Let $(x_i)_{i \in \mathbb{N}}$ be a family of independent $\pm 1$ centered Bernoulli random variables, and let $p, q > 1$. There exists a constant C such that for every (finite) linear combination $Y$ of the $x_i$'s, $$ \frac{1}{C} \|Y\|_q \le \|Y\|_p \le C \|Y\|_q. $$ A proof of this can be obtained from the hypercontractivity of the "heat" semi-group, see for instance Theorem V.2 in the lecture notes [1].
Q1: Is there a more direct way to prove this?
One thing I tried is to observe that linear combinations of $x_i$'s are those functions that are stable under the convolution with $$ g(x) = \sum_i x_i $$ (say that there are only a finite number $n$ of $x_i$'s, but that you want an inequality that does not depend on $n$). [I liked this idea because it looked very much the same as what one does with Littlewood-Paley decompositions in $\mathbb{R}^d$ (for fixed $d$)]. Then some Young inequality sounded attractive, but in fact fails because the constant one gets does depend on $n$.
From hypercontractivity, one gets that the inequality on top is in fact valid for any $Y$ that is a homogeneous polynomial (the constant $C$ depending on the degree).
Q2: does the "direct proof" from Q1 extend nicely to the more general form?
Q3: How optimal is the constant $C$ one gets from hypercontractivity? From the direct method, if any?
Remark: if the $x_i$'s are Gaussian, then the question is about elements of homogeneous Wiener chaoses; but in this case Q1 is trivial since Y is a standard Gaussian.
Ref: [1] Garban, Steif. http://arxiv.org/abs/1102.5761
