Dahlberg degeneracy for homogeneous Besov and Triebel-Lizorkin spaces

We consider the composition operators T-f : g bar right arrow f og acting on the real-valued homogeneous Besov or Triebel-Lizorkin spaces, realized as dilation invariant subspaces of S'(R-n), denoted as u(p,q)(s) (R-n). If s > 1 + (1/p) and s not equal n/p, then any function f : R -> R acting by composition on u(p,q)(s) (R-n) is necessarily linear. The above conditions are optimal: (i) in case = n/p, 0 < q <= 1(Besov space), 0 < p <= 1(Triebel-Lizorkin space), u(p,q)(s)(R-n) is a quasi-Banach algebra for thepointwise product, (ii) in case 1 < s <1+(1/p),1< p < infinity, 1 <= q <= infinity, any function such that f ''' is a finite measure, and f(0) = 0, acts by composition on u(p,q)(s)(R-n).