Towards sharp superposition theorems in Besov and Lizorkin-Triebel spaces

This paper is devoted to the study of the superposition operator T-f(g) := f o g in the framework of Lizorkin-Triehel spaces F-p.q(s) (R) and Besov spaces B-p.q(s) (R). For the case s > 1 + (1/p), 1 < p < infinity, 1 <= q <= infinity, it is natural to conjecture the following: the operator T-f takes F-p.q(s) (R) to itself if and only if f(0) = 0 and f belongs locally to F-p.q(s) (R). We establish this conjecture for the following two cases: (1) s - vertical bar s vertical bar > 1/p, (2) s - vertical bar s vertical bar <= 1/p 3/4. For the case p <= 4/3 and s - vertical bar s vertical bar <= 1/p, the conjecture is also proved, but with a restriction oil s, namely vertical bar s - vertical bar s vertical bar + 1/2 - 1/p vertical bar > root 1/p - 3/4. A similar result holds for Besov spaces B-p.q(s) (R), but with some extra restrictions involving q. (C) 2007 Elsevier Ltd. All rights reserved.