Bilinear correlation bound for the box norm

With uniform expectations and real-valued functions,Apply the Cauchy-Schwarz inequality first in $y$. Expand the remaining square, and apply the Cauchy-Schwarz inequality in $(x,x^{\prime })$ to $u(x)u(x^{\prime })$ and ${\mathbb{E}}_{y}f(x,y)f(x^{\prime },y)$. Their squared averages are $\Vert u{\Vert }_2^4$ and $\Vert f{\Vert }_{\square }^4$, respectively.

Let a tripartite graph have nonempty parts $X,Y,Z$, pair edge density of a bipartite graph values $\alpha ,\beta ,\gamma$ on $XY,YZ,XZ$, and $\Vert G(X,Y)-\alpha {\Vert }_{\square }\le c$. If every $z\in Z$ has exactly $\beta |Y|$ neighbors in $Y$, its normalized triangle count $\tau$ obeysThe constant-degree assumption makes the contribution of the constant $\alpha$ exactly $\alpha \beta \gamma$. For each fixed $z$, apply the bilinear correlation bound for the box norm to the indicator functions of its two vertex neighbourhoods. Their squared $L^2$ norms are $\beta$ and the relative $X$-degree of $z$. Average over $z$ and use the Cauchy-Schwarz inequality to bound the mean square root of that degree by $\sqrt{\gamma }$.