I have a question in proving Meyer's inequality. The proof I read is taken from the book "Malliavin Calculus and related topics" by Nualart.
I just have one equality which I am not sure, I will attach the statement and partly of its proof up to the steps that I am not clear:
$$=E\left(\left|\sum^\infty_{i=1}\sum_{\alpha\in\Bbb N^k_*}(D_iD^k_\alpha F)^2\right|^\frac{p}{2}\right)$$
$$=A_p^{-1}\int_0^1\int_0^1 E\left(\left|\sum^\infty_{i=1}\sum_{\alpha\in\Bbb N^k_*} D_iD^k_\alpha F\gamma_i(s)\right|^p\right)dtds$$
I am not sure about the final equality. It seems that he use Gaussian Formula, i.e. $$E[|\sum_{i} a_{i}\zeta_{i}|^{p}]=A_{p}\left(\sum_{i} a_{i}^{2}\right)^{\frac{p}{2}} $$ where $A_{p}$ is a constants, $\zeta_{i}$ are i.i.d. and $\sum_{i} a_{i}^{2}<\infty$. If it is just a single sum, I understand how to make use the Gaussian Formula, but I am not sure how he can arrive the final equality when it involves a double summation, could someone here suggest a way to arrive it?