Improved log-Sobolev inequalities, hypercontractivity and uncertainty principle on the hypercube

We develop a new class of log-Sobolev inequalities (LSIs) that provide a nonlinear comparison between the entropy and the Dirichlet form. For the hypercube, these LSIs imply a new version of the hypercontractivity for functions of small support. As a consequence, we derive a sharp form of the uncertainty principle for the hypercube: a function whose energy is concentrated on a set of small size, and whose Fourier energy is concentrated on a small Hamming ball must be zero. The tradeoff we derive is asymptotically optimal. We observe that for the Euclidean space, an analogous (asymptotically optimal) tradeoff follows from the sharp form of Young’s inequality due to Beckner. As an application, we show how uncertainty principle implies a new estimate of the metric properties of linear maps F2 → F2 .