Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 168 / 2 / i

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

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