  We consider an one-dimensional lattice system of unbounded and continuous
spins. The Hamiltonian consists of a perturbed strictly-convex single-site
potential and with longe-range interaction. We show that if the interactions
decay algebraically of order 2+a, a>0 then the correlations also decay
algebraically of order 2+\~a for some \~a > 0. For the argument we generalize a
method from Zegarlinski from finite-range to infinite-range interaction to get
a preliminary decay of correlations, which is improved to the correct order by
a recursive scheme based on Lebowitz inequalities. Because the decay of
correlations yields the uniqueness of the Gibbs measure, the main result of
this article yields that the on-phase region of a continuous spin system is at
least as large as for the Ising model.
