= Pair-factor correlation bound for the three-dimensional box norm
{title2=$|\mathbb E\,g\,a_{yz}b_{xz}c_{xy}|\leq\|g\|_{\square^3}$}
For a real <function> $g$ on $X\times Y\times Z$ and <functions> $a(y,z),b(x,z),c(x,y)$ bounded in absolute value by one,
$$
|\mathbb E_{x,y,z}g(x,y,z)a(y,z)b(x,z)c(x,y)|\leq\|g\|_{\square^3}.
$$
First apply <Cauchy-Schwarz inequality> in $(y,z)$ to remove $a$, duplicating $x$. Apply it in $(x_0,x_1,z)$ to remove $b(x_0,z)b(x_1,z)$, duplicating $y$. Finally apply it in $(x_0,x_1,y_0,y_1)$ to remove the four $c$ factors, duplicating $z$. The resulting eighth-power bound is exactly the eight-factor average defining the <three-dimensional box norm>. <Functions> depending on just one coordinate, or constants, can be incorporated in any of the three pair factors.
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