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 Lᵖ -log-Sobolev inequality (p ≥ 1) (p 1) involving a log-concave homogeneous weight on an open convex cone E ⊆ R n E Rⁿ. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the L p Lᵖ -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= Rⁿ. 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.