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Gradual smoothing: strong hypercontractivity and logarithmic Sobolev inequalities

Arturo de Pablo, David Lee, Fernando Quirós, Jorge Ruiz-Cases

TL;DR

This work investigates gradual regularity improvement for parabolic equations driven by general Lévy-type operators on $\mathbb{R}^N$, revealing a deep link between smoothing effects and a family of $p$-logarithmic Sobolev inequalities. Central to the analysis is the model operator $\mathcal{L}=\log(I-\Delta)$, which acts as a near-zero-order, subordinated nonlocal operator with kernel behaving like $|x-y|^{-N}$ at short distances; it exhibits strong hypercontractivity (solutions enter every $L^q$ space for finite time) but not ultracontractivity. The authors establish a robust equivalence between hypercontractivity and $p$-logarithmic Sobolev inequalities, and extend these results to general, possibly non-translation-invariant Lévy operators via kernel comparison. They also derive explicit quantitative hypercontractivity constants and corresponding logarithmic Sobolev inequalities, including sharp (or near-sharp) time thresholds for gradual smoothing and eventual boundedness in the translation-invariant case, thereby advancing understanding of nonlocal diffusion on infinite measure spaces. The findings illuminate the nuanced landscape between instantaneous regularization and gradual smoothing and open avenues for extensions to Nash-type inequalities and Porous Medium-type evolutions.

Abstract

We 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. 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 $L^p$ 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 $\log(I-Δ)$ is strongly hypercontractive, but fails to be supercontractive and, consequently, also fails to be ultracontractive. Finally, in the translation-invariant case, we also prove that solutions get bounded eventually.

Gradual smoothing: strong hypercontractivity and logarithmic Sobolev inequalities

TL;DR

This work investigates gradual regularity improvement for parabolic equations driven by general Lévy-type operators on , revealing a deep link between smoothing effects and a family of -logarithmic Sobolev inequalities. Central to the analysis is the model operator , which acts as a near-zero-order, subordinated nonlocal operator with kernel behaving like at short distances; it exhibits strong hypercontractivity (solutions enter every space for finite time) but not ultracontractivity. The authors establish a robust equivalence between hypercontractivity and -logarithmic Sobolev inequalities, and extend these results to general, possibly non-translation-invariant Lévy operators via kernel comparison. They also derive explicit quantitative hypercontractivity constants and corresponding logarithmic Sobolev inequalities, including sharp (or near-sharp) time thresholds for gradual smoothing and eventual boundedness in the translation-invariant case, thereby advancing understanding of nonlocal diffusion on infinite measure spaces. The findings illuminate the nuanced landscape between instantaneous regularization and gradual smoothing and open avenues for extensions to Nash-type inequalities and Porous Medium-type evolutions.

Abstract

We 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. 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 space with 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 is strongly hypercontractive, but fails to be supercontractive and, consequently, also fails to be ultracontractive. Finally, in the translation-invariant case, we also prove that solutions get bounded eventually.
Paper Structure (20 sections, 29 theorems, 218 equations)

This paper contains 20 sections, 29 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Motivation and goal
  3. Main results
  4. Precedents
  5. Outline of the paper
  6. Preliminaries
  7. Lévy operators
  8. Concept of solution
  9. Bessel potentials spaces
  10. Hypercontractivity and logarithmic Sobolev inequalities
  11. The log(I − ∆) operator
  12. Representation in terms of a Lévy kernel
  13. The heat kernel
  14. Smoothing properties
  15. Quantitative hypercontractivity
  16. ...and 5 more sections

Key Result

Lemma 1

Let $q:[0, t_\infty)\to[2, \infty)$, for some $t_\infty\in(0,\infty]$, be a $\mathcal{C}^1$ function, and let $u\ge0$ be the solution of problem-L with $u_0 \in L^1(\mathbb{R}^N) \cap L^\infty(\mathbb{R}^N)$. Then $\partial_t u(t) u(t)^{q(t)-1}$ and $u(t)^{q(t)} \log(u(t))$ belong to $\mathcal{C}([0

Theorems & Definitions (73)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 1
  • Remark 2
  • Definition 7
  • Definition 8
  • ...and 63 more