Good Bounds in Certain Systems of True Complexity One

Let $\Phi = (\phi_1,\dots,\phi_6)$ be a system of $6$ linear forms in $3$ variables, i.e. $\phi_i \colon \mathbb{Z}^3 \to \mathbb{Z}$ for each $i$. Suppose also that $\Phi$ has Cauchy--Schwarz complexity $2$ and true complexity $1$, in the sense defined by Gowers and Wolf; in fact this is true generically in this setting. Finally let $G = \mathbb{F}_p^n$ for any $p$ prime and $n \ge 1$. Then we show that multilinear averages by $\Phi$ are controlled by the $U^2$-norm, with a polynomial dependence; i.e. if $f_1,\dots,f_6 \colon G \to \mathbb{C}$ are functions with $\|f_i\|_{\infty} \le 1$ for each $i$, then for each $j$, $1 \le j \le 6$: $$\left| \mathbb{E}_{x_1,x_2,x_3 \in G} f_1(\varphi_1(x_1,x_2,x_3)) \dots f_6(\phi_6(x_1,x_2,x_3)) \right| \le \|f_j\|_{U^2}^{1/C}$$ for some $C > 0$ depending on $\Phi$. This recovers and strengthens a result of Gowers and Wolf in these cases. Moreover, the proof uses only multiple applications of the Cauchy--Schwarz inequality, avoiding appeals to the inverse theory of the Gowers norms. We also show that some dependence of $C$ on $\Phi$ is necessary; that is, the constant $C$ can unavoidably become large as the coefficients of $\Phi$ grow.