On a paper by Adams and Frazier 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?
 A: I'm not quite confident about what I'll write, but I think that there is an issue here. I haven't read the paper, but I'd suggest the following amendment:
By Hölder, with exponents $q$ and $q'$, to be chosen later to make everything works fine, I'd write, just after "proof of Lemma 2.4."
$
S_{\theta}(f_1f_2)(x)\le \cdots + D_{q}^{\theta/q}f_1D_{q'}^{\theta/q'}f_2(x).
$
If we choose $\frac{1}{s}=\frac{1}{q'\gamma_1}+\frac{1}{qp_1}$ and $\frac{1}{s'}=\frac{1}{q\gamma_2}+\frac{1}{q'p_2}$, then we get
$
\lVert D_{q}^{\theta/q}f_1D_{q'}^{\theta/q'}f_2\rVert_p\le \lVert D_{q}^{\theta/q}f_1\rVert_{sp}\lVert D_{q'}^{\theta/q'}f_2\rVert_{s'p}\le c\lVert f_1\rVert_{L^{sp}_{\theta/q}}\lVert f_2\rVert_{L^{s'p}_{\theta/q'}}.
$
Now, we have $\lVert f_1\rVert_{L^{sp}_{\theta/q}}\le \lVert f_1\rVert_{L^{\gamma_1p}}^{1/q'}\lVert f_1\rVert_{L^{p_1p}_{\theta}}^{1/q}$ and $\lVert f_2\rVert_{L^{sp}_{\theta/q'}}\le \lVert f_2\rVert_{L^{\gamma_2p}}^{1/q}\lVert f_2\rVert_{L^{p_2p}_{\theta}}^{1/q'}$. Hence
$\begin{align}
\lVert D_{q}^{\theta/q}f_1D_{q'}^{\theta/q'}f_2\rVert_p& \le c\lVert f_1\rVert_{L^{\gamma_1p}}^{1/q'}\lVert f_2\rVert_{L^{p_2p}_{\theta}}^{1/q'}\lVert f_2\rVert_{L^{\gamma_2p}}^{1/q}\lVert f_1\rVert_{L^{p_1p}_{\theta}}^{1/q} \\
&\le c(\lVert f_1\rVert_{L^{\gamma_1p}}\lVert f_2\rVert_{L^{p_2p}_{\theta}}+\lVert f_2\rVert_{L^{\gamma_2p}}\lVert f_1\rVert_{L^{p_1p}_{\theta}})
\end{align}$.
It remains to choose $q$, so that $sp>q$ and $s'p>q'$. Equivalently, we want to get the inequalities
$\begin{align}
p&> \frac{q-1}{\gamma_1}+\frac{1}{p_1}=\frac{q-1}{\gamma_1}+1-\frac{1}{\gamma_2} \\
p&> \frac{q'-1}{\gamma_2}+\frac{1}{p_2}=\frac{q'-1}{\gamma_2}+1-\frac{1}{\gamma_1}.
\end{align}$
Here, you can see what it does happen if you choose $q=\frac{\gamma_1+\gamma_2}{\gamma_2}>1$ and $q'=\frac{\gamma_1+\gamma_2}{\gamma_1}>1$.
Please, look carefully, I'm not sure.
