The $L_p$-discrepancy is a classical quantitative measure for the irregularity of distribution of an $N$-element point set in the $d$-dimensional unit cube. Its inverse for dimension $d$ and error threshold $\varepsilon \in (0,1)$ is the number of points in $[0,1)^d$ that is required such that the minimal normalized $L_p$-discrepancy is less or equal $\varepsilon$. It is well known, that the inverse of $L_2$-discrepancy grows exponentially fast with the dimension $d$, i.e., we have the curse of dimensionality, whereas the inverse of $L_{\infty}$-discrepancy depends exactly linearly on $d$. The behavior of inverse of $L_p$-discrepancy for general $p \not\in \{2,\infty\}$ was an open problem since many years. Recently, the curse of dimensionality for the $L_p$-discrepancy was shown for an infinite sequence of values $p$ in $(1,2]$, but the general result seemed to be out of reach. In the present paper we show that the $L_p$-discrepancy suffers from the curse of dimensionality for all $p$ in $(1,\infty)$ and only the case $p=1$ is still open. This result follows from a more general result that we show for the worst-case error of positive quadrature formulas for an anchored Sobolev space of once differentiable functions in each variable whose first mixed derivative has finite $L_q$-norm, where $q$ is the H\"older conjugate of $p$.

翻译：$L_p$差异是衡量$d$维单位立方体中$N$个点集分布不均匀性的经典定量指标。其关于维度$d$和误差阈值$\varepsilon \in (0,1)$的逆函数，是指使得最小归一化$L_p$差异不超过$\varepsilon$所需的$[0,1)^d$中点的个数。众所周知，$L_2$差异的逆函数随维度$d$呈指数增长，即存在维数灾难，而$L_{\infty}$差异的逆函数则严格线性依赖于$d$。对于一般$p \not\in \{2,\infty\}$情形下$L_p$差异逆函数的行为，多年来一直是一个未解决问题。近期，尽管已证明对于$(1,2]$区间内无穷序列$p$值存在维数灾难，但一般性结果似乎仍遥不可及。在本文中，我们证明对所有$p \in (1,\infty)$，$L_p$差异均遭受维数灾难，仅$p=1$的情形仍待解决。该结论源于我们证明的一个更一般性结果：针对每个变量一阶可微、且一阶混合导数具有有限$L_q$范数（其中$q$为$p$的Hölder共轭指数）的锚定Sobolev空间，正求积公式的最坏情况误差同样存在维数灾难。