  We consider spin systems in the $d$-dimensional lattice $Z^d$ satisfying the
so-called strong spatial mixing condition. We show that the relative entropy
functional of the corresponding Gibbs measure satisfies a family of
inequalities which control the entropy on a given region $V\subset Z^d$ in
terms of a weighted sum of the entropies on blocks $A\subset V$ when each $A$
is given an arbitrary nonnegative weight $\alpha_A$. These inequalities
generalize the well known logarithmic Sobolev inequality for the Glauber
dynamics. Moreover, they provide a natural extension of the classical Shearer
inequality satisfied by the Shannon entropy. Finally, they imply a family of
modified logarithmic Sobolev inequalities which give quantitative control on
the convergence to equilibrium of arbitrary weighted block dynamics of heat
bath type.
