We report the results of an extensive radio-continuum observing campaign of host galaxies of short gamma-ray bursts (GRBs). The goal of this survey was to search for optically obscured star formation, possibly indicative of a population of young short-GRB progenitors. Our sample comprises the hosts and host-galaxy candidates of 16 short-GRBs from 2005 to 2015, corresponding to roughly 1/3 of the presently known ensemble of well-localized short bursts. Eight GRB fields were observed with ATCA (at 5.5 and 9.0 GHz), and eight fields with the VLA (mostly at 5.5 GHz). The observations typically achieved a 1-sigma_rms of 5 to 8 microJy. In most cases they were performed years after the corresponding burst. No new short-GRB host with optically obscured star formation was found. Only one host galaxy was detected, the one of GRB 100206A at z=0.407. However, its starburst nature was already known from optical/IR data. Its measured radio flux can be interpreted as being due to a star formation rate (SFR) of about 60 M_sol/yr. This is in good agreement with earlier expectations based on the observed broad-band spectral energy distribution of this galaxy. The 15 non-detections constrain the SFRs of the suspected host galaxies and provide upper limits on late-time luminosities of the associated radio afterglows and predicted kilonova radio flares. The non-detection of radio emission from GRB explosion sites confirms the intrinsically low luminosity of short-GRB afterglows and places significant constraints on the parameter space of magnetar-powered radio flares. Luminous radio flares from fiducial massive magnetars have not been found.
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