Logarithmic Sobolev Inequalities

After Poincare inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. They contain much more information than Poincare inequalities, and are at the same time sufficiently general to be available in numerous cases of interest, in particular in infinite dimension (as limits of Sobolev inequalities on finite-dimensional spaces). After the basic definition of a logarithmic Sobolev inequality together with its first properties, the first sections of this chapter present the exponential decay in entropy and the fundamental equivalence between the logarithmic Sobolev inequality and smoothing properties of the semigroup in the form of hypercontractivity. Next, integrability properties of eigenvectors and of Lipschitz functions under a logarithmic Sobolev inequality are discussed together with a criterion for measures on the real line to satisfy a logarithmic Sobolev inequality (for the usual gradient). The further sections deal with curvature conditions, first for the local logarithmic Sobolev inequalities for heat kernel measures, then for the invariant measure with an additional dimensional information. Local hypercontractivity and some applications of the local logarithmic Sobolev inequalities towards heat kernel bounds are further presented. Harnack-type inequalities under the infinite-dimensional curvature conditions, linked with reverse local logarithmic Sobolev inequalities complete the chapter.