A bilinear estimate in ${\mathbb{F}}_p$

We improve an $L^2\times L^2\to L^2$ estimate for a certain bilinear operator in the finite field of size $p$, where $p$ is a prime sufficiently large. Our method carefully picks the variables to apply the Cauchy-Schwarz inequality. As a corollary, we show that there exists a quadratic progression $x,x+y,x+y^2$ for nonzero $y$ inside any subset of ${\mathbb{F}}_p$ of density $\gtrsim p^{-1\mathrm{/}8}$