Some calculus of the composition of functions in Besov-type spaces

Abstract

In the Besov-type spaces $B^{s,\tau}_{p,q}(R^n)$, we will prove that thecomposition operator $T_f: g \to f \circ g$ takes both$B^{s}_{\infty,q}(R^n)\cap B^{s,\tau}_{p,q}(R^n)$ and $W^1_{\infty}(R^n)\cap B^{s,\tau}_{p,q}(R^n)$ to $B^{s,\tau}_{p,q}(R^n)$, under somerestrictions on $s, \tau, p,q$, and if the real function $f$ vanishes at the origin and belongs locally to $B^{s+1}_{\infty,q}({R})$.