Seminario EDP UAM-ICMAT
Gradual smoothing: strong hypercontractivity and logarithmic Sobolev inequalities
Ponente: Jorge Ruiz Cases (UAM)Fecha: viernes 20 de marzo de 2026 - 13:45Lugar: Aula 520, Módulo 17, Departamento de Matemáticas, UAM
Resumen:
In this talk we will study the possibility of a gradual improvement as time progresses of the regularity of solutions to evolution problems of parabolic type driven by Lévy operators, not necessarily translation invariant. The model operator for this behaviour is log(I − ∆). In the course of our analysis, we study the equivalence between general smoothing effects and a family of logarithmic Sobolev inequalities. This equivalence allows us to identify a new type of regularization, strong hypercontractivity, characterized by the existence of a time at which solutions belong to every Lp space with p finite.
It can also be used to prove logarithmic Sobolev inequalities in a context not previously seen in the literature. We then show that any purely nonlocal Lévy operator whose kernel is comparable to that of log(I − ∆) satisfies exactly the same improvement in integrability as log(I − ∆) does. Finally, in the translation-invariant case, solutions get bounded after they stop improving in integrability, and get increasingly more differentiable afterwards.
