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

Abstract

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