  Recent work has shown that NIR Hubble Space Telescope (HST) photometry allows
us to disentangle multiple populations (MPs) among M dwarfs of globular
clusters (GCs) and investigate this phenomenon in very low-mass (VLM) stars.
Here, we present the color-magnitude diagrams (CMDs) of nine GCs and the open
cluster NGC 6791 in the F110W and F160W bands of HST, showing that the main
sequences (MSs) below the knee are either broadened or split thus providing
evidence of MPs among VLM stars. In contrast, the MS of NGC 6791 is consistent
with a single population. The color distribution of M-dwarfs dramatically
changes between different GCs and the color width correlates with the cluster
mass. We conclude that the MP ubiquity, variety, and dependence on GC mass are
properties common to VLM and more-massive stars. We combined UV, optical, and
NIR observations of NGC 2808 and NGC 6121 (M 4) to identify MPs along with a
wide range of stellar masses (~ 0.2 - 0.8M ), from the MS turn off to the VLM
regime, and measured, for the first time, their mass functions (MFs). We find
that the fraction of MPs does not depend on the stellar mass and that their MFs
have similar slopes. These findings indicate that the properties of MPs do not
depend on stellar mass. In a scenario where the second generations formed in
higher-density environments than the first generations, the possibility that
the MPs formed with the same initial MF would suggest that it does not depend
on the environment.
