  Motivated by the entropic optimal transport problem in unbounded settings, we
study versions of Hilbert's projective metric for spaces of integrable
functions of bounded growth. These versions of Hilbert's metric originate from
cones which are relaxations of the cone of all non-negative functions, in the
sense that they include all functions having non-negative integral values when
multiplied with certain test functions. We show that kernel integral operators
are contractions with respect to suitable specifications of such metrics even
for kernels which are not bounded away from zero, provided that the decay to
zero of the kernel is controlled. As an application to entropic optimal
transport, we show exponential convergence of Sinkhorn's algorithm in settings
where the marginal distributions have sufficiently light tails compared to the
growth of the cost function.
