## Fractional order derivative of composition

This question is about proposition 13.10.2 from Michael Taylor, "Partial Differential Equations III".

The proposition states that the $H^{s, p}$ norm of a function $F(u)$ can be estimated in terms of the $H^{s, p}$ norm of $u$, where the constant depends on $\|u\|_{L^\infty}$ and the $C^N$-norm of $F'$, where $N>s$ (here, $s$ is non-integer).

This seems strange: in order to estimate a derivative of order less than $N$, a bound on $F^{(n+1)}$ is needed. If $s$ were integer, the chain and product rule would have given a bound for the $H^{s, p}$ norm of $F(u)$ in terms of the $s$-th derivative of $F$.

Is there an easy explanation for this "gap"?

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