  We derive a multiscale generalisation of the Bakry--\'Emery criterion for a
measure to satisfy a Log-Sobolev inequality. Our criterion relies on the
control of an associated PDE well known in renormalisation theory: the
Polchinski equation. It implies the usual Bakry--\'Emery criterion, but we show
that it remains effective for measures which are far from log-concave. Indeed,
using our criterion, we prove that the massive continuum Sine-Gordon model with
$\beta < 6\pi$ satisfies asymptotically optimal Log-Sobolev inequalities for
Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs
recently constructed via regularity structures, but our results are independent
of this theory.
