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

VLBA observations of radio faint Fermi-LAT sources above 10 GeV