On an asymmetric additive energy inequality

Let $d \geq 1$ be an integer, $G$ be an abelian group and $\nu, w_1, \dots, w_{2d}: G \to [0, \infty)$ be functions with finite, non-empty supports. Define the generalised additive energy $$E_{2d, \nu}(w_1, \dots, w_{2d}) = \sum_{y,y' \in G}\sum_{a_1, \dots, a_{2d} \in G } w_1(a_1) \dots w_{2d}(a_{2d}) \nu(y) \nu(y') 1_{\sum_{i=1}^d (a_i - a_{i+d}) = y-y'} .$$ Moreover, for every $1 \leq i \leq 2d$, let $E_{2d, \nu}(w_i) = E_{2d, \nu}(w_i, \dots, w_i)$. A standard Fourier analytic argument delivers the estimate $$E_{2d,\nu}(w_1, \dots, w_{2d}) \leq \prod_{1 \leq i \leq 2d} E_{2d, \nu}(w_i)^{1/2d}.$$ 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.