  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 $b$. We give upper bounds on the norm of the local discrepancy
in Besov spaces of dominating mixed smoothness $S_{pq}^rB([0,1)^s)$, which will
also give us bounds on the $L_p$-discrepancy. Our sequence and point sets will
achieve the known optimal order for the $L_p$- and $S_{pq}^rB$-discrepancy. The
results in this paper generalize several previous results on $L_p$- and
$S_{pq}^rB$-discrepancy estimates and provide a sharp upper bound on the
$S_{pq}^rB$-discrepancy of one-dimensional sequences for $r>0$. We will use the
$b$-adic Haar function system in the proofs.
