  In the current paper we obtain discrepancy estimates in exponential Orlicz
and BMO spaces in arbitrary dimension $d \ge 3$. In particular, we use dyadic
harmonic analysis to prove that for the so-called digital nets of order $2$ the
BMO${}^d$ and $\exp \big( L^{2/(d-1)} \big)$ norms of the discrepancy function
are bounded above by $(\log N)^{\frac{d-1}{2}}$. The latter bound has been
recently conjectured in several papers and is consistent with the best known
low-discrepancy constructions. Such estimates play an important role as an
intermediate step between the well-understood $L_p$ bounds and the notorious
open problem of finding the precise $L_\infty$ asymptotics of the discrepancy
function in higher dimensions, which is still elusive.
