Bilinear quadratic exponential sum fourth-moment bound
= Bilinear quadratic exponential sum fourth-moment bound
Two applications of the <Cauchy-Schwarz inequality> give
$$
\left|\sum_{t,r}b_tc_re(\alpha r^2t^2)\right|
\leq\|b\|_2\|c\|_2
\left(\sum_{t_1,t_2,r_1,r_2}
e\bigl(\alpha(t_1^2-t_2^2)(r_1^2-r_2^2)\bigr)
\right)^{1/4}.
$$
The fourth moment exposes differences of squares that can be factored and estimated by one-dimensional geometric sums.