Functional calculus in certainLizorkin-Triebel spaces (Q1345865)

We prove that every real variable function \(G\) such that \(G(0)=0\) and \(G''\) is a bounded measure acts, via left composition, on the Lizorkin- Triebel space \(F_ p^{s,q} (\mathbb{R}^ n)\), for \(1<q< +\infty\), \(1<p <+\infty\) and \(1<s< 1+(1/p)\). More precisely, there exists a number \(C= C(G,n, s,p, q)>0\) such that \[ \| G\circ f \|_{F_ p^{s,q} (\mathbb{R}^ n)}\leq C\| f \|_{F_ p^{s,q} (\mathbb{R}^ n)}, \] for all real valued \(f\in F_ p^{s,q} (\mathbb{R}^ n)\).