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

In this note we give equivalent characterizations for a fractional Triebel-Lizorkin space $F^s_{p,q}(\Omega)$ 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 finite indices $p,q>1$ as long as the fractional part $\{s\}$ is greater than $d/p-d/q$.