$L^p$ integrability of products of Gaussian variables

Question

Gaussian variables have moments of all orders, so by Hölder's inequality the product of two Gaussian variables $\xi$ and $\eta$ has finite $L^1$-norm:

$$ \|\xi \cdot \eta\|_1 \leq \|\xi\|_2 \cdot \|\eta\|_2 < \infty. $$

Can one strengthen this argument to show that the $L^2$-norm (or, in general, $L^p$ for $p<\infty$) is also finite?

Answer 1

It looks to me that $\xi\cdot\eta$ has a finite $L^p$ norm due to Young's inequality. For instance: $$\|\xi\cdot\eta\|_p^p = \mathbb{E}[|\xi|^p\cdot|\eta|^p]\leq\frac{1}{2}\left(\mathbb{E}[|\xi|^{2p}]+\mathbb{E}[|\eta|^{2p}]\right)=\frac{\|\xi\|_{2p}^{2p}+\|\eta\|_{2p}^{2p}}{2}.$$