We are interested in the superposition operators $T_f(g):= fcirc g$ on vector valued Besov and Lizorkin-Triebel spaces of positive smoothness exponent $s$. As a first step towards the characterization of functions which operate, we establish that the local Lipschitz continuity of $f$ is necessary if the space $B_{{p},{q}}^{s}({mathbb{R}}^n,{mathbb{R}}^m)$ or $F_{{p},{q}}^{s}({mathbb{R}}^n,{mathbb{R}}^m)$ is imbedded into $L_infty ({mathbb{R}}^n,mathbb{R}^m)$, and that the uniform Lipschitz continuity of $f$ is necessary if the space is not imbedded into $L_infty ({mathbb{R}}^n,mathbb{R}^m)$. We prove also that the local membership to the same space is necessary for $mle n$. We finally study the regularity of the superposition operator $T_f$.
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机译：我们对正光滑指数为$ s $的矢量值Besov和Lizorkin-Triebel空间的矢量上的叠加算子$ T_f（g）：= f circ g $感兴趣。作为表征功能的第一步，我们确定如果空间$ B _ {{p}，{q}} ^ {s}（{ mathbb {R} } ^ n，{ mathbb {R}} ^ m）$或$ F _ {{p}，{q}} ^ {s}（{ mathbb {R}} ^ n，{ mathbb {R}} ^ m）$被嵌入到$ L_ infty（{ mathbb {R}} ^ n， mathbb {R} ^ m）$中，并且如果空间没有被嵌入，则$ f $的统一Lipschitz连续性是必要的$ L_ infty（{ mathbb {R}} ^ n， mathbb {R} ^ m）$。我们还证明，对于$ m le n $，必须使用同一空间的本地成员身份。我们最终研究了叠加算子$ T_f $的正则性。
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