Measuring Triebel-Lizorkin fractional smoothness on domains in terms of first-order differences

Abstract

In this note we give equivalent characterizations for a fractional Triebel-Lizorkin space $F_{p,q}^s(\Omega )$ in terms of first-order differences in a uniform domain $\Omega$. The characterization is valid for any positive, non-integer real smoothness $s\in {\mathbb{R}}_{+}\setminus \mathbb{N}$ and {indices $1\le p<\infty$, $1\le q\le \infty$} as long as the fractional part $\{s\}$ is greater than $d/p-d/q$.

25 pages, 3 figures log: misprints fixed, a couple of proofs amended, the range of indices $p$ and $q$ has been extended to the usual endpoints, an appendix is needed to address some background in these cases

References in corpus (2)

• A T(P) theorem for Sobolev spaces on domains
• A T(1) theorem for fractional Sobolev spaces on domains

Cited by in corpus (4)

• Bourgain-Brezis-Mironescu Convergence via Triebel-Lizorkin Spaces
• Uniqueness and Regularity of the Fractional Harmonic Gradient Flow in $S^{n-1}$
• Triebel-Lizorkin regularity and bi-Lipschitz maps: composition operator and inverse function regularity
• Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale