  The first Fermi-LAT High-energy source catalog (1FHL), containing gamma-ray
sources detected above 10 GeV, is an ideal sample to characterize the physical
properties of the most extreme gamma-ray sources. We investigate the pc scale
properties of a sub-sample of radio faint 1FHL sources with the aim to confirm
the proposed blazar associations, by revealing a compact high brightness
temperature radio core, and we propose new low-frequency counterparts for the
unassociated gamma-ray sources (UGS). Moreover, we increase the number of 1FHL
sources with high resolution observations to explore the possible connection
between radio and gamma rays at E >10 GeV. We observed 84 1FHL sources, mostly
blazars of High Synchrotron Peaked (HSP) type, in the northern sky with the
Very Long Baseline Array (VLBA) at 5 GHz. These sources lack high resolution
radio observations and have at least one NVSS counterpart within the 95%
confidence radius. For those sources without a well identified radio
counterpart we exploit the VLBA multiple phase-center correlation capability to
discern among the possible low-frequency candidates. For about 93% of the
sources of our sample we reveal a compact high brightness temperature radio
core, supporting their proposed blazar association. The vast majority of the
detected sources are radio weak, with a median VLBI flux density value of 16.3
mJy. For the detected sources we obtain an average brightness temperature of
the order of $2\times10^{10} \, \rm{K}$. We find a compact component for 16
UGS, for which we propose a new low-frequency association. We find brightness
temperature values which do not require high Doppler factors, and are in
agreement with the expected values for the equipartition of energy between
particles and magnetic field. We find strong indications about the blazar
nature of all of the detected UGS, for which we propose new low-frequency
associations.
