论文标题
立方体公式和Sobolev不平等现象
Cubature formulas and Sobolev inequalities
论文作者
论文摘要
我们研究了球体上的立方公式理论中的一个问题：给定（0，1）$中的$θ\，确定$ \ | c | g |ν\ |_θ= \ sum_ {i = 1}^nν_i^θ$在cuile $ t $ t $ $ n $ n n $ v $ n supe $ c $ n progenth $ t $ ns progenth $ ns $ n $ n $ n n $ n $ n $ n c $ n $ nem $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n q $ n $ n $ nnν_i^θ$。这个问题概括了界限立方体公式的最低基数的经典问题 - 案例$θ= 0 $ - 是在最近的Hang and Wang和Wang（Arxiv：2010.10654）中引入的，后者显示了与Sobolev不平等中最佳常数有关的问题。使用$ s^{n -1} $复制内核希尔伯特空间的基本理论，我们将最低的上限和下限扩展了强度的最低基数 -$ t $ cubature公式到范围，以限制为$ \ | \ | \ cdot \ cdot \ cdot \ |_θ$，以获取任何$θ\ in（0，1）$。特别是，我们完全表征了强度的立方体度量$ 3 $最小化$ \ | \ cdot \ |_θ$，这表明这些恰恰是紧密的球形$ 3 $ -DESIGNS。
We study a problem in the theory of cubature formulas on the sphere: given $θ\in (0, 1)$, determine the infimum of $\|ν\|_θ= \sum_{i = 1}^n ν_i^θ$ over cubature formulas $ν$ of strength $t$, where $ν_i$ are the weights of the formula $ν$. This problem, which generalizes the classical problem of bounding the minimal cardinality of a cubature formula -- the case $θ= 0$ -- was introduced in recent work of Hang and Wang (arXiv:2010.10654), who showed the problem to be related to optimal constants in Sobolev inequalities. Using the elementary theory of reproducing kernel Hilbert spaces on $S^{n - 1}$, we extend the best known upper and lower bounds for the minimal cardinality of strength-$t$ cubature formulas to bounds for the infimum of $\|\cdot\|_θ$ for any $θ\in (0, 1)$. In particular, we completely characterize the cubature measures of strength $3$ minimizing $\|\cdot\|_θ$, showing that these are precisely the tight spherical $3$-designs.
