I am reading a paper by Adams and Frazier (namely Adams, Frazier, Composition operators on potential spaces. Proc. Amer. Math. Soc. 114 (1992), no. 1, 155–165, available here), whose main purpose is to show that, whenever $H\in C^\infty(\mathbb R)$ and $f\in L_\alpha^p\cap\dot L_1^{\alpha p}(\mathbb{R}^n)$, the composition $H(f)$ lies in the same space.
They assume that $\alpha>1$, $1<p<\infty$ and that the first $\lceil\alpha\rceil$ derivatives of $H$ are bounded. The space $L_\alpha^p$ is the inhomogeneous fractional Bessel potential space (which today is often denoted by $H^{\alpha,p}$), and $\dot L_1^{\alpha p}$ is the homogeneous Sobolev space (tempered distributions whose gradient is in $L^{\alpha p}$).
Question: they apply the estimate (2.10), i.e. $\|D_t^\alpha f\|_{L^p}\lesssim\|f\|_{\dot L_\alpha^p}$ (where $D_t^\alpha$ is some nonlinear operator defined as an integral of difference quotients), during the proof of Lemma 2.4 (on page 164). However, they forget to check that $sp>2$ and $s'p>2$, which is necessary in order to invoke (2.10). It seems that this can fail in general, also looking at how this Lemma 2.4 is applied in the proof of the main theorem (unless we assume e.g. $p>2$).
Did anybody else come across this? Am I missing something?