Asymptotic Behaviour of fractional seminorms

TL;DR

This work provides asymptotically sharp identifications among fractional Sobolev spaces $W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces $\dot{F}^{s}_{p,q}$, along with BBM-type stability results as $s \to 1$. By leveraging harmonic extension seminorms via the Poisson kernel and a refined Littlewood-Paley theory for mixed $L^{p}$ spaces, the authors establish sharp upper and lower bounds linking these spaces, including new results even in the $p=q$ case. The paper also clarifies the precise relationship between $W^{s}_{p,q}$ and $E^{s}_{p,q}$ in a stable regime (notably at $q=2$) and provides BBM-type convergence theorems that connect fractional seminorms to classical Sobolev norms. Collectively, these results advance the understanding of fractional functional spaces and their interconnections, with potential implications for fractional PDEs and related harmonic analysis frameworks.

Abstract

We obtain asymptotically sharp identification of fractional Sobolev spaces $ W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces $\dot{F}^s_{p,q}$. In particular we obtain for $W^{s}_{p,q}$ and $E^{s}_{p,q}$ a stability theory a la Bourgain-Brezis-Mironescu as $s \to 1$, answering a question raised by Brazke--Schikorra--Yung. Part of the results are new even for $p=q$.

Theorems & Definitions (101)

• Theorem 1.1: BBM1
• Theorem 1.2: BBM2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10