We report the results from our analysis of a large set of archival data acquired with the X-ray telescope (XRT) onboard Swift, covering the sky region surrounding objects from the first Fermi Large Area Telescope (LAT) catalogue of high-energy sources (1FHL), which still lack an association. Of the 23 regions analysed, ten did not show any evidence of X-ray emission, but 13 were characterised by the presence of one or more objects emitting in the 0.3-10 keV band. Only in a couple of cases is the X-ray counterpart located outside the Fermi positional uncertainty, while in all other cases the associations found are compatible with the high-energy error ellipses. All counterparts we found have been studied in detail by means of a multi-waveband approach to evaluate their nature or class; in most cases, we have been able to propose a likely or possible association except for one Fermi source whose nature remains doubtful at the moment. The majority of the likely associations are extragalactic in nature, most probably blazars of the BL Lac type.
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