Dahlberg degeneracy for homogeneous Besov and Triebel–Lizorkin spaces

We consider the composition operators Tf:g↦f∘g$T_f: g\mapsto f\circ g$ acting on the real‐valued homogeneous Besov or Triebel–Lizorkin spaces, realized as dilation invariant subspaces of S′(Rn)$\mathcal {S}^{\prime }({\mathbb {R}}^n)$, denoted as Ap,qs(Rn)$\mathfrak {A}^s_{p,q}({\mathbb {R}}^n)$. If s>1+(1/p)$s>1+ (1/p)$ and s≠n/p$s\not= n/p$, then any function f:R→R$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ acting by composition on Ap,qs(Rn)$\mathfrak {A}^s_{p,q}({\mathbb {R}}^n)$ is necessarily linear. The above conditions are optimal: (i) in case s=n/p$s=n/p$, 0