Bilinear correlation bound for the box norm

For real-valued functions or complex functions on finite nonempty sets, with uniform expectations,To keep the complex case valid, retain every complex conjugate. Apply the Cauchy-Schwarz inequality first in $y$ and writeA second Cauchy-Schwarz inequality in $(x,x^{\prime })$ bounds the last expectation by $\Vert u{\Vert }_2^2({\mathbb{E}}_{x,x^{\prime }}|K(x,x^{\prime }){|}^2{)}^{1/2}$. The final factor is $\Vert f{\Vert }_{\square }^2$, by the box norm definition. Taking square roots proves the result. For real-valued functions, the complex conjugates can be omitted.

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 $\sqrt{\gamma }$.