Towards sharp superposition theorems in Besov and Lizorkin–Triebel spaces

This paper is devoted to the study of the superposition operator Tf(g)≔f∘gTf(g)≔f∘g in the framework of Lizorkin–Triebel spaces Fp,qs(R) and Besov spaces Bp,qs(R). For the case s>1+(1/p)s>1+(1/p), 11/ps−[s]>1/p, (2) s−[s]≤1/p<3/4s−[s]≤1/p<3/4. For the case p≤4/3p≤4/3 and s−[s]≤1/ps−[s]≤1/p, the conjecture is also proved, but with a restriction on ss, namely |s−[s]+12−1p|>1p−34. A similar result holds for Besov spaces Bp,qs(R), but with some extra restrictions involving qq.