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Remarks on the symbolic calculus in vector valued Besov spaces. (Remarques sur le calcul symbolique dans certains espaces de Besov à valeurs vectorielles.) (French) Zbl 1182.46019
Summary: We are interested in the superposition operators ${T}_{f}\left(g\right):=f\circ g$ 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 ${B}_{p,q}^{s}\left({ℝ}^{n},ℝ{}^{m}\right)$ or ${F}_{p,q}^{s}\left(ℝ{}^{n},ℝ{}^{m}\right)$ is imbedded into ${L}_{\infty }\left({ℝ}^{n},{ℝ}^{m}\right)$, and that the uniform Lipschitz continuity of $f$ is necessary if the space is not imbedded into ${L}_{\infty }\left(ℝ{}^{n},{ℝ}^{m}\right)$. We also prove that the local membership to the same space is necessary for $m\le n$. We finally study the regularity of the superposition operator ${T}_{f}$.
##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47H30 Particular nonlinear operators