  Planets around young stars are thought to undergo atmospheric evaporation due
to the high magnetic activity of the host stars. Here we report on X-ray
observations of V1298 Tau, a young star with four transiting exoplanets. We use
X-ray observations of the host star with Chandra and ROSAT to measure the
current high-energy irradiation level of the planets, and employ a model for
the stellar activity evolution together with exoplanetary mass loss to estimate
the possible evolution of the planets. We find that V1298 Tau is X-ray bright
with $\log L_X$ [erg/s] $=30.1$ and has a mean coronal temperature of $\approx
9$ MK. This places the star amongst the more X-ray luminous ones at this
stellar age. We estimate the radiation-driven mass loss of the exoplanets, and
find that it depends sensitively on the possible evolutionary spin-down tracks
of the star as well as on the current planetary densities. Assuming the planets
are of low density due to their youth, we find that the innermost two planets
can lose significant parts of their gaseous envelopes, and could be evaporated
down to their rocky cores depending on the stellar spin evolution. However, if
the planets are heavier and follow the mass-radius relation of older planets,
then even in the highest XUV irradiation scenario none of the planets is
expected to cross the radius gap into the rocky regime until the system reaches
an age of 5 Gyr.
