  We are interested in the Logarithmic Sobolev Inequality for the infinite
volume Gibbs measure with no quadratic interactions. We consider unbounded spin
systems on the one dimensional Lattice with interactions that go beyond the
usual strict convexity and without uniform bound on the second derivative. We
assume that the one dimensional single-site measure with boundaries satisfies
the Log-Sobolev inequality uniformly on the boundary conditions and we
determine conditions under which the Log-Sobolev Inequality can be extended to
the infinite volume Gibbs measure.
