  A small contribution of molecular Bremsstrahlung radiation to the
air-fluorescence yield in the UV range is estimated based on an approach
previously developed in the framework of the radio-detection of showers in the
gigahertz frequency range. First, this approach is shown to provide an estimate
of the main contribution of the fluorescence yield due to the de-excitation of
the C $^3\Pi_{\mathrm{u}}$ electronic level of nitrogen molecules to the B
$^3\Pi_{\mathrm{g}}$ one amounting to $Y_{[337]}=(6.05\pm 1.50)~$ MeV$^{-1}$ at
800 hPa pressure and 293 K temperature conditions, which compares well to
previous dedicated works and to experimental results. Then, under the same
pressure and temperature conditions, the fluorescence yield induced by
molecular Bremsstrahlung radiation is found to be
$Y_{[330-400]}^{\mathrm{MBR}}=0.10~$ MeV$^{-1}$ in the wavelength range of
interest for the air-fluorescence detectors used to detect extensive air
showers induced in the atmosphere by ultra-high energy cosmic rays. This means
that out of $\simeq 175~$ photons with wavelength between 330 and 400 nm
detected by fluorescence detectors, one of them has been produced by molecular
Bremsstrahlung radiation. Although small, this contribution is not negligible
in regards to the total budget of systematic uncertainties when considering the
absolute energy scale of fluorescence detectors.
