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On boundedness of superposition operators in spaces of Triebel-Lizorkin type. (English) Zbl 0693.46039
This paper contains certain results announced in the previous note [W. Sickel, Seminar analysis 1987, Teubner Texte Math. 106, 319-326 (1988)]. Therefore the notations introduced there will here be employed with no explication. Letting $$0<p<\infty$$, $$0<q\leq \infty$$, $$\sigma_{\rho}+1<s<n/p$$ and $$\rho >\sigma_{\rho}$$, we observe first that $$T_ G$$ is a bounded mapping $$\cup_{0<q\leq \infty}\tilde F^ s_{p,q}\to \cap_{0<r\leq \infty}F^{\rho}_{p,r}$$ and that for any $$0<r$$, a constant c is given such that $$\| G(f)| F^{\rho}_{p,r}\| \leq c(\| f| F^ s_{p,\infty}\| +\| f| F^ s_{p,\infty}\|^{\rho}),$$ $$f\in \tilde F^ s_{p,\infty}$$ (Theorem 1).
Particular interests are taken in the case where $$G(t)=t^{\mu}$$. Then putting $$T_ G=T_{\mu}$$, $$s_{\mu}=s-(\mu -1)((n/p)-s)$$ and letting $$\mu =m=2,3,4,...$$, $$n\cdot \max (0,(1/p)-(1/m))<s<n/p$$, one observes that for any $$0<r\leq \infty$$, there exists a constant c such that $$\| f^ m| F^{s_ m}_{p,r}\| \leq c\| f| F^ s_{p,q}\|^ m$$, $$f\in F^ s_{p,q}$$ (Theorem 4).
In case $$\mu >1$$ is not necessarily an integer and $$0<s<n/p$$, $$\sigma_{\rho}<s_{\mu}<\mu$$, it is seen that $$T_{\mu}$$ is a bounded mapping $$\cup_{0<q\leq \infty}\tilde F^ s_{p,q}\to \cap_{0<r\leq \infty}F^{s_{\mu}}_{p,r}$$ and for any $$0<r\leq \infty$$ the inequality $$\| f^{\mu}| F^{s_{\mu}}_{p,r}\| \leq c\| f| F^ s_{p,\infty}\|^{\mu},$$ $$f\in \tilde F^ s_{p,\infty}$$, holds true for some constant c. If, in particular, $$\max (1,\sigma_{\rho})<\mu <n/p$$ and $$s=1+((\mu -1)/\mu)(n/p)$$, then $$T_{\mu}$$ is a bounded mapping $$\cup_{0<q\leq \infty}\tilde F^ s_{p,q}\to F^{\mu}_{p,\infty}$$ and there exists a constant c such that $$\| f^{\mu}| F^{\mu}_{p,\infty}\| \leq c\| f| F^ s_{p,\infty}\|^{\mu}$$, $$f\in \tilde F^ s_{p,\infty}$$ (Theorem 5).
As for weakening the differentiability of G, there are given particular studies, e.g. Theorem 2, 3. Furthermore, we are given 28 remarks, most of which are devoted to the examination of various inequalities that are assumed among parameters in each theorem.
Reviewer: K.Yoshinaga

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47B38 Linear operators on function spaces (general)
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