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Measuring Triebel-Lizorkin fractional smoothness on domains in terms of first-order differences. (English) Zbl 1428.42041
Summary: 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 $$1 \leqslant p < \infty, 1 \leqslant q \leqslant \infty$$ as long as the fractional part $$\{ s \}$$ is greater than $$d / p -d / q$$.
MSC:
 42B35 Function spaces arising in harmonic analysis 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Keywords:
Triebel-Lizorkin space
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