We report on a systematic search for hard X-ray and gamma-ray emission in coincidence with fast radio bursts (FRBs) observed by the AGILE satellite. We used 13 years of AGILE archival data searching for time coincidences between exposed FRBs and events detectable by the MCAL (0.4-100 MeV) and GRID (50 MeV-30 GeV) detectors at timescales ranging from milliseconds to days/weeks. The current AGILE sky coverage allowed us to extend the search for high-energy emission preceding and following the FRB occurrence. We considered all FRBs sources currently included in catalogues, and identified a sub-sample (15 events) for which a good AGILE exposure either with MCAL or GRID was obtained. In this paper we focus on non-repeating FRBs, compared to a few nearby repeating sources. We did not detect significant MeV or GeV emission from any event. Our hard X-ray upper limits (ULs) in the MeV energy range were obtained for timescales from sub-millisecond to seconds, and in the GeV range from minutes to weeks around event times. We focus on a sub-set of 5 non-repeating and 2 repeating FRB sources whose distances are most likely smaller than that of 180916.J0158+65 (150 Mpc). For these sources, our MeV ULs translate into ULs on the isotropically-emitted energy of about 3x10^46 erg, comparable to that observed in the 2004 giant flare from the Galactic magnetar SGR 1806-20. On average, these nearby FRBs emit radio pulses of energies significantly larger than the recently detected SGR 1935+2154 and are not yet associated with intense MeV flaring.
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