THE COMPOSITION IN LIZORKIN-TRIEBEL SPACES VIA PARA-DIFFERENTIAL OPERATORS

We will use the para.-differential operators for the study of the composition operator T(f) : u -> f o u on Lizorkin-Triebel space f(p,q)(s)(R(n)), in the following sense: Let f : R -> R be a function which belongs locally to Besov space B(infinity,q)(s+1)(R) such that f(0) = 0, then T(f) takes either B(infinity,q)(s)(R(n)) boolean AND F(p,q)(s)(R(n)) (in case s > 1) or W(infinity)(1)(R(n)) boolean AND F(p,q)(s)(R(n)) (in case 0 < s < 1) to F(p,q)(s) (R(n)).