 Bilinear quadratic exponential sum fourth-moment bound

ID: bilinear-quadratic-exponential-sum-fourth-moment-bound

Two applications of the Cauchy-Schwarz inequality give

\sum_{t, r} b_{t} c_{r} e (α r^{2} t^{2}) \leq ∥ b ∥_{2} ∥ c ∥_{2} (\sum_{t_{1}, t_{2}, r_{1}, r_{2}} e (α (t_{1}^{2} - t_{2}^{2}) (r_{1}^{2} - r_{2}^{2})))^{1/4} .

The fourth moment exposes differences of squares that can be factored and estimated by one-dimensional geometric sums.

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