Towards sharp superposition theorems in Besov and Lizorkin–Triebel spaces

✍ Scribed by Gérard Bourdaud; Madani Moussai; Winfried Sickel

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 517 KB
• Volume: 68
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2007.02.035

✦ Synopsis

This paper is devoted to the study of the superposition operator T f (g) := f • g in the framework of Lizorkin-Triebel spaces F s p,q (R) and Besov spaces B s p,q (R). For the case s > 1+(1/ p), 1 < p < ∞, 1 ≤ q ≤ ∞, it is natural to conjecture 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 establish this conjecture for the following two cases: (1) s -[s] > 1/ p, (2) s -[s] ≤ 1/ p < 3/4. For the case p ≤ 4/3 and s -[s] ≤ 1/ p, the conjecture is also proved, but with a restriction on s, namely

A similar result holds for Besov spaces B s p,q (R), but with some extra restrictions involving q.