  In this article we consider a lattice system of unbounded continuos spins.
Otto & Reznikoff used the two-scale approach to show that exponential decay of
correlations yields a logarithmic Sobolev inequality (LSI) with uniform
constant in the system size. We improve their statement by weakening the
assumptions. For the proof a more detailed analysis based on two new
ingredients is needed. The two new ingredients are a new basic covariance
estimate and a uniform moment estimate. We additionally provide a comparison
principle for covariances showing that the correlations for the conditioned
Gibbs measures are controlled by the correlations of the original Gibbs measure
with ferromagnetic interaction. The latter simplifies the application of the
main result. As an application, we show how decay of correlations combined with
the uniform LSI yields the uniqueness of the infinite-volume Gibbs measure,
generalizing a result of Yoshida form finite-range to infinite-range
interaction.
