Bounded linear extensions in the endpoint and quasi-Banach Lizorkin–Triebel ranges

Determine whether there exists a bounded linear extension operator from the trace space of the Lizorkin–Triebel space F^s_{p,q}(\mathbb{R}^n) on a closed d-thick set to F^s_{p,q}(\mathbb{R}^n) when p=1 or q<1, under the parameter conditions s>(n-d)/p and 0<q≤∞.

Background

The paper constructs bounded extension operators for traces of F^s_{p,q}(\mathbb{R}^n) to closed d-thick sets throughout the stated parameter range. When p>1 and q≥1, the extension operator can additionally be chosen bounded and linear by using integral averages as local approximations.

For the remaining cases p=1 or q<1, the construction based on almost-best local approximations does not yield a bounded linear operator. The authors explain that, when the local approximation exponent satisfies σ<1 and the relevant measures are nonatomic, linear local almost-best approximations are impossible because the continuous dual of L_σ for 0<σ<1 is trivial. They explicitly note that this is only an obstruction to their construction and does not settle whether another bounded linear extension operator exists.