I am reading the file https://arxiv.org/abs/1101.2913.
Specifically, I am interested in the part about Hypercontractivity and the log-Sobolev inequality.
Our world is the boolean cube, $F_2 ^n$. We then have the $L_q$ norms by $\mathbb{E}[f^q]^{1/q}$
The file proves that if the optimal log-sobolev constant is $\alpha$, then the $T_\rho$ is hypercontractive from $L_p$ to $L_q$ if $\rho^{-2\alpha n} \geq \frac{q-1}{p-1}$ (in fact he says it's iff, but as he mentions only proves the forward direction).
The proof is a mess of expressions that seem to magically cancel out at the end. Moreover, he only proves it for $p=2$. It is not immediate to me how to extend it beyond $p=2$.
In my experience most things in math have an intuitive and motivating explanation\proof, at the worst cases it is in retrospect but still valuable (for instance holder being the weak norm of integrating vs a function $g$).
I am asking thus for any motivation on where this result and\or proof are coming from.
I will try very lamely to motivate the proof:
We're trying to bound $|T_\rho(f)|_{2 \to q}$ .
We want to say this if $q \geq f(\rho)$ for some $f$ we'll find. We clearly need $f(1)=2$, where the inequality is clear.
It's then reasonable (although not clear how it would lead to tightness) to take the derivative of $||T_\rho f||_q$ with respect to $\rho$ (and having $q$ fixed), so we'll see for which $q, \rho$ this derivative is nonegative.
This derivative expression is an absolute mess.
Letting $G = ||||T_\rho f||_q ^q$ it's:
$q^-2 G^{\frac{1}{q}-1} ( qG' - q'Gln(G))$ (who even continues past this point??)
Next, we explicitly compute $G'$, apparently the operator $T_\rho$ is simple enough, making the answer simple. This where an expression of the form $\mathbb{E} (T_\rho f)ln(T_\rho f)$ arises, and together with the $q'Gln(G)$ from the above will eventually simplify to entropy $(T_\rho f)^q$. This suggests that if we get the Dirichlet energy between $(T_\rho f)^{q/2}$ and itself we'll be use the log-sobolev constant.
After much simplifying the rest of the expression one ends up with $\mathbb{D}[(T_\rho f)^{q-1}, (T_\rho f)$, and then via an easy pointwise inequality converts it to $\mathbb{D}[(T_\rho f)^{q/2}, (T_\rho f)^{q/2}$
I felt like stuff just simplified with no good reason, help please.