Non-local Torsion functions and Embeddings

TL;DR

This paper investigates the embedding properties of fractional Sobolev spaces, establishing conditions for compactness and continuity related to fractional torsion functions and employing non-local Hardy inequalities.

Contribution

It introduces new links between fractional torsion functions and embedding properties of fractional Sobolev spaces, including compactness and continuity criteria.

Findings

Established compactness of embeddings for certain q values.

Connected embedding continuity to fractional torsion function summability.

Utilized non-local Hardy inequalities involving fractional torsion functions.

Abstract

Given $s \in (0,1)$, we discuss the embedding of $\mathcal D^{s,p}_0(\Omega)$ in $L^q(\Omega)$. In particular, for $1\le q < p$ we deduce its compactness on all open sets $\Omega\subset \mathbb R^N$ on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in $\Omega$ in a suitable weak sense, for every open set $\Omega$. The proofs make use of a non-local Hardy-type inequality in $\mathcal D^{s,p}_0(\Omega)$, involving the fractional torsion function as a weight.