Functions which operate on Besov spaces (Q1906495)

We determine a new class of real variable functions which act, via left composition, on the Besov space \(B^s_{p, q}(\mathbb{R}^n)\), for \(0< s< 1+ (1/p)\). \(F^{1+ (1/p)}_p\) is the set of functions \(F\) such that there exists a constant \(C> 0\) for which \[ \int_{\mathbb{R}} \Biggl(\sup_{|\rho|\leq \rho} |F'(x+ h)- F'(x)|\Biggr)^p dx\leq Ch, \] whatever be \(h> 0\). Every function \(F\in F^{1+ (1/p)}_p\) such that \(F(0)= 0\) and \(F'\in L^\infty\) acts on \(B^s_{p, q}(\mathbb{R}^n)\). The space \(F^{1+ (1/p)}_p\) includes the Besov spaces \(B^t_{p, q}(\mathbb{R})\), for \(t> 1+ (1/ p)\), and the function classes known as acting on \(B^s_{p, q}(\mathbb{R}^n)\) (works of Yves Meyer and the first author).