When $f : \{-1,1\}^{n} \to \mathbb{C}$ is Walsh--Rademacher chaoes of degree $k$, i.e., $$ f(x) = \sum_{1\leq j_{1}<\ldots<j_{k}\leq n} a_{j_{1}\ldots j_{k}}x_{j_{1}}\cdots x_{j_{k}} \quad (*) $$ where $x = (x_{1}, \ldots, x_{n}) \in \{-1,1\}^{n}$ then one can improve by square root the bound in Theorem 22.
Namely, jointly with Tomasz Tkocz, we have obtained the following result
Theorem. For any $f(x) = \sum_{1\leq j_{1}<\ldots<j_{k}\leq n} a_{j_{1}\ldots j_{k}}x_{j_{1}}\cdots x_{j_{k}}$ we have $$\|f\|_{2}\leq e^{k/2}\|f\|_{1}.$$
The argument uses complex hypercontractivity instead of real one. Using Beckner's result in the proof of Hausdorff--Young inequality with sharp constants, one has $\|T_{i\sqrt{p-1}} h\|_{q}\leq\|h\|_{p}$, for any $h :\{-1,1\}^{n} \to \mathbb{C}$, and any $1\leq p \leq q <\infty$, $\frac{1}{p}+\frac{1}{q}=1$. This immediately implies that we have strong hypercontractivity for Rademacher chaoses of degree $k$ on the Hamming cube $$ \left(\sqrt{\frac{p}{q}}\right)^{k}\|f\|_{q} = \|T_{\sqrt{\frac{p}{q}}}f\|_{q}\leq \|f\|_{p} \quad (**) $$ whenever $q\geq p$, and $p,q$ are dual exponents. Using Holder's inequality one can show $$ \|f\|_{2} \leq \|f\|_{1} \left(\frac{\|f\|_{q}}{\|f\|_{p}}\right)^{\frac{1}{2(1/p-1/q)}} $$ Finally applying (**), and then taking the limit $p \to 2-$ we obtain the theorem.
Our starting point in these questions was Pelczinski's conjecture which says that for degree k=2 chaoses we have the sharp estimate $\|f\|_{2}\leq2\|f\|_{1}$, and as you can see our theorem gives $e$ instead of $2$ (real hypercontractivity would give $e^{2}>7.3...$). It is also interesting to remark that for degree 1 chaoses (Khinchin case) the sharp bound is $\sqrt{2}$ due to Szarek, however, our theorem gives $\sqrt{e}$ for free.
