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For $1<p\le q<\infty$ and $0\le \rho \le \sqrt{(p-1)/(q-1)}$, the noise operator on the Boolean hypercube satisfies $\Vert T_{\rho }f{\Vert }_q\le \Vert f{\Vert }_p$ for the uniform probability measure. In particular,Here $\hat{f}(S)$ are Fourier-Walsh transform coefficients. The endpoint $\rho =0$ follows by averaging. This hypercontractive inequality on the Boolean hypercube controls Fourier-Walsh transform coefficients of discrete derivatives of a Boolean function and is a principal ingredient in the Kahn-Kalai-Linial theorem.

Decompose $f={\sum }_{j=0}^n{f}^{(=j)}$ into its homogeneous Fourier levels. The Bonami lemma and the triangle inequality giveApply this estimate to the $m$-fold tensor power $f^{\otimes m}$. Tensor products multiply both relevant norms and commute with the noise operator on the Boolean hypercube, soTaking $m$th roots and the limit $m\to \infty$ proves the hypercontractive inequality on the Boolean hypercube

The noise operators are self-adjoint and satisfy $T_{\rho }{T}_{\sigma }=T_{\rho \sigma }$. By the duality of Lp spaces,Consequently