The Lₚ-discrepancy for finite $p>1$ suffers from the curse of dimensionality

The Lₚ-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 ε ∈ (0,1) is the number of points in [0,1)ᵈ that is required such that the minimal normalized Lₚ-discrepancy is less or equal ε. It is well known, that the inverse of L₂-discrepancy grows exponentially fast with the dimension d, i.e., we have the curse of dimensionality, whereas the inverse of L∞-discrepancy depends exactly linearly on d. The behavior of inverse of Lₚ-discrepancy for general p ∈ \2,∞\ was an open problem since many years. Recently, the curse of dimensionality for the Lₚ-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 Lₚ-discrepancy suffers from the curse of dimensionality for all p in (1,∞) 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 Lq-norm, where q is the H\"older conjugate of p.