Three-dimensional box norm ($\Vert f{\Vert }_{{\square }^3}$)

For a complex function on $X_1\times X_2\times X_3$, its eighth power is the average of the conjugated product over the eight vertices of a coordinate cube. Three successive applications of the Cauchy-Schwarz inequality show that it bounds correlation with a product of three bounded functions, each omitting one coordinate. This is the analytic estimate behind the tetrahedron counting lemma.

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,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.