The composition in Lizorkin-Triebel spaces via para-differential operators (Q2891119)

Let \(E_{p,q}^s(\mathbb R^n)\) denote either the real Besov space \(B_{p,q}^s(\mathbb R^n)\) or the Lizorkin-Triebel space \(F_{p,q}^s(\mathbb R^n)\), where \(0<s\neq1\) and \(1\leq p,q\leq\infty\) with \(p<\infty\) in the case of the \(F\)-space. Let \(\mathcal K\) denote the Sobolev space \(W^1_\infty(\mathbb R^n)\) if \(0<s<1\) and the Besov space \(B_{\infty,q}^s(\mathbb R^n)\) if \(s>1\). Consider the composition operator \(T_f:u\mapsto f\circ u\) on \(E_{p,q}^s(\mathbb R^n)\), where \(f:\mathbb R\to\mathbb R\) is a function belonging to the Besov space \(B_{\infty,q}^{s+1,\text{loc}}(\mathbb R)\) such that \(f(0)=0\). Using the para-differential operators technique, the author proves that \(T_f\) maps \(\mathcal K\cap E_{p,q}^s(\mathbb R^n)\) into \(E_{p,q}^s(\mathbb R^n)\). An analogous result is proved for the composition operator modulo a para-product. The novelty in the paper concerns mainly the \(F\)-spaces.