Regularity of the symbolic calculus in Besov algebras

We consider Besov and Lizorkin-Triebel algebras, that is, the real-valued function spaces B-p,q(s) (R-n) boolean AND L-infinity(R-n) and F-p,q(s) (R-n) n L-infinity (R-n) for all s > 0. To each function f : R -> R one can associate the composition operator T-f which takes a real-valued function g to the composite function f circle g. We give necessary conditions and sufficient conditions on f for the continuity, local Lipschitz continuity, and differentiability of any order of T-f as a map acting in Besov and Lizorkin-Triebel algebras. In some cases, such as for n = 1, such conditions turn out to be necessary and sufficient.