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Zbl 1194.46055
Schneider, Cornelia; Vyb{\'\i}ral, Jan
On dilation operators in Triebel-Lizorkin spaces.
(English)
[J] Funct. Approximatio, Comment. Math. 41, No. 2, 139-162 (2009). ISSN 0208-6573

The behaviour of the dilation operators $T_k$, $$T_k f(x) = f (2^k x), \quad x \in \Bbb R^n, \quad k \in \bbfN,$$ is well understood in the spaces $B^s_{pq} (\Bbb R^n)$ and $F^s_{pq} (\Bbb R^n)$ with $0<p,q \le \infty$ ($p< \infty$ for $F$-spaces), $s > \frac{n}{\min (p,1)} -n$. But it was an open problem what happens in the limiting cases $1 \le p < \infty$, $s=0$ and $0<p<1$, $s = \frac{n}{p} - n$. Whereas the $B$-case has been treated by the authors in earlier publications, the paper under review deals with the $F$-case, producing some unexpected effects. As a consequence, these limiting $F$-spaces do not coincide with corresponding spaces defined by other means (moduli of smoothness, subatomic decompositions).
[Hans Triebel (Jena)]
MSC 2000:
*46E35 Sobolev spaces and generalizations
47A20 Extensions and related concepts of linear operators

Keywords: Triebel-Lizorkin spaces; Besov spaces; dilation operators; moment conditions

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