Asymptotic Behaviour of fractional seminorms

TL;DR

This paper characterizes the asymptotic behavior of fractional Sobolev, extension, and Triebel-Lizorkin spaces, providing a stability theory as the fractional parameter approaches 1, and addresses open questions in the field.

Contribution

It offers a sharp asymptotic identification of these function spaces and develops a stability theory as the fractional parameter tends to 1, answering prior open questions.

Findings

Established asymptotic equivalences for fractional Sobolev and extension spaces as s approaches 1.

Developed a stability theory similar to Bourgain-Brezis-Mironescu for these spaces.

Provided new results even in the case p=q.

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 $p=q$.