We study tractability properties of the weighted $L_p$-discrepancy. The concept of {\it weighted} discrepancy was introduced by Sloan and Wo\'{z}\-nia\-kowski in 1998 in order to prove a weighted version of the Koksma-Hlawka inequality for the error of quasi-Monte Carlo integration rules. The weights have the aim to model the influence of different coordinates of integrands on the error. A discrepancy is said to be tractable if the information complexity, i.e., the minimal number $N$ of points such that the discrepancy is less than the initial discrepancy times an error threshold $\varepsilon$, does not grow exponentially fast with the dimension. In this case there are various notions of tractabilities used in order to classify the exact rate. For even integer parameters $p$ there are sufficient conditions on the weights available in literature, which guarantee the one or other notion of tractability. In the present paper we prove matching sufficient conditions (upper bounds) and neccessary conditions (lower bounds) for polynomial and weak tractability for all $p \in (1, \infty)$. The proofs of the lower bounds are based on a general result for the information complexity of integration with positive quadrature formulas for tensor product spaces. In order to demonstrate this lower bound we consider as a second application the integration of tensor products of polynomials of degree at most 2.

翻译：我们研究加权$L_p$-偏差的可解性性质。加权偏差的概念由Sloan和Woźniakowski于1998年提出，旨在证明拟蒙特卡洛积分规则误差的加权Koksma-Hlawka不等式。权重用于模拟被积函数不同坐标对误差的影响程度。若信息复杂度（即使偏差小于初始偏差乘以误差阈值$\varepsilon$所需的最小点数$N$）不随维度呈指数增长，则称该偏差具有可解性。在此情形下，存在多种用于分类精确速率的可解性概念。对于偶数参数$p$，文献中已有关于保证某种可解性概念的充分条件。本文对所有$p \in (1, \infty)$情形，证明多项式可解性和弱可解性的匹配充分条件（上界）与必要条件（下界）。下界的证明基于张量积空间中正求积公式积分信息复杂度的通用结论。为演示该下界，我们以二次多项式张量积的积分为第二个应用案例。