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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 (g):=fg 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 ( n , m ) or F p,q s ( n , m ) is imbedded into L ( n , m ), and that the uniform Lipschitz continuity of f is necessary if the space is not imbedded into L ( n , m ). We also prove that the local membership to the same space is necessary for mn. We finally study the regularity of the superposition operator T f .
MSC:
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47H30Particular nonlinear operators