An Analogue of Bonami's Lemma for Functions on Spaces of Linear Maps, and 2-2 Games

We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on L(V, W), where V and W are vector spaces over a finite field. This inequality is useful for functions on L(V, W) whose 'generalised influences' are small, in an appropriate sense. It leads to a significant shortening of the proof of a recent seminal result by Khot, Minzer and Safra [18] that pseudorandom sets in Grassmann graphs have near-perfect expansion, which (in combination with the work of Dinur, Khot, Kindler, Minzer and Safra [5]) implies the 2-2 Games conjecture (the variant, that is, with imperfect completeness).