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：$L_p$差异度是衡量$d$维单位立方体中$N$个点集分布不规则性的经典定量指标。对于维度$d$和误差阈值$\varepsilon \in (0,1)$，其逆量定义为：使得归一化最小$L_p$差异度不超过$\varepsilon$所需的$[0,1)^d$中的点数。众所周知，$L_2$差异度的逆量随维度$d$呈指数增长，即存在维数灾难，而$L_{\infty}$差异度的逆量则恰好与$d$呈线性关系。对于一般$p \not\in \{2,\infty\}$，$L_p$差异度逆量的行为多年来一直是一个开放问题。近期，$L_p$差异度的维数灾难已在$(1,2]$区间内的无穷序列值$p$上得到证明，但一般结果似乎难以企及。本文证明，对于所有$p \in (1,\infty)$，$L_p$差异度均遭受维数灾难，仅$p=1$的情况仍悬而未决。这一结果源自我们针对一个各变量一阶可导、且其混合一阶导数具有有限$L_q$范数的锚定Sobolev空间（其中$q$为$p$的Hölder共轭指数）上正求积公式的最坏情况误差所证明的更一般结论。