Bilinear correlation bound for the box norm
= Bilinear correlation bound for the box norm
With uniform <expectations> and <real-valued functions>,
$$
|\mathbb E_{x,y}f(x,y)u(x)v(y)|\leq\|f\|_{\square}\|u\|_2\|v\|_2.
$$
Apply the <Cauchy-Schwarz inequality> first in $y$. Expand the remaining square, and apply the <Cauchy-Schwarz inequality> in $(x,x')$ to $u(x)u(x')$ and $\mathbb E_y f(x,y)f(x',y)$. Their squared averages are $\|u\|_2^4$ and $\|f\|_{\square}^4$, respectively.