rademacher_l2_le_l1_of_l4_le_l2sq

de la Peña–Montgomery-Smith 1995 (arXiv:math/9309211), the L4 → L2 → L1 moment-transfer device from the proof of Lemma 2, stated on the symmetric Rademacher ($\pm 1$) fiber rademacherExpectation (the uniform $(1/2)^N$ measure on sign assignments).

For any real statistic $F$ of the sign configuration and any $K \ge 0$, the hypercontractive control $\mathbb{E}[F^4] \le K\,(\mathbb{E}[F^2])^2$ implies the L2–L1 control $\mathbb{E}[F^2] \le K\,(\mathbb{E}|F|)^2$. Squaring the moment form, this is dlP–MS Lemma 2's line $\|\xi\|_4 \le c\|\xi\|_2 \Rightarrow \|\xi\|_2 \le c^2\|\xi\|_1$ with $K = c^4$.

The proof is two applications of Cauchy–Schwarz on the probability measure: $(\mathbb{E}F^2)^2 \le (\mathbb{E}|F|)(\mathbb{E}|F|^3)$ and $(\mathbb{E}|F|^3)^2 \le (\mathbb{E}F^2)(\mathbb{E}F^4)$, combined with the hypothesis. The uniform sign weights are unconditionally nonnegative and sum to $1$ by the binomial theorem, so no $p\in[0,1]$ hypothesis is needed (unlike the Bernoulli analogue).

This is the Rademacher-fiber analogue of bernoulli_l2_le_l1_of_l4_le_l2sq; it is the more source-faithful form, since dlP–MS Lemma 2 and the Bonami–Beckner inequality are naturally stated on the $\pm 1$ cube. Source: de la Peña–Montgomery-Smith 1995, Lemma 2; Bonami 1970 / O'Donnell, Analysis of Boolean Functions, Ch. 9.