Fractional Sobolev logarithmic inequalities

TL;DR

This work extends Sobolev logarithmic inequalities to nonlocal Euclidean settings by deriving a fractional Sobolev logarithmic inequality for $sp<d$ with explicit constants and showing consistency with the local case as $s\to1$. The authors build a fractional Caffarelli–Kohn–Nirenberg-type interpolation inequality, connect its best constant to a variational problem, and use this to obtain sharp, explicit constants in the fractional and weighted fractional log-Sobolev inequalities. They further generalize to weighted fractional Sobolev spaces, obtaining weighted log-Sobolev inequalities with explicit constants and the corresponding interpolation framework. The optimal constant is characterized variationally in a dedicated section, linking scaling analysis to a precise minimization, and the results recover classical inequalities in the appropriate limits, providing new analytical tools for fractional Sobolev spaces and their applications in PDEs.

Abstract

We establish new Euclidean Sobolev logarithmic inequalities in the framework of fractional Sobolev spaces and their weighted version. Our approach relies on a interpolation inequality, which can be viewed as a fractional Caffarelli-Kohn-Nirenberg type inequality. We further relate the optimal constant in this interpolation inequality to a corresponding variational problem. These results extend classical Sobolev logarithmic inequalities to the nonlocal Euclidean framework and provide new tools for analysis in fractional Sobolev spaces.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['Theorem: Frac log Sob ineq']}