On an asymmetric additive energy inequality

Let $d\ge 1$ be an integer, $G$ be an abelian group and $\nu ,w_1,\dots ,w_{2d}:G\to [0,\mathrm{\infty })$ be functions with finite, non-empty supports. Define the generalised additive energy $$E_{2d,\nu }(w_1,\dots ,w_{2d})=\sum _{y,y^{\mathrm{\prime }}\in G}\sum _{a_1,\dots ,a_{2d}\in G}{w}_1(a_1)\dots w_{2d}(a_{2d})\nu (y)\nu (y^{\mathrm{\prime }})1_{\sum _{i=1}^d(a_i-a_{i+d})=y-y^{\mathrm{\prime }}}\mathrm{.}$$ Moreover, for every $1\le i\le 2d$, let $E_{2d,\nu }(w_i)=E_{2d,\nu }(w_i,\dots ,w_i)$. A standard Fourier analytic argument delivers the estimate $$E_{2d,\nu }(w_1,\dots ,w_{2d})\le \prod _{1\le i\le 2d}{E}_{2d,\nu }(w_i{)}^{1\mathrm{/}2d}\mathrm{.}$$ In this note, we present a purely combinatorial proof of the above inequality. In particular, our proof does not use any Fourier or spectral analysis and relies on repeated applications of Cauchy–Schwarz inequality combined with a discrete convexity extension type argument. We also record a variation of this upper bound in the non-abelian setting via spectral inequalities following work of Hatami on graph norms, as well as a relevant sumset analogue obtained via iterative applications of the Pl"{u}nnecke–Ruzsa inequality.