  We consider the Helmholtz transmission problem with one penetrable
star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of
the wavenumbers, we prove bounds on the solution in terms of the data, with
these bounds explicit in all parameters. In particular, the (weighted) $H^1$
norm of the solution is bounded by the $L^2$ norm of the source term,
independently of the wavenumber. These bounds then imply the existence of a
resonance-free strip beneath the real axis. The main novelty is that the only
comparable results currently in the literature are for smooth, convex obstacles
with strictly positive curvature, while here we assume only Lipschitz
regularity and star-shapedness with respect to a point. Furthermore, our bounds
are obtained using identities first introduced by Morawetz (essentially
integration by parts), whereas the existing bounds use the much-more
sophisticated technology of microlocal analysis and propagation of
singularities. We also recap existing results that show that if the assumption
on the wavenumbers is lifted, then no bound with polynomial dependence on the
wavenumber is possible.
