  We prove logarithmic Sobolev inequality for measures $$
q^n(x^n)=\text{dist}(X^n)=\exp\bigl(-V(x^n)\bigr), \quad x^n\in \Bbb R^n, $$
under the assumptions that: (i) the conditional distributions $$ Q_i(\cdot|
x_j, j\neq i)=\text{dist}(X_i| X_j= x_j, j\neq i) $$ satisfy a logarithmic
Sobolev inequality with a common constant $\rho$, and (ii) they also satisfy
some condition expressing that the mixed partial derivatives of the Hamiltonian
$V$ are not too large relative to $\rho$. \bigskip Condition (ii) has the form
that the norms of some matrices defined in terms of the mixed partial
derivatives of $V$ do not exceed $1/2\cdot\rho\cdot(1-\de)$. The logarithmic
Sobolev constant of $q^n$ can then be estimated from below by
$1/2\cdot\rho\cdot\delta$. This improves on earlier results by Th. Bodineau and
B. Helffer, by giving an explicit bound, for the logarithmic Sobolev constant
for $q^n$.
