  We use the dynamical vertex approximation (D$\Gamma$A) with a Moriyaesque $%
\lambda$ correction for studying the impact of antiferromagnetic fluctuations
on the spectral function of the Hubbard model in two and three dimensions. Our
results show the suppression of the quasiparticle weight in three dimensions
and dramatically stronger impact of spin fluctuations in two dimensions where
the pseudogap is formed at low enough temperatures. Even in the presence of the
Hubbard subbands, the origin of the pseudogap at weak-to-intermediate coupling
is in the splitting of the quasiparticle peak. At stronger coupling (closer to
the insulating phase) the splitting of Hubbard subbands is expected instead.
The $\mathbf{k}$-dependence of the self energy appears to be also much more
pronounced in two dimensions as can be observed in the $\mathbf{k}$-resolved
D$\Gamma$A spectra, experimentally accessible by angular resolved photoemission
spectroscopy in layered correlated systems.
