Bonami lemma

The Bonami lemma states that a function of Fourier degree at most $k$ satisfies $\Vert f{\Vert }_4\le 3^{k/2}\Vert f{\Vert }_2$.

One form of the hypercontractive inequality is $\Vert T_{1/\sqrt{3}}f{\Vert }_4\le \Vert f{\Vert }_2$, where $T_{\rho }$ is the noise operator on the Boolean hypercube.

If a nonzero function on the Boolean hypercube has degree at most $k$, then its support has measure at least $2^{-k}$. This sharp bound follows by induction on the dimension.