Let $ell geqslant 1$. Let us consider $(g_n)_{n in mathbb{N}}$ identically distributed idependent real gaussian variables and real number $(a_{n_1,dots n_{ell}})_{(n_1, dots, n_{ell}s)inmathbb{N}^{ell}}$. We consider the random variable

$$G(omega)=sum_{n_1, dots, n_{ell}}a_{n_1, dots, n_{ell}} g_{n_1}(omega) cdots g_{n_{ell}}(omega)$$

where we write $omega in Omega$ the probability space at hand.

It is known as Gaussian polynomial chaos (or Wiener chaos)

$$|G(omega)|_{L^p_{Omega}} lesssim p^{frac{ell}{2}} |G(omega)|_{L^2_{Omega}}.$$

The standard proof of this results uses hypercontractivity of the Ornstein-Uhlenbeck semigroup.

My question is the following: in the case $ell =2$ and $p=2k$ an even integer,

Is it possible to prove “directly by hand” that

$$mathbb{E}(|G(omega)|^{2k}) leqslant C k^{2k}mathbb{E}(|G(omega)|^2)^k.$$

If yes, is there a reference book/article to find it? Or is it possible to finish my attempted proof?

This does not seem impossible. At least, when one expands the left-hand side, we have

$$mathbb{E}(|G(omega)|^{2k}) = sum_{substack{n_1,dots ,n_k\ m_1, dots, m_k}}a_{n_1,m_1} dots a_{n_{2k},m_{2k}}mathbb{E}(g_{n_1}g_{m_1}cdots g_{n_{2k}}g_{m_{2k}}).$$

From there, one can use the fact that $mathbb{E}(g^{2m+1})=0$ for any $m$ to infer that in the above, the expectation is non-vanishing only if the indices ${n_i, m_i}$ appear an even number of time. So in particular, all $n_i, m_i$ are paired.

The right hand side reads

$$k^{2k}mathbb{E}(|G(omega)|^2)^k = k^{2k}sum_{substack{n_1,dots ,n_k\ m_1, dots, m_k}}a_{n_1,m_1} dots a_{n_{2k},m_{2k}}prod_{i=1}^kmathbb{E}(g_{n_i}g_{m_i}g_{n_{k+i}}g_{m_{k+i}}).$$

Also, remark that in the LHS, when all the $g_i$ are equal, then we have $mathbb{E}(g^{4k}) lesssim k^{2k}$, which is the correct magnitude which we look for.

However I am not able to prove that $LHS leqslant RHS$. But I remain convinced that there should be somewhere to find such a proof but I did not locate it in the litterature.