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This paper is devoted to the study of the composition operator T f (g) := f • g on Lizorkin-Triebel spaces F s p,q (R). In case s > 1 + (1/p), 1 < p < ∞, and 1 q ∞ we will prove the following: the operator T f takes F s p,q (R) to itself if and only if f (0) = 0 and f belongs locally to F s p,q (R).

We study the composition operator T f (g) := f • g on Besov spaces B s p,q (R). In case 1 < p < +∞, 0 < q ≤ +∞ and s > 1 + (1/ p), we will prove that the operator T f maps B s p,q (R) to itself if, and only if, f (0) = 0 and f belongs locally to B s p,q (R). For the case p = q, i.e., in case of Slobodeckij spaces, we can extend our results from the real line to R n . KeywordsHomogeneous and inhomogeneous Besov spaces on the real line • Slobodeckij spaces on R n • Functions of bounded p-variation • Composition operators • Optimal inequalities Mathematics Subject Classification (2000) 46E35 • 47H30 1 IntroductionThe present paper is a continuation of our earlier investigations, see [13][14][15] as well as [20], on composition operators, i.e., mappings

We study the dilation commuting realizations of the homogeneous Besov spaces [Formula: see text] or the homogeneous Triebel–Lizorkin spaces [Formula: see text] in the case p, q > 0, and either s - (n/p) ∉ ℕ0 or s - (n/p) ∈ ℕ0 and 0 < q ≤ 1 (0 < p ≤ 1 in the [Formula: see text]-case). We present an application to pointwise multiplication if s ≤ n/p.

In this paper we consider Besov algebras on R, that is Besov spaces B s p,q (R) for s > 1/p. For s > 1 + (1/p), p > 4/3, and q p we prove that the above algebras have a maximal symbolic calculus in the following sense: for any function f belonging locally to B s p,q (R) and such that f (0) = 0, the associated superposition operator T f (g) := f • g takes B s p,q (R) to itself.