  Our closest neighbours, the Local Group dwarf spheroidal (dSph) galaxies, are
extremely quiescent and dim objects, where thermal and non-thermal diffuse
emissions lack, so far, of detection. In order to possibly study the dSph
interstellar medium, deep observations are required. They could reveal
non-thermal emissions associated with the very-low level of star formation, or
to particle dark matter annihilating or decaying in the dSph halo. In this
work, we employ radio observations of six dSphs, conducted with the Australia
Telescope Compact Array in the frequency band 1.1-3.1 GHz, to test the presence
of a diffuse component over typical scales of few arcmin and at an rms
sensitivity below 0.05 mJy/beam. We observed the dSph fields with both a
compact array and long baselines. Short spacings led to a synthesized beam of
about 1 arcmin and were used for the extended emission search. The
high-resolution data mapped background sources, which in turn were subtracted
in the short-baseline maps, to reduce their confusion limit. We found no
significant detection of a diffuse radio continuum component. After a detailed
discussion on the modelling of the cosmic-ray (CR) electron distribution and on
the dSph magnetic properties, we present bounds on several physical quantities
related to the dSphs, such that the total radio flux, the angular shape of the
radio emissivity, the equipartition magnetic field, and the injection and
equilibrium distributions of CR electrons. Finally, we discuss the connection
to far-infrared and X-ray observations.
