  Narrow-line Seyfert 1 (NLS1) galaxies are active galactic nuclei (AGN)
believed to be in the early stages of their evolution. A fraction of them have
been found to host relativistic jets. Due to the lack of large-scale diffuse
radio emission they are believed to be experiencing one of their first activity
cycles, and can offer us an opportunity to study the early evolution of more
powerful AGN, such as radio galaxies and flat-spectrum radio quasars. Recently,
a group of intriguing jetted NLS1s was discovered: based on high radio
frequency data they host relativistic jets, but in the JVLA observations they
all showed steep radio spectra at least up to 9.0~GHz, indicating very strong
absorption at these frequencies. In this paper we study a subset of these
sources in detail by employing spatially resolved radio spectral index maps at
central frequencies of 1.6, 5.2, and 9.0~GHz. With spectral index maps we can
disentangle the different radio emission components over the radio-emitting
region, and get insights into the production mechanisms of radio emission. In
addition, we study their host galaxies in relation to the radio emission to
investigate if the host can provide us additional information regarding the
origin of the radio emission, or the launching mechanism of the jets. It is
fascinating how different the sources studied are, and certainly more,
especially wide frequency-range, and high-resolution observations will be
needed to understand their history and current properties, such as the reason
behind the extraordinary radio spectra.
