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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
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