LpLp- and Sp,qrB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases

We study the local discrepancy of a symmetrized version of the well-known van der Corput sequence and of modified two-dimensional Hammersley point sets in arbitrary base bb. We give upper bounds on the norm of the local discrepancy in Besov spaces of dominating mixed smoothness Sp,qrB([0,1)s), which will also give us bounds on the LpLp-discrepancy. Our sequence and point sets will achieve the known optimal order for the LpLp- and Sp,qrB-discrepancy. The results in this paper generalize several previous results on LpLp- and Sp,qrB-discrepancy estimates and provide a sharp upper bound on the Sp,qrB-discrepancy of one-dimensional sequences for r>0r>0. We will use the bb-adic Haar function system in the proofs.