  Aims. The main goal of this paper is to derive observational constraints on
the halo mass fuction (HMF) by performing a tomographic analysis of the
magnification bias signal on a sample of background submillimeter galaxies. The
results can then be compared with those from a non-tomographic study. Methods.
We measure the cross-correlation function between a sample of foreground GAMA
galaxies with spectroscopic redshifts in the range $0.1 < z < 0.8$ (and divided
up into four bins) and a sample of background submillimeter galaxies from
H-ATLAS with photometric redshifts in the range $1.2 < z < 4.0$. We model the
weak lensing signal within the halo model formalism and carry out a Markov
chain Monte Carlo algorithm to obtain the posterior distribution of all HMF
parameters, which we assume to follow the Sheth and Tormen (ST) three-parameter
and two-parameter fits. Results. While the observational constraints on the HMF
from the non-tomographic analysis are not stringent, there is a remarkable
improvement in terms of uncertainty reduction when tomography is adopted.
Moreover, with respect to the traditional ST triple of values from numerical
simulations, the results from the three-parameter fit predict a higher number
density of halos at masses below $10^{12}M_{\odot}/h$ at 95% credibility. The
two-parameter fit yields even more restricting results, with a larger number
density of halos below $10^{13}M_{\odot}/h$ and a lower one above
$10^{14}M_{\odot}/h$, this time at more than 3$\sigma$ credibility. Our results
are therefore in disagreement with the standard N-body values for the ST fit at
2$\sigma$ and 3$\sigma$, respectively.
