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Zbl 1182.46019
Allaoui, Salah Eddine
Remarks on the symbolic calculus in vector valued Besov spaces. (Remarques sur le calcul symbolique dans certains espaces de Besov à valeurs vectorielles.) (French)
[J] Ann. Math. Blaise Pascal 16, No. 2, 399-429 (2009). ISSN 1259-1734

Summary: We are interested in the superposition operators $T_{ f }(g):=f\circ g$ 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 $B_{ p,q}^s(\Bbb R^{{ n }},\Bbb R{^{ m }})$ or $F_{ p,q}^s(\Bbb R{^{ n }},\Bbb R{^{ m }})$ is imbedded into $L_{ \infty }(\Bbb R^{{ n }},\Bbb R^{ m })$, and that the uniform Lipschitz continuity of $f$ is necessary if the space is not imbedded into $L_{ \infty }(\Bbb R{^{ n }},\Bbb R^{ m })$. 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 2000:: *46E35 Sobolev spaces and generalizations
47H30 Particular nonlinear operators

Keywords: Lizorkin-Triebel spaces; Besov spaces; continuity and differentiability of superposition operators

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