表时间：
2008
影响因子：
1.8
通讯作者：
A. Volberg
中科院分区：
文献类型：
--
作者：
V. Eiderman;F. Nazarov;A. Volberg
我们给出了在d ≠ 1的情况下，Borel测度ν的Riesz变换很大时，在d ≠ 1的情况下，点集的大小的精确上界。该尺寸通过具有各种规范函数的Hausdorff内容来测量。我们开始讨论ν是N个质点的线性组合的情况。除此之外，我们描述了所有的规范函数，其估计不会随着N趋于无穷大而爆炸。在这种情况下，一个例行的限制参数允许我们扩展我们的边界到所有有限的Borel措施。我们还展示了我们的技术可以应用于某些能力的估计。
We give sharp upper bounds for the size of the set of points in ℝd, with d ⩾ 1, where the Riesz transform of a Borel measure ν is large. This size is measured by the Hausdorff content with various gauge functions. We begin with the case when ν is a linear combination of N point masses. Among other things, we characterize all gauge functions for which the estimates do not blow up as N tends to infinity. In this case a routine limiting argument allows us to extend our bounds to all finite Borel measures. We also show how our techniques can be applied to estimates for certain capacities.