Sharp weighted log-Sobolev inequalities: Characterization of equality cases and applications

By using optimal mass transport theory, we provide a direct proof to the sharp L p L^p -log-Sobolev inequality ( p ≥ 1 ) (p≥ 1) involving a log-concave homogeneous weight on an open convex cone E ⊆ R n E⊆ R^n . The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the L p L^p -log-Sobolev inequality. The characterization of the equality cases is new for p ≥ n p≥ n even in the unweighted setting and E = R n E= R^n . As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.