log-Sobolev inequalities in Simon's Harmonic Analysis I'm having quite a time reading about log-Sobolev inequalities and hypercontractivity in Simon's Harmonic Analysis.  Proposition 6.6.16 is the culprit.  
For context, $(X,\mu)$ is a probability space and $A$ generates a Markov semi-group on $L^{2}$ that extends to a contractive semi-group on $L^{p}$ for each $p$.  Also, $\mathcal{D}_{++} = \bigcup_{s > 0} e^{-sA} \{u \in L^{\infty} \, \mid \, \exists \epsilon > 0 \, \, \text{so} \, \, \epsilon \mathbf{1} \leq u\}$. 
The statement is:
(a) For $f \in \mathcal{D}_{++}$, $G(p) = \|f\|_{p}^{p}$ is $C^{1}$ in $p$ on $[0,\infty)$ and
$$G'(p) = \int_{X} f(x)^{p} \log(f(x)) \, \mu(dx)$$
(b)  For $f \in \mathcal{D}_{++}$, $Q(p) = \|f\|_{q}$ is $C^{1}$ and 
$$Q'(p) = (p\|f\|_{p}^{p - 1})^{-1} \left(\int_{X} f(x)^{p} \log(f(x)) - \|f\|^{p}_{p} \log(\|f\|_{p})\right)$$
(c) For $f \in \mathcal{D}_{++}$ and $p$ fixed, $F(s) = \|e^{-sA} f\|_{p}^{p}$ is $C^{1}$ on $[0,\infty)$ and (with $f_{s} = e^{-sA}f$)
$$F'(s) = - \int_{X} f_{s}(x)^{p-1} (Af_{s})(x) \, \mu(dx)$$
(d) If $f \in \mathcal{D}_{++}$ and $p$ is fixed and $R(s) = \|e^{-sA}f\|_{p}$, then 
$$R'(s) = - (p\|f\|_{p}^{p - 1})^{-1} \int_{X} f_{s}(x)^{p - 1} (Af_{s})(x) \, \mu(dx)$$
(e) If $p(s) \in [1,\infty)$ is a $C^{1}$-function of $s$ and $M(s) \in (-\infty,\infty)$ is a $C^{1}$-function of $s$, then for $f \in \mathcal{D}_{++}$
$$H(s) = \log(e^{-M(s)} \|f_{s}\|_{p(s)}); \quad f(s) = e^{-sA}f$$
is $C^{1}$ and 
\begin{align*}
\frac{dH}{ds} &= - M'(s) \\
&\quad + \|f_{s}\|_{p(s)}^{-p(s)} p(s)^{-1} p'(s) \left\{ \int_{X} f_{s}(x)^{p(s)} \log(f_{s}(x)) \, \mu(dx) \right. \\
&\quad \left. - \|f_{s}\|_{p(s)}^{p(s)} \log(\|f_{s}\|_{p(s)}) - \int_{X} f_{s}(x)^{p-1} (Af_{s})(x) \, \mu(dx)\right\}
\end{align*}
(a) and (b) are correct.  I'm convinced (c) is missing a factor of $p$ in $F'(s)$ (i.e. it should be $-p \int_{X}$...)  This propagates to (d), which should not have a $p$ in the first factor appearing in $R'(s)$.  This error propagates to (e), where, in addition, there's a new factor of $p'(s)$ in front of the third summand in the brackets...  In other words, I fear the third term in the brackets should not be in the brackets at all and, instead, the following holds:
\begin{align*}
\frac{dH}{ds} &= - M'(s) -\|f_{s}\|_{p(s)}^{-p(s)} \int_{X} f_{s}(x)^{p-1} (Af_{s})(x) \, \mu(dx) \\
&\quad + \|f_{s}\|_{p(s)}^{-p(s)} p(s)^{-1} p'(s) \left\{ \int_{X} f_{s}(x)^{p(s)} \log(f_{s}(x)) \, \mu(dx) \right. \\
&\quad \left. - \|f_{s}\|_{p(s)}^{p(s)} \log(\|f_{s}\|_{p(s)}) \right\}
\end{align*}
However, if this is the case, it messes with later results in the chapter.  Am I seeing this correctly or, if not, where am I erring?
 A: For the sake of others who might be wading into these waters as well:
The errors I pointed out are indeed errors.  In some sense, this is good since they need to be!  Consider $H$ defined in 6.6.166 of the textbook.  As I correctly pointed out, the derivative of $H$ satisfies
\begin{align*}
H'(s) = -M'(s) + \left(\|f_{s}\|_{\tilde{p}(s)}^{\tilde{p}(s)} \tilde{p}(s) \right)^{-1} \tilde{p}'(s) \left\{ \int_{X} f_{s}(x)^{\tilde{p}(s)} \log(f_{s}(x)) \mu(dx) - \|f_{s}\|^{\tilde{p}(s)}{\tilde{p}(s)}\right\} - \|f_{s}\|^{-\tilde{p}(s)}_{\tilde{p}(s)} \int_{X} f_{s}(x)^{\tilde{p}(s) - 1} (Af_{s})(x) \, \mu(dx)
\end{align*}
As Simon points out, $M'(s) = \Gamma(\tilde{p}(s)) \tilde{p}'(s) \tilde{p}(s)^{-1}$.  Also, observe that $\tilde{c}(s) = \tilde{p}(s) \tilde{p}'(s)^{-1}$.  Therefore, we can write
\begin{align*}
H'(s) = \left(\|f_{s}\|_{\tilde{p}(s)}^{\tilde{p}(s)} \tilde{p}(s) \right)^{-1} \tilde{p}'(s) \left\{ \int_{X} f_{s}(x)^{\tilde{p}(s)} \log(f_{s}(x)) \mu(dx) - \|f_{s}\|^{\tilde{p}(s)}{\tilde{p}(s)} - \tilde{c}(s) \int_{X} f_{s}(x)^{\tilde{p}(s) - 1} (Af_{s})(x) \, \mu(dx) - \Gamma(\tilde{p}(s)) \|f_{s}\|^{\tilde{p}(s)}_{\tilde{p}(s)}\right\}
\end{align*}
Now we can conclude that $H'(s) \leq 0$ if and only if $A$ obeys a $L^{\tilde{p}(s)}$-logarithmic Sobolev inequality with local constant $c(\tilde{p}(s))$ and local norm $\Gamma(\tilde{p}(s))$.  This is what Simon was trying to say: he simply made a bunch of typos (even as far back as Proposition 6.6.16) and neglected the local constant entirely...
